Essential Norms of Volterra Type Operators between Zygmund ...
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Research ArticleEssential Norms of Volterra Type Operators betweenZygmund Type Spaces
Shanli Ye1 and Caishu Lin2
1School of Science Zhejiang University of Science and Technology Hangzhou 310023 China2Department of Mathematics Fujian Normal University Fuzhou 350007 China
Correspondence should be addressed to Shanli Ye ye_shanlialiyuncom
Received 5 January 2017 Revised 4 March 2017 Accepted 7 March 2017 Published 11 April 2017
Academic Editor Ruhan Zhao
Copyright copy 2017 Shanli Ye and Caishu Lin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We investigate the boundedness of some Volterra type operators between 119885119910119892119898119906119899119889 type spacesThen we give the essential normsof such operators in terms of 119892 120593 their derivatives and the nth power 120593119899 of 120593
1 Introduction
Let 119863 = 119911 |119911| lt 1 be the open unit disk in the complexplane C and let 120597119863 = 119911 |119911| = 1 be its boundary and119867(119863)denote the set of all analytic functions on119863
For every 0 lt 120572 lt infin we denote by B120572 the Bloch typespace of all functions 119891 isin 119867(119863) satisfying
119887120572 (119891) = sup119911isin119863
(1 minus |119911|2)120572 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt infin (1)
endowed with the norm 119891B120572 = |119891(0)| + 119887120572(119891) The littleBloch type space B1205720 consists of all 119891 isin B120572 satisfyinglim|119911|rarr1minus(1minus|119911|2)120572|1198911015840(119911)| = 0 andB1205720 is obviously the closedsubspace ofB120572 When 120572 = 1 we get the classical Bloch spaceB1 = B and little Bloch space B10 = B0 It is well knownthat for 0 lt 120572 lt 1 B120572 is a subspace of 119867infin the Banachspace of bounded analytic functions on 119863 Some sources forresults and references about the Bloch type functions are thepapers of Yoneda [1] Stevic [2 3] and the first author [4ndash7]
For 0 lt 120572 lt infin we denote byZ120572 the Zygmund type spaceof those functions 119891 isin 119867(119863) satisfying
sup119911isin119863
(1 minus |119911|2)120572 1003816100381610038161003816100381611989110158401015840 (119911)10038161003816100381610038161003816 lt infin (2)
and the little Zygmund type spaceZ1205720 consists of all 119891 isin Z120572
satisfying lim|119911|rarr1minus(1minus|119911|2)120572|11989110158401015840(119911)| = 0 andZ1205720 is obviouslythe closed subspace ofZ120572 It can easily be proved thatZ120572 isa Banach space under the norm
10038171003817100381710038171198911003817100381710038171003817Z120572 = 1003816100381610038161003816119891 (0)1003816100381610038161003816 + 100381610038161003816100381610038161198911015840 (0)10038161003816100381610038161003816 + sup119911isin119863
(1 minus |119911|2)120572 1003816100381610038161003816100381611989110158401015840 (119911)10038161003816100381610038161003816 (3)
and thatZ1205720 is a closed subspace ofZ120572 When 120572 = 1 we getthe classical Zygmund spaceZ1 = Z and the little ZygmundspaceZ10 = Z0 It is clear that 119891 isin Z if and only if 1198911015840 isin B1
We consider the weighted Banach spaces of analyticfunctions
119867infinV = 119891 isin 119867 (119863) 119865V = sup119911isin119863
V (119911) 1003816100381610038161003816119891 (119911)1003816100381610038161003816 lt infin (4)
endowed with norm sdot V where the weight V 119863 rarr 119877+is a continuous strictly positive and bounded function Theweight V is called radial if V(119911) = V(|119911|) for all 119911 isin 119863 For aweight V the associated weight V is defined by
V (119911) = (sup 1003816100381610038161003816119891 (119911)1003816100381610038161003816 119891 isin 119867infinV 10038171003817100381710038171198911003817100381710038171003817V le 1)minus1 119911 isin 119863 (5)
HindawiJournal of Function SpacesVolume 2017 Article ID 1409642 14 pageshttpsdoiorg10115520171409642
2 Journal of Function Spaces
We notice the standard weights V120572(119911) = (1minus |119911|2)120572 where 0 lt120572 lt infin and it is well known that V120572 = V120572 We also considerthe logarithmic weight
Vlog = (log 21 minus |119911|2)minus1 119911 isin 119863 (6)
It is straightforward to show that Vlog = VlogFor an analytic self-map 120593 of119863 and a function 119906 isin 119867(119863)
we define the weighted composition operator as 119906119862120593119891 =119906 sdot (119891 ∘ 120593) for 119891 isin 119867(119863) Weighted composition operatorshave been extensively studied recently It is interesting toprovide a function theoretic characterization when 120593 and119906 induce a bounded or compact composition operator onvarious function spaces Some results on the boundednessand compactness of concrete operators between some spacesof analytic functions one of which is of Zygmund type can befound for example in [8ndash19]
Suppose that 119892 119863 rarr C is an analytic map Let 119879119892 and 119868119892denote the Volterra type operators with the analytic symbol119892 on119863 respectively
119879119892119891 (119911) = int1199110119891 (120585) 1198921015840 (120585) 119889120585 119911 isin 119863
119868119892119891 (119911) = int11991101198911015840 (120585) 119892 (120585) 119889120585 119911 isin 119863
(7)
If 119892(119911) = 119911 then 119879119892 is an integral operator While119892(119911) = ln(1(1minus119911)) then119879119892 is Cesaro operator Pommerenkeintroduced the 119881119900119897119905119890119903119903119886 type operator 119879119892 and characterizedthe boundedness of 119879119892 between 1198672 spaces in [20] Morerecently boundedness and compactness of 119881119900119897119905119890119903119903119886 typeoperators between several spaces of analytic functions havebeen studied by many authors one may see [21 22]
In this paper we consider the following integral typeoperators which were introduced by Li and Stevic (see eg[10 23]) they can be defined by
(119862120593119879119892119891) (119911) = int120593(119911)0
119891 (120585) 1198921015840 (120585) 119889120585(119862120593119868119892119891) (119911) = int120593(119911)
01198911015840 (120585) 119892 (120585) 119889120585
(119879119892119862120593119891) (119911) = int1199110119891 (120593 (120585)) 1198921015840 (120585) 119889120585
(119868119892119862120593119891) (119911) = int11991101198911015840 (120593 (120585)) 119892 (120585) 119889120585
(8)
We will characterize the boundedness of those integral typeoperators between Zygmund type spaces and also estimatetheir essential norms The boundedness and compactness ofthese operators on the logarithmic Bloch space have beencharacterized in [22]
Recall that essential norm 119879119890119883rarr119884 of a bounded linearoperator 119879 119883 rarr 119884 is defined as the distance from 119879
to K(119883 119884) the space of compact operators from 119883 to 119884namely
119879119890119883rarr119884= inf 119879 + 119870119883rarr119884 119870 119883 rarr 119884 is compact (9)
It provides a measure of noncompactness of 119879 Clearly 119879 iscompact if and only if 119879119890119883rarr119884 = 0
Throughout this paper constants are denoted by 119862 theyare positive and may differ from one occurrence to the otherThe notation 119886 ≍ 119887 means that there are positive constants1198621 1198622 such that 1198621119886 le 119887 le 11986221198862 Boundedness
In order to prove themain results of this paperWe need someauxiliary results
Lemma 1 (see [8 13]) For 0 lt 120572 lt 2 and let 119891119899 be a boundedsequence in Z120572 which converges to 0 uniformly on compactsubsets of119863 Then lim119899rarrinfinsup119911isin119863|119891119899(119911)| = 0Lemma 2 (see [8 13]) For every 119891 isin Z120572 where 120572 gt 0 onehas
(i) |1198911015840(119911)| le (2(1 minus 120572))119891Z120572 and |119891(119911)| le (2(1 minus120572))119891Z120572 for every 0 lt 120572 lt 1(ii) |1198911015840(119911)| le 2 log(2(1 minus |119911|))119891Z120572 and |119891(119911)| le 119891Z120572
for 120572 = 1(iii) |1198911015840(119911)| le (2(1 minus 120572))(119891Z120572(1 minus |119911|)120572minus1) for every120572 gt 1(iv) |119891(119911)| le (2(120572 minus 1)(2 minus 120572))119891Z120572 for every 1 lt 120572 lt 2(v) |119891(119911)| le 2 log(2(1 minus |119911|))119891Z120572 for every 120572 = 2(vi) |119891(119911)| le (2(120572 minus 1)(120572 minus 2))(119891Z120572(1 minus |119911|)120572minus2) for
every 120572 gt 2Lemma 3 (see [8]) Let 0 lt 120572 lt infin and V a radial non-increasing weight tending to 0 at boundary of 119863 and let theweighted composition operator 119906119862120593 Z120572 rarr 119867infinV be bounded
(i) If 0 lt 120572 lt 2 then 119906119862120593 is a compact operator(ii)1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890Z2rarr119867infinV ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171199061205931198991003817100381710038171003817V
≍ lim sup|120593(119911)|rarr1
V (119911) |119906 (119911)| log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(10)
(iii) If 120572 gt 2 then1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890Z120572rarr119867infinV ≍ lim sup
119899rarrinfin(119899 + 1)120572minus2 10038171003817100381710038171199061205931198991003817100381710038171003817V
≍ lim sup|120593(119911)|rarr1
V (119911) |119906 (119911)|(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
(11)
The following lemma is due to [24 25]
Journal of Function Spaces 3
Lemma 4 Let V and 119908 be radial nonincreasing weightstending to zero at the boundary of119863 Then
(i) Theweighted composition operator 119906119862120593 maps119867infinV into119867infin119908 if and only if
sup119899ge0
10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V ≍ sup119911isin119863
119908 (119911)V (120593 (119911)) |119906 (119911)| lt infin (12)
with norm comparable to the above supermum
(ii)
1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890119867infinV rarr119867infin119908 = lim sup119899rarrinfin
10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V= lim sup|120593(119911)|rarr1
119908 (119911)V (120593 (119911)) |119906 (119911)|
(13)
Lemma 5 (see [26]) For every 0 lt 120572 lt infin one has
lim119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171199111198991003817100381710038171003817V120572 = (2120572119890 )120572 (14)
lim119899rarrinfin
(log 119899) 10038171003817100381710038171199111198991003817100381710038171003817Vlog = 1 (15)
Theorem6 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 1198921015840 isin 119867infinV120573 andsup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (16)
(ii) If 120572 = 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin(17)
sup119899ge0
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (18)
(iii) If 120572 gt 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (19)
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (20)
Proof Suppose that 119868119892119862120593 is bounded fromZ120572 toZ120573 Usingthe test functions 119891(119911) = 119911 and 119891(119911) = 1199112 we have
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119868119892119862120593119911)10158401015840100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (21)
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1198681198921198621205931199112)10158401015840100381610038161003816100381610038161003816= (1 minus |119911|2)120573 1003816100381610038161003816100381621205931015840 (119911) 119892 (119911) + 2120593 (119911) 1198921015840 (119911)10038161003816100381610038161003816 lt infin (22)
Since 120593 is a self-map we get that 1198921015840 isin 119867infinV120573 1205931015840119892 isin 119867infinV120573 For every 0 lt 120572 lt infin and given nonzero 119886 isin 119863 we take
the test functions
119891119886 (119911) = 11198862 [[
(1 minus |119886|2)2(1 minus 119886119911)120572 minus 1 minus |119886|2(1 minus 119886119911)120572minus1]] (23)
ℎ119886 (119911) = 1119886 int1199110
1 minus |119886|2(1 minus 119886119908)120572 119889119908 (24)
119892119886 (119911) = 119891119886 (119911) minus ℎ119886 (119911) (25)
for every 119911 isin 119863 One can show that 119891119886 ℎ119886 and 119892119886 are inZ1205720 sup12lt119886lt1119891119886Z120572 lt infin and sup12lt119886lt1ℎ119886Z120572 lt infin Since1198921015840119886(119886) = 0 11989210158401015840119886 (119886) = 120572(1 minus |119886|2)120572 it follows that for all 119911 isin 119863with |120593(119911)| gt 12 we have
+infin gt 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 ge 10038171003817100381710038171003817119868119892119862120593 (119892120593(119911))10038171003817100381710038171003817Z120573ge (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989210158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
minus (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840120593(119911) (120593 (119911))10038161003816100381610038161003816= (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161003816100381610038161003816120572
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(26)
4 Journal of Function Spaces
Then
sup119911isin119863
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
le sup|120593(119911)|le12
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
+ sup|120593(119911)|gt12
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
le (43)120572 10038171003817100381710038171003817120593101584011989210038171003817100381710038171003817V120573 + 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 lt infin
(27)
Now we use (14) and Lemma 4 to conclude that
sup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (28)
which shows that (16) is necessary for all caseConversely suppose that 1198921015840 isin 119867infinV120573 and (16) holds As-
sume that 119891 isin Z120572 From Lemma 2 it follows that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119868119892119862120593119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573sdot 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911)) + 1198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816le (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911))10038161003816100381610038161003816 + (1 minus |119911|2)120573
sdot 100381610038161003816100381610038161198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816 le (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + 119862 (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572 10038161003816100381610038161003816119868119892119862120593 (119891) (0)10038161003816100381610038161003816 = 0
100381610038161003816100381610038161003816(119868119892119862120593119891)1015840 (0)100381610038161003816100381610038161003816 = 100381610038161003816100381610038161198911015840 (120593 (0)) 119892 (0)10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572sdot 1003816100381610038161003816119892 (120593 (0))1003816100381610038161003816
(29)
which implies that 119868119892119862120593 is boundedThis completes the proofof (i)
Next we will prove (ii)The necessity in condition (17) hasbeen proved above Fixing 119886 isin 119863 with |119886| gt 12 we take thefunction
119896119886 (119911) = 119901 (119886119911)119886 (log 11 minus |119886|)minus1 (30)
for 119911 isin 119863 where
119901 (119911) = (119911 minus 1) ((1 + log 11 minus 119911)2 + 1) (31)
Then we have sup12lt|119886|lt1119896119886Z120572 le 119862 by [11] Let 119886 = 120593(119911) Itfollows that10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573
ge (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198961015840120593(119911) (120593 (119911))10038161003816100381610038161003816minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989610158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(32)
Since (17) holds and 119868119892119862120593 is bounded we obtain that
sup|120593(119911)|gt12
(1 minus |119911|2)120573 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ sup|120593(119911)|gt12
10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573 lt infin(33)
Noting 1198921015840 isin 119867infinV120573 and together with (15) and Lemma 4 weconclude that (18) holds
The converse implication can be shown as in the proof of(i)
Finally we will prove (iii) We have proved that (19) holdsabove To prove (20) we take function 119891120593(119911) defined in (23)for every 119911 isin 119863 with |120593(119911)| gt 12 and obtain that10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573
ge (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989110158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 110038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816100381610038162120593 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(34)
Since 119868119892119862120593 is bounded and (19) holds we obtain that
sup|120593(119911)|gt12
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le 2 sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ 2 sup|120593(119911)|gt12
10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(35)
therefore we deduce that (20) holds by (14) and Lemma 4
Journal of Function Spaces 5
The converse implication can be shown as in the proof of(i)
Theorem 7 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((119892∘120593)(12059310158401015840)+(1198921015840∘120593)(1205931015840)2) isin 119867infinV120573and
sup119899ge0
(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)
(ii) If 120572 = 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin
(37)
(iii) If 120572 gt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if (36) holds and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)
The proof is similar to that of Theorem 6 and the detailsare omitted
Theorem8 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119879119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573
(ii) If 120572 = 1 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)
(iii) If 1 lt 120572 lt 2 then 119862120593119879119892 Z120572 rarr Z120573 is a boundedoperator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)
(iv) If 120572 = 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)
(v) If 120572 gt 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if (40) holds and
sup119899ge0
(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2
+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)
6 Journal of Function Spaces
Proof Suppose that 119862120593119879119892 is bounded fromZ120572 toZ120573 space
(i) Case 0 lt 120572 lt 1 Using functions 119891 = 1 isin Z120572 and 119891 = 119911 isinZ120572 we obtain
sup119911isin119863
(1 minus |119911|2)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infinsup119911isin119863
(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)
Then we obtain that ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and(1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 are necessary for all caseFor the converse implication suppose that ((1198921015840 ∘120593)(12059310158401015840)+(11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 For 119891 isin Z120572 it
follows from Lemma 2 that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2sdot 1198921015840 (120593 (119911)) 1198911015840 (120593 (119911))+ (11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))sdot 1198911015840 (120593 (119911))100381610038161003816100381610038161003816 + (1 minus |119911|2)120573sdot 100381610038161003816100381610038161003816(11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + (1 minus |119911|2)120573 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572
10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)
0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816
le max|120577|le|120593(0)|
1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 le 21 minus 120572 10038171003817100381710038171198911003817100381710038171003817Z120572sdot max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572
sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816
(45)
Then 119862120593119879119892 is bounded This completes the proof of (i)
(ii) Case 120572 = 1 We consider the test function 119896120593(119911)(119911) definedin (30) for every 119911 isin 119863 with |120593(119911)| gt 12 It follows that
10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 ge log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 minus (1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816
(46)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 andsup|120593(119911)|gt12119862120593119879119892119896120593(119911)Z120573 le 119862 we get
sup|120593(119911)|gt12
log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12
(1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816 + 10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 lt infin
(47)
Then we have
sup119911isin119863
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1
minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le 119862 + log(43) 100381710038171003817100381710038171003817(1198921015840∘ 120593) (1205931015840)2100381710038171003817100381710038171003817V120573 lt infin
(48)
On the other hand from (15) and Lemma 4 we have
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)
Hence (39) holdsThe converse implication can be shown as in the proof of
(i)
(iii) Case 1 lt 120572 lt 2 ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 hasbeen proved above We take the test function 119891120593(119911) in (23) for
Journal of Function Spaces 7
every 119911 isin 119863with |120593(119911)| gt 12 by the same way as (ii) we canobtain that (40) holds
The converse implication can be shown as in the proof of(i)
(iv) Case 120572 = 2 We have proved that (41) holds aboveTo prove (42) we consider another test function 119905119886(119911) =log(2(1 minus 119886119911)) Clearly 119905119886 isin Z2 and sup12lt|119886|lt1119905119886Z2 lt infinFor every 119911 isin 119863 with |120593(119911)| gt 12 it follows that
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816
(50)
Applying (41) we get
sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin
(51)
Noting ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and usingLemma 4 and (15) we conclude that (42) holds
(v) Case 120572 gt 2 We have proved that (40) holds aboveApplying test function 119891120593(119911) in (23) for every 119911 isin 119863 with|120593(119911)| gt 12 we have
1198781 = sup|120593(119911)|gt12
(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup
|120593(119911)|gt12
10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)
With the same calculation for test function 119905120593(119911)(120593(119911)) =(1 minus |120593(119911)|2)2(1 minus 120593(119911)119911)120572 with |120593(119911)| gt 12 thensup|120593(119911)|gt(12)119905120593(119911)Z120572 le 119862 and we have that
1198782 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1100381610038161003816100381610038161003816100381610038161003816100381610038161003816 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(119862120593119879119892 (119905120593(119911)))10158401015840 (120593 (119911))100381610038161003816100381610038161003816 le 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817Z120572rarrZ120573sdot sup|120593(119911)|gt12
10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin
(53)
Therefore
sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782
+ 120572 sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572
sdot sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))10038161003816100381610038161003816100381610038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 1205721198781lt infin
(54)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 we conclude that(43) holds
Theorem9 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 11989210158401205931015840 isin 119867infinV120573 and 11989210158401015840 isin 119867infinV120573 (ii) If 120572 = 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
if and only if 11989210158401015840 isin 119867infinV120573 andsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin
(55)
8 Journal of Function Spaces
(iii) If 1 lt 120572 lt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a boundedoperator if and only if 11989210158401015840 isin 119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)
(iv) If 120572 = 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin
(57)
(v) If 120572 gt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if (56) holds and
sup119899ge0
(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)
The proof is similar to that of Theorem 8 and the detailsare omitted
3 Essential Norms
In this section we estimate the essential norms of theseintegral type operators on 119885119910119892119898119906119899119889 type spaces in terms of119892 120593 their derivatives and the nth power 120593119899 of 120593
Let Z120572 = 119891 isin Z120572 119891(0) = 1198911015840(0) = 0 and B120572 =119891 isin B120572 119891(0) = 0 We note that every compact operator119879 isin K(Z120572Z120573) can be extended to a compact operator119870 isinK(Z120572Z120573) In fact for every 119891 isin Z120572 119891 minus 119891(0) minus 1198911015840(0)119911 isinZ120572 and we can define119870(119891) = 119879(119891minus119891(0) minus1198911015840(0)119911) +119891(0) +1198911015840(0)119911
For 119903 isin (0 1) we consider the compact operator 119870119903 Z120572 rarr Z120573 defined by 119870119903119891(119911) = 119891(119903119911)Lemma 10 If 119883(119868119892119862120593 119862120593119868119892 119862120593119879119892 119879119892119862120593) is a bounded oper-ator fromZ120572 toZ120573 space then
119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)
Proof Clearly 119883119890Z120572rarrZ120573 ge 119883119890Z120572rarrZ120573 Then we prove119883119890Z120572rarrZ120573 le 119883119890Z120572rarrZ120573 Let 119879 isin 119870(Z120572Z120573) be given Let 119903119899 be an increasing
sequence in (0 1) converging to 1 andA = ℎ | ℎ = 119886 + 119887119911the closed subspace ofZ120572 Then
119883 minus 119879Z120572rarrZ120573 = sup119891isinZ120572119891Z120572le1
1003817100381710038171003817119883 (119891) minus 119879 (119891)1003817100381710038171003817Z120573le sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)minus 119879 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573+ sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup
119892isinZ120572119892Z120572le1
1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup
ℎisinAℎZ120572le1
1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573
(60)
Hence
inf119879isin119870(Z120572 Z120573)
119883 minus 119879Z120572rarrZ120573le inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|Z1205721003817100381710038171003817Z120572rarrZ120573+ inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|1205721003817100381710038171003817Z120572rarrZ120573le 119883119890Z120572rarrZ120573 + lim
119899rarrinfin
10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573
(61)
Since lim119899rarrinfin119883(119868minus119870119903119899)Z120572rarrZ120573 = 0 we have 119883119890Z120572rarrZ120573 le119883119890Z120572rarrZ120573 and the proof is finished
Let119863120572 Z120572 rarr B120572 and 119878120572 B120572 rarr 119867infinV120572 be the derivativeoperatorsThen clearly119863120572 and 119878120572 are linear isometries on Z120572and B120572 respectively and
119878120573119863120573119868119892119862120593119863minus1120572 119878minus1120572 = 119892 (1205931015840) 119862120593 + 1198921015840119862120593119878minus1120572 (62)
Therefore
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890B120572rarr119867infinV120573+ 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
(63)
Similarly
119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593
+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)
Journal of Function Spaces 9
1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)
119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572
+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)
119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573
(69)
Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)
Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572
into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume
to be identically zero Then it follows from Theorem 6 andLemma 1 that
lim119896rarrinfin
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin
sup119911isin119863
10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)
which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816
= lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(74)
For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin
1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin
1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
(75)
Now we use (14) and Lemma 4 to obtain that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1
10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
= 120572119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(76)
Hence (70) holds
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces
We notice the standard weights V120572(119911) = (1minus |119911|2)120572 where 0 lt120572 lt infin and it is well known that V120572 = V120572 We also considerthe logarithmic weight
Vlog = (log 21 minus |119911|2)minus1 119911 isin 119863 (6)
It is straightforward to show that Vlog = VlogFor an analytic self-map 120593 of119863 and a function 119906 isin 119867(119863)
we define the weighted composition operator as 119906119862120593119891 =119906 sdot (119891 ∘ 120593) for 119891 isin 119867(119863) Weighted composition operatorshave been extensively studied recently It is interesting toprovide a function theoretic characterization when 120593 and119906 induce a bounded or compact composition operator onvarious function spaces Some results on the boundednessand compactness of concrete operators between some spacesof analytic functions one of which is of Zygmund type can befound for example in [8ndash19]
Suppose that 119892 119863 rarr C is an analytic map Let 119879119892 and 119868119892denote the Volterra type operators with the analytic symbol119892 on119863 respectively
119879119892119891 (119911) = int1199110119891 (120585) 1198921015840 (120585) 119889120585 119911 isin 119863
119868119892119891 (119911) = int11991101198911015840 (120585) 119892 (120585) 119889120585 119911 isin 119863
(7)
If 119892(119911) = 119911 then 119879119892 is an integral operator While119892(119911) = ln(1(1minus119911)) then119879119892 is Cesaro operator Pommerenkeintroduced the 119881119900119897119905119890119903119903119886 type operator 119879119892 and characterizedthe boundedness of 119879119892 between 1198672 spaces in [20] Morerecently boundedness and compactness of 119881119900119897119905119890119903119903119886 typeoperators between several spaces of analytic functions havebeen studied by many authors one may see [21 22]
In this paper we consider the following integral typeoperators which were introduced by Li and Stevic (see eg[10 23]) they can be defined by
(119862120593119879119892119891) (119911) = int120593(119911)0
119891 (120585) 1198921015840 (120585) 119889120585(119862120593119868119892119891) (119911) = int120593(119911)
01198911015840 (120585) 119892 (120585) 119889120585
(119879119892119862120593119891) (119911) = int1199110119891 (120593 (120585)) 1198921015840 (120585) 119889120585
(119868119892119862120593119891) (119911) = int11991101198911015840 (120593 (120585)) 119892 (120585) 119889120585
(8)
We will characterize the boundedness of those integral typeoperators between Zygmund type spaces and also estimatetheir essential norms The boundedness and compactness ofthese operators on the logarithmic Bloch space have beencharacterized in [22]
Recall that essential norm 119879119890119883rarr119884 of a bounded linearoperator 119879 119883 rarr 119884 is defined as the distance from 119879
to K(119883 119884) the space of compact operators from 119883 to 119884namely
119879119890119883rarr119884= inf 119879 + 119870119883rarr119884 119870 119883 rarr 119884 is compact (9)
It provides a measure of noncompactness of 119879 Clearly 119879 iscompact if and only if 119879119890119883rarr119884 = 0
Throughout this paper constants are denoted by 119862 theyare positive and may differ from one occurrence to the otherThe notation 119886 ≍ 119887 means that there are positive constants1198621 1198622 such that 1198621119886 le 119887 le 11986221198862 Boundedness
In order to prove themain results of this paperWe need someauxiliary results
Lemma 1 (see [8 13]) For 0 lt 120572 lt 2 and let 119891119899 be a boundedsequence in Z120572 which converges to 0 uniformly on compactsubsets of119863 Then lim119899rarrinfinsup119911isin119863|119891119899(119911)| = 0Lemma 2 (see [8 13]) For every 119891 isin Z120572 where 120572 gt 0 onehas
(i) |1198911015840(119911)| le (2(1 minus 120572))119891Z120572 and |119891(119911)| le (2(1 minus120572))119891Z120572 for every 0 lt 120572 lt 1(ii) |1198911015840(119911)| le 2 log(2(1 minus |119911|))119891Z120572 and |119891(119911)| le 119891Z120572
for 120572 = 1(iii) |1198911015840(119911)| le (2(1 minus 120572))(119891Z120572(1 minus |119911|)120572minus1) for every120572 gt 1(iv) |119891(119911)| le (2(120572 minus 1)(2 minus 120572))119891Z120572 for every 1 lt 120572 lt 2(v) |119891(119911)| le 2 log(2(1 minus |119911|))119891Z120572 for every 120572 = 2(vi) |119891(119911)| le (2(120572 minus 1)(120572 minus 2))(119891Z120572(1 minus |119911|)120572minus2) for
every 120572 gt 2Lemma 3 (see [8]) Let 0 lt 120572 lt infin and V a radial non-increasing weight tending to 0 at boundary of 119863 and let theweighted composition operator 119906119862120593 Z120572 rarr 119867infinV be bounded
(i) If 0 lt 120572 lt 2 then 119906119862120593 is a compact operator(ii)1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890Z2rarr119867infinV ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171199061205931198991003817100381710038171003817V
≍ lim sup|120593(119911)|rarr1
V (119911) |119906 (119911)| log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(10)
(iii) If 120572 gt 2 then1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890Z120572rarr119867infinV ≍ lim sup
119899rarrinfin(119899 + 1)120572minus2 10038171003817100381710038171199061205931198991003817100381710038171003817V
≍ lim sup|120593(119911)|rarr1
V (119911) |119906 (119911)|(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
(11)
The following lemma is due to [24 25]
Journal of Function Spaces 3
Lemma 4 Let V and 119908 be radial nonincreasing weightstending to zero at the boundary of119863 Then
(i) Theweighted composition operator 119906119862120593 maps119867infinV into119867infin119908 if and only if
sup119899ge0
10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V ≍ sup119911isin119863
119908 (119911)V (120593 (119911)) |119906 (119911)| lt infin (12)
with norm comparable to the above supermum
(ii)
1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890119867infinV rarr119867infin119908 = lim sup119899rarrinfin
10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V= lim sup|120593(119911)|rarr1
119908 (119911)V (120593 (119911)) |119906 (119911)|
(13)
Lemma 5 (see [26]) For every 0 lt 120572 lt infin one has
lim119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171199111198991003817100381710038171003817V120572 = (2120572119890 )120572 (14)
lim119899rarrinfin
(log 119899) 10038171003817100381710038171199111198991003817100381710038171003817Vlog = 1 (15)
Theorem6 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 1198921015840 isin 119867infinV120573 andsup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (16)
(ii) If 120572 = 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin(17)
sup119899ge0
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (18)
(iii) If 120572 gt 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (19)
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (20)
Proof Suppose that 119868119892119862120593 is bounded fromZ120572 toZ120573 Usingthe test functions 119891(119911) = 119911 and 119891(119911) = 1199112 we have
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119868119892119862120593119911)10158401015840100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (21)
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1198681198921198621205931199112)10158401015840100381610038161003816100381610038161003816= (1 minus |119911|2)120573 1003816100381610038161003816100381621205931015840 (119911) 119892 (119911) + 2120593 (119911) 1198921015840 (119911)10038161003816100381610038161003816 lt infin (22)
Since 120593 is a self-map we get that 1198921015840 isin 119867infinV120573 1205931015840119892 isin 119867infinV120573 For every 0 lt 120572 lt infin and given nonzero 119886 isin 119863 we take
the test functions
119891119886 (119911) = 11198862 [[
(1 minus |119886|2)2(1 minus 119886119911)120572 minus 1 minus |119886|2(1 minus 119886119911)120572minus1]] (23)
ℎ119886 (119911) = 1119886 int1199110
1 minus |119886|2(1 minus 119886119908)120572 119889119908 (24)
119892119886 (119911) = 119891119886 (119911) minus ℎ119886 (119911) (25)
for every 119911 isin 119863 One can show that 119891119886 ℎ119886 and 119892119886 are inZ1205720 sup12lt119886lt1119891119886Z120572 lt infin and sup12lt119886lt1ℎ119886Z120572 lt infin Since1198921015840119886(119886) = 0 11989210158401015840119886 (119886) = 120572(1 minus |119886|2)120572 it follows that for all 119911 isin 119863with |120593(119911)| gt 12 we have
+infin gt 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 ge 10038171003817100381710038171003817119868119892119862120593 (119892120593(119911))10038171003817100381710038171003817Z120573ge (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989210158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
minus (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840120593(119911) (120593 (119911))10038161003816100381610038161003816= (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161003816100381610038161003816120572
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(26)
4 Journal of Function Spaces
Then
sup119911isin119863
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
le sup|120593(119911)|le12
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
+ sup|120593(119911)|gt12
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
le (43)120572 10038171003817100381710038171003817120593101584011989210038171003817100381710038171003817V120573 + 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 lt infin
(27)
Now we use (14) and Lemma 4 to conclude that
sup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (28)
which shows that (16) is necessary for all caseConversely suppose that 1198921015840 isin 119867infinV120573 and (16) holds As-
sume that 119891 isin Z120572 From Lemma 2 it follows that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119868119892119862120593119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573sdot 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911)) + 1198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816le (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911))10038161003816100381610038161003816 + (1 minus |119911|2)120573
sdot 100381610038161003816100381610038161198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816 le (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + 119862 (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572 10038161003816100381610038161003816119868119892119862120593 (119891) (0)10038161003816100381610038161003816 = 0
100381610038161003816100381610038161003816(119868119892119862120593119891)1015840 (0)100381610038161003816100381610038161003816 = 100381610038161003816100381610038161198911015840 (120593 (0)) 119892 (0)10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572sdot 1003816100381610038161003816119892 (120593 (0))1003816100381610038161003816
(29)
which implies that 119868119892119862120593 is boundedThis completes the proofof (i)
Next we will prove (ii)The necessity in condition (17) hasbeen proved above Fixing 119886 isin 119863 with |119886| gt 12 we take thefunction
119896119886 (119911) = 119901 (119886119911)119886 (log 11 minus |119886|)minus1 (30)
for 119911 isin 119863 where
119901 (119911) = (119911 minus 1) ((1 + log 11 minus 119911)2 + 1) (31)
Then we have sup12lt|119886|lt1119896119886Z120572 le 119862 by [11] Let 119886 = 120593(119911) Itfollows that10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573
ge (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198961015840120593(119911) (120593 (119911))10038161003816100381610038161003816minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989610158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(32)
Since (17) holds and 119868119892119862120593 is bounded we obtain that
sup|120593(119911)|gt12
(1 minus |119911|2)120573 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ sup|120593(119911)|gt12
10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573 lt infin(33)
Noting 1198921015840 isin 119867infinV120573 and together with (15) and Lemma 4 weconclude that (18) holds
The converse implication can be shown as in the proof of(i)
Finally we will prove (iii) We have proved that (19) holdsabove To prove (20) we take function 119891120593(119911) defined in (23)for every 119911 isin 119863 with |120593(119911)| gt 12 and obtain that10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573
ge (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989110158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 110038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816100381610038162120593 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(34)
Since 119868119892119862120593 is bounded and (19) holds we obtain that
sup|120593(119911)|gt12
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le 2 sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ 2 sup|120593(119911)|gt12
10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(35)
therefore we deduce that (20) holds by (14) and Lemma 4
Journal of Function Spaces 5
The converse implication can be shown as in the proof of(i)
Theorem 7 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((119892∘120593)(12059310158401015840)+(1198921015840∘120593)(1205931015840)2) isin 119867infinV120573and
sup119899ge0
(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)
(ii) If 120572 = 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin
(37)
(iii) If 120572 gt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if (36) holds and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)
The proof is similar to that of Theorem 6 and the detailsare omitted
Theorem8 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119879119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573
(ii) If 120572 = 1 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)
(iii) If 1 lt 120572 lt 2 then 119862120593119879119892 Z120572 rarr Z120573 is a boundedoperator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)
(iv) If 120572 = 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)
(v) If 120572 gt 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if (40) holds and
sup119899ge0
(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2
+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)
6 Journal of Function Spaces
Proof Suppose that 119862120593119879119892 is bounded fromZ120572 toZ120573 space
(i) Case 0 lt 120572 lt 1 Using functions 119891 = 1 isin Z120572 and 119891 = 119911 isinZ120572 we obtain
sup119911isin119863
(1 minus |119911|2)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infinsup119911isin119863
(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)
Then we obtain that ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and(1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 are necessary for all caseFor the converse implication suppose that ((1198921015840 ∘120593)(12059310158401015840)+(11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 For 119891 isin Z120572 it
follows from Lemma 2 that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2sdot 1198921015840 (120593 (119911)) 1198911015840 (120593 (119911))+ (11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))sdot 1198911015840 (120593 (119911))100381610038161003816100381610038161003816 + (1 minus |119911|2)120573sdot 100381610038161003816100381610038161003816(11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + (1 minus |119911|2)120573 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572
10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)
0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816
le max|120577|le|120593(0)|
1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 le 21 minus 120572 10038171003817100381710038171198911003817100381710038171003817Z120572sdot max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572
sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816
(45)
Then 119862120593119879119892 is bounded This completes the proof of (i)
(ii) Case 120572 = 1 We consider the test function 119896120593(119911)(119911) definedin (30) for every 119911 isin 119863 with |120593(119911)| gt 12 It follows that
10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 ge log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 minus (1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816
(46)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 andsup|120593(119911)|gt12119862120593119879119892119896120593(119911)Z120573 le 119862 we get
sup|120593(119911)|gt12
log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12
(1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816 + 10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 lt infin
(47)
Then we have
sup119911isin119863
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1
minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le 119862 + log(43) 100381710038171003817100381710038171003817(1198921015840∘ 120593) (1205931015840)2100381710038171003817100381710038171003817V120573 lt infin
(48)
On the other hand from (15) and Lemma 4 we have
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)
Hence (39) holdsThe converse implication can be shown as in the proof of
(i)
(iii) Case 1 lt 120572 lt 2 ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 hasbeen proved above We take the test function 119891120593(119911) in (23) for
Journal of Function Spaces 7
every 119911 isin 119863with |120593(119911)| gt 12 by the same way as (ii) we canobtain that (40) holds
The converse implication can be shown as in the proof of(i)
(iv) Case 120572 = 2 We have proved that (41) holds aboveTo prove (42) we consider another test function 119905119886(119911) =log(2(1 minus 119886119911)) Clearly 119905119886 isin Z2 and sup12lt|119886|lt1119905119886Z2 lt infinFor every 119911 isin 119863 with |120593(119911)| gt 12 it follows that
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816
(50)
Applying (41) we get
sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin
(51)
Noting ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and usingLemma 4 and (15) we conclude that (42) holds
(v) Case 120572 gt 2 We have proved that (40) holds aboveApplying test function 119891120593(119911) in (23) for every 119911 isin 119863 with|120593(119911)| gt 12 we have
1198781 = sup|120593(119911)|gt12
(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup
|120593(119911)|gt12
10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)
With the same calculation for test function 119905120593(119911)(120593(119911)) =(1 minus |120593(119911)|2)2(1 minus 120593(119911)119911)120572 with |120593(119911)| gt 12 thensup|120593(119911)|gt(12)119905120593(119911)Z120572 le 119862 and we have that
1198782 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1100381610038161003816100381610038161003816100381610038161003816100381610038161003816 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(119862120593119879119892 (119905120593(119911)))10158401015840 (120593 (119911))100381610038161003816100381610038161003816 le 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817Z120572rarrZ120573sdot sup|120593(119911)|gt12
10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin
(53)
Therefore
sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782
+ 120572 sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572
sdot sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))10038161003816100381610038161003816100381610038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 1205721198781lt infin
(54)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 we conclude that(43) holds
Theorem9 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 11989210158401205931015840 isin 119867infinV120573 and 11989210158401015840 isin 119867infinV120573 (ii) If 120572 = 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
if and only if 11989210158401015840 isin 119867infinV120573 andsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin
(55)
8 Journal of Function Spaces
(iii) If 1 lt 120572 lt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a boundedoperator if and only if 11989210158401015840 isin 119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)
(iv) If 120572 = 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin
(57)
(v) If 120572 gt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if (56) holds and
sup119899ge0
(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)
The proof is similar to that of Theorem 8 and the detailsare omitted
3 Essential Norms
In this section we estimate the essential norms of theseintegral type operators on 119885119910119892119898119906119899119889 type spaces in terms of119892 120593 their derivatives and the nth power 120593119899 of 120593
Let Z120572 = 119891 isin Z120572 119891(0) = 1198911015840(0) = 0 and B120572 =119891 isin B120572 119891(0) = 0 We note that every compact operator119879 isin K(Z120572Z120573) can be extended to a compact operator119870 isinK(Z120572Z120573) In fact for every 119891 isin Z120572 119891 minus 119891(0) minus 1198911015840(0)119911 isinZ120572 and we can define119870(119891) = 119879(119891minus119891(0) minus1198911015840(0)119911) +119891(0) +1198911015840(0)119911
For 119903 isin (0 1) we consider the compact operator 119870119903 Z120572 rarr Z120573 defined by 119870119903119891(119911) = 119891(119903119911)Lemma 10 If 119883(119868119892119862120593 119862120593119868119892 119862120593119879119892 119879119892119862120593) is a bounded oper-ator fromZ120572 toZ120573 space then
119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)
Proof Clearly 119883119890Z120572rarrZ120573 ge 119883119890Z120572rarrZ120573 Then we prove119883119890Z120572rarrZ120573 le 119883119890Z120572rarrZ120573 Let 119879 isin 119870(Z120572Z120573) be given Let 119903119899 be an increasing
sequence in (0 1) converging to 1 andA = ℎ | ℎ = 119886 + 119887119911the closed subspace ofZ120572 Then
119883 minus 119879Z120572rarrZ120573 = sup119891isinZ120572119891Z120572le1
1003817100381710038171003817119883 (119891) minus 119879 (119891)1003817100381710038171003817Z120573le sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)minus 119879 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573+ sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup
119892isinZ120572119892Z120572le1
1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup
ℎisinAℎZ120572le1
1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573
(60)
Hence
inf119879isin119870(Z120572 Z120573)
119883 minus 119879Z120572rarrZ120573le inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|Z1205721003817100381710038171003817Z120572rarrZ120573+ inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|1205721003817100381710038171003817Z120572rarrZ120573le 119883119890Z120572rarrZ120573 + lim
119899rarrinfin
10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573
(61)
Since lim119899rarrinfin119883(119868minus119870119903119899)Z120572rarrZ120573 = 0 we have 119883119890Z120572rarrZ120573 le119883119890Z120572rarrZ120573 and the proof is finished
Let119863120572 Z120572 rarr B120572 and 119878120572 B120572 rarr 119867infinV120572 be the derivativeoperatorsThen clearly119863120572 and 119878120572 are linear isometries on Z120572and B120572 respectively and
119878120573119863120573119868119892119862120593119863minus1120572 119878minus1120572 = 119892 (1205931015840) 119862120593 + 1198921015840119862120593119878minus1120572 (62)
Therefore
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890B120572rarr119867infinV120573+ 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
(63)
Similarly
119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593
+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)
Journal of Function Spaces 9
1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)
119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572
+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)
119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573
(69)
Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)
Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572
into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume
to be identically zero Then it follows from Theorem 6 andLemma 1 that
lim119896rarrinfin
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin
sup119911isin119863
10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)
which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816
= lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(74)
For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin
1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin
1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
(75)
Now we use (14) and Lemma 4 to obtain that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1
10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
= 120572119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(76)
Hence (70) holds
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
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Journal of Function Spaces 3
Lemma 4 Let V and 119908 be radial nonincreasing weightstending to zero at the boundary of119863 Then
(i) Theweighted composition operator 119906119862120593 maps119867infinV into119867infin119908 if and only if
sup119899ge0
10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V ≍ sup119911isin119863
119908 (119911)V (120593 (119911)) |119906 (119911)| lt infin (12)
with norm comparable to the above supermum
(ii)
1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890119867infinV rarr119867infin119908 = lim sup119899rarrinfin
10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V= lim sup|120593(119911)|rarr1
119908 (119911)V (120593 (119911)) |119906 (119911)|
(13)
Lemma 5 (see [26]) For every 0 lt 120572 lt infin one has
lim119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171199111198991003817100381710038171003817V120572 = (2120572119890 )120572 (14)
lim119899rarrinfin
(log 119899) 10038171003817100381710038171199111198991003817100381710038171003817Vlog = 1 (15)
Theorem6 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 1198921015840 isin 119867infinV120573 andsup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (16)
(ii) If 120572 = 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin(17)
sup119899ge0
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (18)
(iii) If 120572 gt 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (19)
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (20)
Proof Suppose that 119868119892119862120593 is bounded fromZ120572 toZ120573 Usingthe test functions 119891(119911) = 119911 and 119891(119911) = 1199112 we have
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119868119892119862120593119911)10158401015840100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (21)
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1198681198921198621205931199112)10158401015840100381610038161003816100381610038161003816= (1 minus |119911|2)120573 1003816100381610038161003816100381621205931015840 (119911) 119892 (119911) + 2120593 (119911) 1198921015840 (119911)10038161003816100381610038161003816 lt infin (22)
Since 120593 is a self-map we get that 1198921015840 isin 119867infinV120573 1205931015840119892 isin 119867infinV120573 For every 0 lt 120572 lt infin and given nonzero 119886 isin 119863 we take
the test functions
119891119886 (119911) = 11198862 [[
(1 minus |119886|2)2(1 minus 119886119911)120572 minus 1 minus |119886|2(1 minus 119886119911)120572minus1]] (23)
ℎ119886 (119911) = 1119886 int1199110
1 minus |119886|2(1 minus 119886119908)120572 119889119908 (24)
119892119886 (119911) = 119891119886 (119911) minus ℎ119886 (119911) (25)
for every 119911 isin 119863 One can show that 119891119886 ℎ119886 and 119892119886 are inZ1205720 sup12lt119886lt1119891119886Z120572 lt infin and sup12lt119886lt1ℎ119886Z120572 lt infin Since1198921015840119886(119886) = 0 11989210158401015840119886 (119886) = 120572(1 minus |119886|2)120572 it follows that for all 119911 isin 119863with |120593(119911)| gt 12 we have
+infin gt 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 ge 10038171003817100381710038171003817119868119892119862120593 (119892120593(119911))10038171003817100381710038171003817Z120573ge (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989210158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
minus (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840120593(119911) (120593 (119911))10038161003816100381610038161003816= (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161003816100381610038161003816120572
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(26)
4 Journal of Function Spaces
Then
sup119911isin119863
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
le sup|120593(119911)|le12
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
+ sup|120593(119911)|gt12
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
le (43)120572 10038171003817100381710038171003817120593101584011989210038171003817100381710038171003817V120573 + 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 lt infin
(27)
Now we use (14) and Lemma 4 to conclude that
sup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (28)
which shows that (16) is necessary for all caseConversely suppose that 1198921015840 isin 119867infinV120573 and (16) holds As-
sume that 119891 isin Z120572 From Lemma 2 it follows that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119868119892119862120593119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573sdot 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911)) + 1198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816le (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911))10038161003816100381610038161003816 + (1 minus |119911|2)120573
sdot 100381610038161003816100381610038161198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816 le (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + 119862 (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572 10038161003816100381610038161003816119868119892119862120593 (119891) (0)10038161003816100381610038161003816 = 0
100381610038161003816100381610038161003816(119868119892119862120593119891)1015840 (0)100381610038161003816100381610038161003816 = 100381610038161003816100381610038161198911015840 (120593 (0)) 119892 (0)10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572sdot 1003816100381610038161003816119892 (120593 (0))1003816100381610038161003816
(29)
which implies that 119868119892119862120593 is boundedThis completes the proofof (i)
Next we will prove (ii)The necessity in condition (17) hasbeen proved above Fixing 119886 isin 119863 with |119886| gt 12 we take thefunction
119896119886 (119911) = 119901 (119886119911)119886 (log 11 minus |119886|)minus1 (30)
for 119911 isin 119863 where
119901 (119911) = (119911 minus 1) ((1 + log 11 minus 119911)2 + 1) (31)
Then we have sup12lt|119886|lt1119896119886Z120572 le 119862 by [11] Let 119886 = 120593(119911) Itfollows that10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573
ge (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198961015840120593(119911) (120593 (119911))10038161003816100381610038161003816minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989610158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(32)
Since (17) holds and 119868119892119862120593 is bounded we obtain that
sup|120593(119911)|gt12
(1 minus |119911|2)120573 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ sup|120593(119911)|gt12
10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573 lt infin(33)
Noting 1198921015840 isin 119867infinV120573 and together with (15) and Lemma 4 weconclude that (18) holds
The converse implication can be shown as in the proof of(i)
Finally we will prove (iii) We have proved that (19) holdsabove To prove (20) we take function 119891120593(119911) defined in (23)for every 119911 isin 119863 with |120593(119911)| gt 12 and obtain that10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573
ge (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989110158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 110038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816100381610038162120593 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(34)
Since 119868119892119862120593 is bounded and (19) holds we obtain that
sup|120593(119911)|gt12
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le 2 sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ 2 sup|120593(119911)|gt12
10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(35)
therefore we deduce that (20) holds by (14) and Lemma 4
Journal of Function Spaces 5
The converse implication can be shown as in the proof of(i)
Theorem 7 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((119892∘120593)(12059310158401015840)+(1198921015840∘120593)(1205931015840)2) isin 119867infinV120573and
sup119899ge0
(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)
(ii) If 120572 = 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin
(37)
(iii) If 120572 gt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if (36) holds and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)
The proof is similar to that of Theorem 6 and the detailsare omitted
Theorem8 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119879119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573
(ii) If 120572 = 1 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)
(iii) If 1 lt 120572 lt 2 then 119862120593119879119892 Z120572 rarr Z120573 is a boundedoperator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)
(iv) If 120572 = 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)
(v) If 120572 gt 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if (40) holds and
sup119899ge0
(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2
+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)
6 Journal of Function Spaces
Proof Suppose that 119862120593119879119892 is bounded fromZ120572 toZ120573 space
(i) Case 0 lt 120572 lt 1 Using functions 119891 = 1 isin Z120572 and 119891 = 119911 isinZ120572 we obtain
sup119911isin119863
(1 minus |119911|2)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infinsup119911isin119863
(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)
Then we obtain that ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and(1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 are necessary for all caseFor the converse implication suppose that ((1198921015840 ∘120593)(12059310158401015840)+(11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 For 119891 isin Z120572 it
follows from Lemma 2 that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2sdot 1198921015840 (120593 (119911)) 1198911015840 (120593 (119911))+ (11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))sdot 1198911015840 (120593 (119911))100381610038161003816100381610038161003816 + (1 minus |119911|2)120573sdot 100381610038161003816100381610038161003816(11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + (1 minus |119911|2)120573 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572
10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)
0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816
le max|120577|le|120593(0)|
1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 le 21 minus 120572 10038171003817100381710038171198911003817100381710038171003817Z120572sdot max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572
sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816
(45)
Then 119862120593119879119892 is bounded This completes the proof of (i)
(ii) Case 120572 = 1 We consider the test function 119896120593(119911)(119911) definedin (30) for every 119911 isin 119863 with |120593(119911)| gt 12 It follows that
10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 ge log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 minus (1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816
(46)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 andsup|120593(119911)|gt12119862120593119879119892119896120593(119911)Z120573 le 119862 we get
sup|120593(119911)|gt12
log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12
(1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816 + 10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 lt infin
(47)
Then we have
sup119911isin119863
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1
minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le 119862 + log(43) 100381710038171003817100381710038171003817(1198921015840∘ 120593) (1205931015840)2100381710038171003817100381710038171003817V120573 lt infin
(48)
On the other hand from (15) and Lemma 4 we have
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)
Hence (39) holdsThe converse implication can be shown as in the proof of
(i)
(iii) Case 1 lt 120572 lt 2 ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 hasbeen proved above We take the test function 119891120593(119911) in (23) for
Journal of Function Spaces 7
every 119911 isin 119863with |120593(119911)| gt 12 by the same way as (ii) we canobtain that (40) holds
The converse implication can be shown as in the proof of(i)
(iv) Case 120572 = 2 We have proved that (41) holds aboveTo prove (42) we consider another test function 119905119886(119911) =log(2(1 minus 119886119911)) Clearly 119905119886 isin Z2 and sup12lt|119886|lt1119905119886Z2 lt infinFor every 119911 isin 119863 with |120593(119911)| gt 12 it follows that
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816
(50)
Applying (41) we get
sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin
(51)
Noting ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and usingLemma 4 and (15) we conclude that (42) holds
(v) Case 120572 gt 2 We have proved that (40) holds aboveApplying test function 119891120593(119911) in (23) for every 119911 isin 119863 with|120593(119911)| gt 12 we have
1198781 = sup|120593(119911)|gt12
(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup
|120593(119911)|gt12
10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)
With the same calculation for test function 119905120593(119911)(120593(119911)) =(1 minus |120593(119911)|2)2(1 minus 120593(119911)119911)120572 with |120593(119911)| gt 12 thensup|120593(119911)|gt(12)119905120593(119911)Z120572 le 119862 and we have that
1198782 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1100381610038161003816100381610038161003816100381610038161003816100381610038161003816 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(119862120593119879119892 (119905120593(119911)))10158401015840 (120593 (119911))100381610038161003816100381610038161003816 le 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817Z120572rarrZ120573sdot sup|120593(119911)|gt12
10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin
(53)
Therefore
sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782
+ 120572 sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572
sdot sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))10038161003816100381610038161003816100381610038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 1205721198781lt infin
(54)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 we conclude that(43) holds
Theorem9 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 11989210158401205931015840 isin 119867infinV120573 and 11989210158401015840 isin 119867infinV120573 (ii) If 120572 = 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
if and only if 11989210158401015840 isin 119867infinV120573 andsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin
(55)
8 Journal of Function Spaces
(iii) If 1 lt 120572 lt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a boundedoperator if and only if 11989210158401015840 isin 119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)
(iv) If 120572 = 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin
(57)
(v) If 120572 gt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if (56) holds and
sup119899ge0
(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)
The proof is similar to that of Theorem 8 and the detailsare omitted
3 Essential Norms
In this section we estimate the essential norms of theseintegral type operators on 119885119910119892119898119906119899119889 type spaces in terms of119892 120593 their derivatives and the nth power 120593119899 of 120593
Let Z120572 = 119891 isin Z120572 119891(0) = 1198911015840(0) = 0 and B120572 =119891 isin B120572 119891(0) = 0 We note that every compact operator119879 isin K(Z120572Z120573) can be extended to a compact operator119870 isinK(Z120572Z120573) In fact for every 119891 isin Z120572 119891 minus 119891(0) minus 1198911015840(0)119911 isinZ120572 and we can define119870(119891) = 119879(119891minus119891(0) minus1198911015840(0)119911) +119891(0) +1198911015840(0)119911
For 119903 isin (0 1) we consider the compact operator 119870119903 Z120572 rarr Z120573 defined by 119870119903119891(119911) = 119891(119903119911)Lemma 10 If 119883(119868119892119862120593 119862120593119868119892 119862120593119879119892 119879119892119862120593) is a bounded oper-ator fromZ120572 toZ120573 space then
119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)
Proof Clearly 119883119890Z120572rarrZ120573 ge 119883119890Z120572rarrZ120573 Then we prove119883119890Z120572rarrZ120573 le 119883119890Z120572rarrZ120573 Let 119879 isin 119870(Z120572Z120573) be given Let 119903119899 be an increasing
sequence in (0 1) converging to 1 andA = ℎ | ℎ = 119886 + 119887119911the closed subspace ofZ120572 Then
119883 minus 119879Z120572rarrZ120573 = sup119891isinZ120572119891Z120572le1
1003817100381710038171003817119883 (119891) minus 119879 (119891)1003817100381710038171003817Z120573le sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)minus 119879 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573+ sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup
119892isinZ120572119892Z120572le1
1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup
ℎisinAℎZ120572le1
1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573
(60)
Hence
inf119879isin119870(Z120572 Z120573)
119883 minus 119879Z120572rarrZ120573le inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|Z1205721003817100381710038171003817Z120572rarrZ120573+ inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|1205721003817100381710038171003817Z120572rarrZ120573le 119883119890Z120572rarrZ120573 + lim
119899rarrinfin
10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573
(61)
Since lim119899rarrinfin119883(119868minus119870119903119899)Z120572rarrZ120573 = 0 we have 119883119890Z120572rarrZ120573 le119883119890Z120572rarrZ120573 and the proof is finished
Let119863120572 Z120572 rarr B120572 and 119878120572 B120572 rarr 119867infinV120572 be the derivativeoperatorsThen clearly119863120572 and 119878120572 are linear isometries on Z120572and B120572 respectively and
119878120573119863120573119868119892119862120593119863minus1120572 119878minus1120572 = 119892 (1205931015840) 119862120593 + 1198921015840119862120593119878minus1120572 (62)
Therefore
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890B120572rarr119867infinV120573+ 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
(63)
Similarly
119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593
+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)
Journal of Function Spaces 9
1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)
119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572
+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)
119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573
(69)
Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)
Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572
into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume
to be identically zero Then it follows from Theorem 6 andLemma 1 that
lim119896rarrinfin
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin
sup119911isin119863
10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)
which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816
= lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(74)
For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin
1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin
1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
(75)
Now we use (14) and Lemma 4 to obtain that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1
10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
= 120572119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(76)
Hence (70) holds
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Function Spaces
Then
sup119911isin119863
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
le sup|120593(119911)|le12
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
+ sup|120593(119911)|gt12
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
le (43)120572 10038171003817100381710038171003817120593101584011989210038171003817100381710038171003817V120573 + 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 lt infin
(27)
Now we use (14) and Lemma 4 to conclude that
sup119899ge0
(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (28)
which shows that (16) is necessary for all caseConversely suppose that 1198921015840 isin 119867infinV120573 and (16) holds As-
sume that 119891 isin Z120572 From Lemma 2 it follows that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119868119892119862120593119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573sdot 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911)) + 1198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816le (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911))10038161003816100381610038161003816 + (1 minus |119911|2)120573
sdot 100381610038161003816100381610038161198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816 le (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + 119862 (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572 10038161003816100381610038161003816119868119892119862120593 (119891) (0)10038161003816100381610038161003816 = 0
100381610038161003816100381610038161003816(119868119892119862120593119891)1015840 (0)100381610038161003816100381610038161003816 = 100381610038161003816100381610038161198911015840 (120593 (0)) 119892 (0)10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572sdot 1003816100381610038161003816119892 (120593 (0))1003816100381610038161003816
(29)
which implies that 119868119892119862120593 is boundedThis completes the proofof (i)
Next we will prove (ii)The necessity in condition (17) hasbeen proved above Fixing 119886 isin 119863 with |119886| gt 12 we take thefunction
119896119886 (119911) = 119901 (119886119911)119886 (log 11 minus |119886|)minus1 (30)
for 119911 isin 119863 where
119901 (119911) = (119911 minus 1) ((1 + log 11 minus 119911)2 + 1) (31)
Then we have sup12lt|119886|lt1119896119886Z120572 le 119862 by [11] Let 119886 = 120593(119911) Itfollows that10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573
ge (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198961015840120593(119911) (120593 (119911))10038161003816100381610038161003816minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989610158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(32)
Since (17) holds and 119868119892119862120593 is bounded we obtain that
sup|120593(119911)|gt12
(1 minus |119911|2)120573 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ sup|120593(119911)|gt12
10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573 lt infin(33)
Noting 1198921015840 isin 119867infinV120573 and together with (15) and Lemma 4 weconclude that (18) holds
The converse implication can be shown as in the proof of(i)
Finally we will prove (iii) We have proved that (19) holdsabove To prove (20) we take function 119891120593(119911) defined in (23)for every 119911 isin 119863 with |120593(119911)| gt 12 and obtain that10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573
ge (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989110158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816
= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 110038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816100381610038162120593 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(34)
Since 119868119892119862120593 is bounded and (19) holds we obtain that
sup|120593(119911)|gt12
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le 2 sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ 2 sup|120593(119911)|gt12
10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(35)
therefore we deduce that (20) holds by (14) and Lemma 4
Journal of Function Spaces 5
The converse implication can be shown as in the proof of(i)
Theorem 7 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((119892∘120593)(12059310158401015840)+(1198921015840∘120593)(1205931015840)2) isin 119867infinV120573and
sup119899ge0
(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)
(ii) If 120572 = 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin
(37)
(iii) If 120572 gt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if (36) holds and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)
The proof is similar to that of Theorem 6 and the detailsare omitted
Theorem8 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119879119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573
(ii) If 120572 = 1 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)
(iii) If 1 lt 120572 lt 2 then 119862120593119879119892 Z120572 rarr Z120573 is a boundedoperator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)
(iv) If 120572 = 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)
(v) If 120572 gt 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if (40) holds and
sup119899ge0
(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2
+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)
6 Journal of Function Spaces
Proof Suppose that 119862120593119879119892 is bounded fromZ120572 toZ120573 space
(i) Case 0 lt 120572 lt 1 Using functions 119891 = 1 isin Z120572 and 119891 = 119911 isinZ120572 we obtain
sup119911isin119863
(1 minus |119911|2)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infinsup119911isin119863
(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)
Then we obtain that ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and(1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 are necessary for all caseFor the converse implication suppose that ((1198921015840 ∘120593)(12059310158401015840)+(11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 For 119891 isin Z120572 it
follows from Lemma 2 that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2sdot 1198921015840 (120593 (119911)) 1198911015840 (120593 (119911))+ (11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))sdot 1198911015840 (120593 (119911))100381610038161003816100381610038161003816 + (1 minus |119911|2)120573sdot 100381610038161003816100381610038161003816(11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + (1 minus |119911|2)120573 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572
10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)
0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816
le max|120577|le|120593(0)|
1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 le 21 minus 120572 10038171003817100381710038171198911003817100381710038171003817Z120572sdot max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572
sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816
(45)
Then 119862120593119879119892 is bounded This completes the proof of (i)
(ii) Case 120572 = 1 We consider the test function 119896120593(119911)(119911) definedin (30) for every 119911 isin 119863 with |120593(119911)| gt 12 It follows that
10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 ge log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 minus (1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816
(46)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 andsup|120593(119911)|gt12119862120593119879119892119896120593(119911)Z120573 le 119862 we get
sup|120593(119911)|gt12
log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12
(1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816 + 10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 lt infin
(47)
Then we have
sup119911isin119863
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1
minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le 119862 + log(43) 100381710038171003817100381710038171003817(1198921015840∘ 120593) (1205931015840)2100381710038171003817100381710038171003817V120573 lt infin
(48)
On the other hand from (15) and Lemma 4 we have
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)
Hence (39) holdsThe converse implication can be shown as in the proof of
(i)
(iii) Case 1 lt 120572 lt 2 ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 hasbeen proved above We take the test function 119891120593(119911) in (23) for
Journal of Function Spaces 7
every 119911 isin 119863with |120593(119911)| gt 12 by the same way as (ii) we canobtain that (40) holds
The converse implication can be shown as in the proof of(i)
(iv) Case 120572 = 2 We have proved that (41) holds aboveTo prove (42) we consider another test function 119905119886(119911) =log(2(1 minus 119886119911)) Clearly 119905119886 isin Z2 and sup12lt|119886|lt1119905119886Z2 lt infinFor every 119911 isin 119863 with |120593(119911)| gt 12 it follows that
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816
(50)
Applying (41) we get
sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin
(51)
Noting ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and usingLemma 4 and (15) we conclude that (42) holds
(v) Case 120572 gt 2 We have proved that (40) holds aboveApplying test function 119891120593(119911) in (23) for every 119911 isin 119863 with|120593(119911)| gt 12 we have
1198781 = sup|120593(119911)|gt12
(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup
|120593(119911)|gt12
10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)
With the same calculation for test function 119905120593(119911)(120593(119911)) =(1 minus |120593(119911)|2)2(1 minus 120593(119911)119911)120572 with |120593(119911)| gt 12 thensup|120593(119911)|gt(12)119905120593(119911)Z120572 le 119862 and we have that
1198782 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1100381610038161003816100381610038161003816100381610038161003816100381610038161003816 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(119862120593119879119892 (119905120593(119911)))10158401015840 (120593 (119911))100381610038161003816100381610038161003816 le 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817Z120572rarrZ120573sdot sup|120593(119911)|gt12
10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin
(53)
Therefore
sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782
+ 120572 sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572
sdot sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))10038161003816100381610038161003816100381610038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 1205721198781lt infin
(54)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 we conclude that(43) holds
Theorem9 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 11989210158401205931015840 isin 119867infinV120573 and 11989210158401015840 isin 119867infinV120573 (ii) If 120572 = 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
if and only if 11989210158401015840 isin 119867infinV120573 andsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin
(55)
8 Journal of Function Spaces
(iii) If 1 lt 120572 lt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a boundedoperator if and only if 11989210158401015840 isin 119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)
(iv) If 120572 = 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin
(57)
(v) If 120572 gt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if (56) holds and
sup119899ge0
(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)
The proof is similar to that of Theorem 8 and the detailsare omitted
3 Essential Norms
In this section we estimate the essential norms of theseintegral type operators on 119885119910119892119898119906119899119889 type spaces in terms of119892 120593 their derivatives and the nth power 120593119899 of 120593
Let Z120572 = 119891 isin Z120572 119891(0) = 1198911015840(0) = 0 and B120572 =119891 isin B120572 119891(0) = 0 We note that every compact operator119879 isin K(Z120572Z120573) can be extended to a compact operator119870 isinK(Z120572Z120573) In fact for every 119891 isin Z120572 119891 minus 119891(0) minus 1198911015840(0)119911 isinZ120572 and we can define119870(119891) = 119879(119891minus119891(0) minus1198911015840(0)119911) +119891(0) +1198911015840(0)119911
For 119903 isin (0 1) we consider the compact operator 119870119903 Z120572 rarr Z120573 defined by 119870119903119891(119911) = 119891(119903119911)Lemma 10 If 119883(119868119892119862120593 119862120593119868119892 119862120593119879119892 119879119892119862120593) is a bounded oper-ator fromZ120572 toZ120573 space then
119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)
Proof Clearly 119883119890Z120572rarrZ120573 ge 119883119890Z120572rarrZ120573 Then we prove119883119890Z120572rarrZ120573 le 119883119890Z120572rarrZ120573 Let 119879 isin 119870(Z120572Z120573) be given Let 119903119899 be an increasing
sequence in (0 1) converging to 1 andA = ℎ | ℎ = 119886 + 119887119911the closed subspace ofZ120572 Then
119883 minus 119879Z120572rarrZ120573 = sup119891isinZ120572119891Z120572le1
1003817100381710038171003817119883 (119891) minus 119879 (119891)1003817100381710038171003817Z120573le sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)minus 119879 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573+ sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup
119892isinZ120572119892Z120572le1
1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup
ℎisinAℎZ120572le1
1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573
(60)
Hence
inf119879isin119870(Z120572 Z120573)
119883 minus 119879Z120572rarrZ120573le inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|Z1205721003817100381710038171003817Z120572rarrZ120573+ inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|1205721003817100381710038171003817Z120572rarrZ120573le 119883119890Z120572rarrZ120573 + lim
119899rarrinfin
10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573
(61)
Since lim119899rarrinfin119883(119868minus119870119903119899)Z120572rarrZ120573 = 0 we have 119883119890Z120572rarrZ120573 le119883119890Z120572rarrZ120573 and the proof is finished
Let119863120572 Z120572 rarr B120572 and 119878120572 B120572 rarr 119867infinV120572 be the derivativeoperatorsThen clearly119863120572 and 119878120572 are linear isometries on Z120572and B120572 respectively and
119878120573119863120573119868119892119862120593119863minus1120572 119878minus1120572 = 119892 (1205931015840) 119862120593 + 1198921015840119862120593119878minus1120572 (62)
Therefore
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890B120572rarr119867infinV120573+ 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
(63)
Similarly
119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593
+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)
Journal of Function Spaces 9
1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)
119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572
+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)
119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573
(69)
Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)
Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572
into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume
to be identically zero Then it follows from Theorem 6 andLemma 1 that
lim119896rarrinfin
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin
sup119911isin119863
10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)
which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816
= lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(74)
For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin
1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin
1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
(75)
Now we use (14) and Lemma 4 to obtain that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1
10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
= 120572119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(76)
Hence (70) holds
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 5
The converse implication can be shown as in the proof of(i)
Theorem 7 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((119892∘120593)(12059310158401015840)+(1198921015840∘120593)(1205931015840)2) isin 119867infinV120573and
sup119899ge0
(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)
(ii) If 120572 = 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin
(37)
(iii) If 120572 gt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if (36) holds and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)
The proof is similar to that of Theorem 6 and the detailsare omitted
Theorem8 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119879119892 Z120572 rarr Z120573 is a bounded
operator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573
(ii) If 120572 = 1 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)
(iii) If 1 lt 120572 lt 2 then 119862120593119879119892 Z120572 rarr Z120573 is a boundedoperator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)
(iv) If 120572 = 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)
(v) If 120572 gt 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if (40) holds and
sup119899ge0
(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2
+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)
6 Journal of Function Spaces
Proof Suppose that 119862120593119879119892 is bounded fromZ120572 toZ120573 space
(i) Case 0 lt 120572 lt 1 Using functions 119891 = 1 isin Z120572 and 119891 = 119911 isinZ120572 we obtain
sup119911isin119863
(1 minus |119911|2)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infinsup119911isin119863
(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)
Then we obtain that ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and(1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 are necessary for all caseFor the converse implication suppose that ((1198921015840 ∘120593)(12059310158401015840)+(11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 For 119891 isin Z120572 it
follows from Lemma 2 that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2sdot 1198921015840 (120593 (119911)) 1198911015840 (120593 (119911))+ (11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))sdot 1198911015840 (120593 (119911))100381610038161003816100381610038161003816 + (1 minus |119911|2)120573sdot 100381610038161003816100381610038161003816(11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + (1 minus |119911|2)120573 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572
10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)
0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816
le max|120577|le|120593(0)|
1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 le 21 minus 120572 10038171003817100381710038171198911003817100381710038171003817Z120572sdot max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572
sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816
(45)
Then 119862120593119879119892 is bounded This completes the proof of (i)
(ii) Case 120572 = 1 We consider the test function 119896120593(119911)(119911) definedin (30) for every 119911 isin 119863 with |120593(119911)| gt 12 It follows that
10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 ge log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 minus (1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816
(46)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 andsup|120593(119911)|gt12119862120593119879119892119896120593(119911)Z120573 le 119862 we get
sup|120593(119911)|gt12
log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12
(1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816 + 10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 lt infin
(47)
Then we have
sup119911isin119863
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1
minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le 119862 + log(43) 100381710038171003817100381710038171003817(1198921015840∘ 120593) (1205931015840)2100381710038171003817100381710038171003817V120573 lt infin
(48)
On the other hand from (15) and Lemma 4 we have
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)
Hence (39) holdsThe converse implication can be shown as in the proof of
(i)
(iii) Case 1 lt 120572 lt 2 ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 hasbeen proved above We take the test function 119891120593(119911) in (23) for
Journal of Function Spaces 7
every 119911 isin 119863with |120593(119911)| gt 12 by the same way as (ii) we canobtain that (40) holds
The converse implication can be shown as in the proof of(i)
(iv) Case 120572 = 2 We have proved that (41) holds aboveTo prove (42) we consider another test function 119905119886(119911) =log(2(1 minus 119886119911)) Clearly 119905119886 isin Z2 and sup12lt|119886|lt1119905119886Z2 lt infinFor every 119911 isin 119863 with |120593(119911)| gt 12 it follows that
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816
(50)
Applying (41) we get
sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin
(51)
Noting ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and usingLemma 4 and (15) we conclude that (42) holds
(v) Case 120572 gt 2 We have proved that (40) holds aboveApplying test function 119891120593(119911) in (23) for every 119911 isin 119863 with|120593(119911)| gt 12 we have
1198781 = sup|120593(119911)|gt12
(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup
|120593(119911)|gt12
10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)
With the same calculation for test function 119905120593(119911)(120593(119911)) =(1 minus |120593(119911)|2)2(1 minus 120593(119911)119911)120572 with |120593(119911)| gt 12 thensup|120593(119911)|gt(12)119905120593(119911)Z120572 le 119862 and we have that
1198782 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1100381610038161003816100381610038161003816100381610038161003816100381610038161003816 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(119862120593119879119892 (119905120593(119911)))10158401015840 (120593 (119911))100381610038161003816100381610038161003816 le 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817Z120572rarrZ120573sdot sup|120593(119911)|gt12
10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin
(53)
Therefore
sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782
+ 120572 sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572
sdot sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))10038161003816100381610038161003816100381610038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 1205721198781lt infin
(54)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 we conclude that(43) holds
Theorem9 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 11989210158401205931015840 isin 119867infinV120573 and 11989210158401015840 isin 119867infinV120573 (ii) If 120572 = 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
if and only if 11989210158401015840 isin 119867infinV120573 andsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin
(55)
8 Journal of Function Spaces
(iii) If 1 lt 120572 lt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a boundedoperator if and only if 11989210158401015840 isin 119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)
(iv) If 120572 = 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin
(57)
(v) If 120572 gt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if (56) holds and
sup119899ge0
(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)
The proof is similar to that of Theorem 8 and the detailsare omitted
3 Essential Norms
In this section we estimate the essential norms of theseintegral type operators on 119885119910119892119898119906119899119889 type spaces in terms of119892 120593 their derivatives and the nth power 120593119899 of 120593
Let Z120572 = 119891 isin Z120572 119891(0) = 1198911015840(0) = 0 and B120572 =119891 isin B120572 119891(0) = 0 We note that every compact operator119879 isin K(Z120572Z120573) can be extended to a compact operator119870 isinK(Z120572Z120573) In fact for every 119891 isin Z120572 119891 minus 119891(0) minus 1198911015840(0)119911 isinZ120572 and we can define119870(119891) = 119879(119891minus119891(0) minus1198911015840(0)119911) +119891(0) +1198911015840(0)119911
For 119903 isin (0 1) we consider the compact operator 119870119903 Z120572 rarr Z120573 defined by 119870119903119891(119911) = 119891(119903119911)Lemma 10 If 119883(119868119892119862120593 119862120593119868119892 119862120593119879119892 119879119892119862120593) is a bounded oper-ator fromZ120572 toZ120573 space then
119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)
Proof Clearly 119883119890Z120572rarrZ120573 ge 119883119890Z120572rarrZ120573 Then we prove119883119890Z120572rarrZ120573 le 119883119890Z120572rarrZ120573 Let 119879 isin 119870(Z120572Z120573) be given Let 119903119899 be an increasing
sequence in (0 1) converging to 1 andA = ℎ | ℎ = 119886 + 119887119911the closed subspace ofZ120572 Then
119883 minus 119879Z120572rarrZ120573 = sup119891isinZ120572119891Z120572le1
1003817100381710038171003817119883 (119891) minus 119879 (119891)1003817100381710038171003817Z120573le sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)minus 119879 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573+ sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup
119892isinZ120572119892Z120572le1
1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup
ℎisinAℎZ120572le1
1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573
(60)
Hence
inf119879isin119870(Z120572 Z120573)
119883 minus 119879Z120572rarrZ120573le inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|Z1205721003817100381710038171003817Z120572rarrZ120573+ inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|1205721003817100381710038171003817Z120572rarrZ120573le 119883119890Z120572rarrZ120573 + lim
119899rarrinfin
10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573
(61)
Since lim119899rarrinfin119883(119868minus119870119903119899)Z120572rarrZ120573 = 0 we have 119883119890Z120572rarrZ120573 le119883119890Z120572rarrZ120573 and the proof is finished
Let119863120572 Z120572 rarr B120572 and 119878120572 B120572 rarr 119867infinV120572 be the derivativeoperatorsThen clearly119863120572 and 119878120572 are linear isometries on Z120572and B120572 respectively and
119878120573119863120573119868119892119862120593119863minus1120572 119878minus1120572 = 119892 (1205931015840) 119862120593 + 1198921015840119862120593119878minus1120572 (62)
Therefore
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890B120572rarr119867infinV120573+ 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
(63)
Similarly
119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593
+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)
Journal of Function Spaces 9
1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)
119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572
+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)
119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573
(69)
Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)
Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572
into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume
to be identically zero Then it follows from Theorem 6 andLemma 1 that
lim119896rarrinfin
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin
sup119911isin119863
10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)
which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816
= lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(74)
For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin
1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin
1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
(75)
Now we use (14) and Lemma 4 to obtain that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1
10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
= 120572119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(76)
Hence (70) holds
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Function Spaces
Proof Suppose that 119862120593119879119892 is bounded fromZ120572 toZ120573 space
(i) Case 0 lt 120572 lt 1 Using functions 119891 = 1 isin Z120572 and 119891 = 119911 isinZ120572 we obtain
sup119911isin119863
(1 minus |119911|2)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infinsup119911isin119863
(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)
Then we obtain that ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and(1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 are necessary for all caseFor the converse implication suppose that ((1198921015840 ∘120593)(12059310158401015840)+(11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 For 119891 isin Z120572 it
follows from Lemma 2 that
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2sdot 1198921015840 (120593 (119911)) 1198911015840 (120593 (119911))+ (11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))sdot 1198911015840 (120593 (119911))100381610038161003816100381610038161003816 + (1 minus |119911|2)120573sdot 100381610038161003816100381610038161003816(11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + (1 minus |119911|2)120573 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572
10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)
0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816
le max|120577|le|120593(0)|
1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 le 21 minus 120572 10038171003817100381710038171198911003817100381710038171003817Z120572sdot max|120577|le|120593(0)|
100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572
sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816
(45)
Then 119862120593119879119892 is bounded This completes the proof of (i)
(ii) Case 120572 = 1 We consider the test function 119896120593(119911)(119911) definedin (30) for every 119911 isin 119863 with |120593(119911)| gt 12 It follows that
10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 ge log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 minus (1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816
(46)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 andsup|120593(119911)|gt12119862120593119879119892119896120593(119911)Z120573 le 119862 we get
sup|120593(119911)|gt12
log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12
(1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816 + 10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 lt infin
(47)
Then we have
sup119911isin119863
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2
sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1
minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le 119862 + log(43) 100381710038171003817100381710038171003817(1198921015840∘ 120593) (1205931015840)2100381710038171003817100381710038171003817V120573 lt infin
(48)
On the other hand from (15) and Lemma 4 we have
sup119899ge0
(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863
(1 minus |119911|2)120573
sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)
Hence (39) holdsThe converse implication can be shown as in the proof of
(i)
(iii) Case 1 lt 120572 lt 2 ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 hasbeen proved above We take the test function 119891120593(119911) in (23) for
Journal of Function Spaces 7
every 119911 isin 119863with |120593(119911)| gt 12 by the same way as (ii) we canobtain that (40) holds
The converse implication can be shown as in the proof of(i)
(iv) Case 120572 = 2 We have proved that (41) holds aboveTo prove (42) we consider another test function 119905119886(119911) =log(2(1 minus 119886119911)) Clearly 119905119886 isin Z2 and sup12lt|119886|lt1119905119886Z2 lt infinFor every 119911 isin 119863 with |120593(119911)| gt 12 it follows that
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816
(50)
Applying (41) we get
sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin
(51)
Noting ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and usingLemma 4 and (15) we conclude that (42) holds
(v) Case 120572 gt 2 We have proved that (40) holds aboveApplying test function 119891120593(119911) in (23) for every 119911 isin 119863 with|120593(119911)| gt 12 we have
1198781 = sup|120593(119911)|gt12
(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup
|120593(119911)|gt12
10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)
With the same calculation for test function 119905120593(119911)(120593(119911)) =(1 minus |120593(119911)|2)2(1 minus 120593(119911)119911)120572 with |120593(119911)| gt 12 thensup|120593(119911)|gt(12)119905120593(119911)Z120572 le 119862 and we have that
1198782 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1100381610038161003816100381610038161003816100381610038161003816100381610038161003816 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(119862120593119879119892 (119905120593(119911)))10158401015840 (120593 (119911))100381610038161003816100381610038161003816 le 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817Z120572rarrZ120573sdot sup|120593(119911)|gt12
10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin
(53)
Therefore
sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782
+ 120572 sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572
sdot sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))10038161003816100381610038161003816100381610038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 1205721198781lt infin
(54)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 we conclude that(43) holds
Theorem9 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 11989210158401205931015840 isin 119867infinV120573 and 11989210158401015840 isin 119867infinV120573 (ii) If 120572 = 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
if and only if 11989210158401015840 isin 119867infinV120573 andsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin
(55)
8 Journal of Function Spaces
(iii) If 1 lt 120572 lt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a boundedoperator if and only if 11989210158401015840 isin 119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)
(iv) If 120572 = 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin
(57)
(v) If 120572 gt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if (56) holds and
sup119899ge0
(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)
The proof is similar to that of Theorem 8 and the detailsare omitted
3 Essential Norms
In this section we estimate the essential norms of theseintegral type operators on 119885119910119892119898119906119899119889 type spaces in terms of119892 120593 their derivatives and the nth power 120593119899 of 120593
Let Z120572 = 119891 isin Z120572 119891(0) = 1198911015840(0) = 0 and B120572 =119891 isin B120572 119891(0) = 0 We note that every compact operator119879 isin K(Z120572Z120573) can be extended to a compact operator119870 isinK(Z120572Z120573) In fact for every 119891 isin Z120572 119891 minus 119891(0) minus 1198911015840(0)119911 isinZ120572 and we can define119870(119891) = 119879(119891minus119891(0) minus1198911015840(0)119911) +119891(0) +1198911015840(0)119911
For 119903 isin (0 1) we consider the compact operator 119870119903 Z120572 rarr Z120573 defined by 119870119903119891(119911) = 119891(119903119911)Lemma 10 If 119883(119868119892119862120593 119862120593119868119892 119862120593119879119892 119879119892119862120593) is a bounded oper-ator fromZ120572 toZ120573 space then
119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)
Proof Clearly 119883119890Z120572rarrZ120573 ge 119883119890Z120572rarrZ120573 Then we prove119883119890Z120572rarrZ120573 le 119883119890Z120572rarrZ120573 Let 119879 isin 119870(Z120572Z120573) be given Let 119903119899 be an increasing
sequence in (0 1) converging to 1 andA = ℎ | ℎ = 119886 + 119887119911the closed subspace ofZ120572 Then
119883 minus 119879Z120572rarrZ120573 = sup119891isinZ120572119891Z120572le1
1003817100381710038171003817119883 (119891) minus 119879 (119891)1003817100381710038171003817Z120573le sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)minus 119879 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573+ sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup
119892isinZ120572119892Z120572le1
1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup
ℎisinAℎZ120572le1
1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573
(60)
Hence
inf119879isin119870(Z120572 Z120573)
119883 minus 119879Z120572rarrZ120573le inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|Z1205721003817100381710038171003817Z120572rarrZ120573+ inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|1205721003817100381710038171003817Z120572rarrZ120573le 119883119890Z120572rarrZ120573 + lim
119899rarrinfin
10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573
(61)
Since lim119899rarrinfin119883(119868minus119870119903119899)Z120572rarrZ120573 = 0 we have 119883119890Z120572rarrZ120573 le119883119890Z120572rarrZ120573 and the proof is finished
Let119863120572 Z120572 rarr B120572 and 119878120572 B120572 rarr 119867infinV120572 be the derivativeoperatorsThen clearly119863120572 and 119878120572 are linear isometries on Z120572and B120572 respectively and
119878120573119863120573119868119892119862120593119863minus1120572 119878minus1120572 = 119892 (1205931015840) 119862120593 + 1198921015840119862120593119878minus1120572 (62)
Therefore
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890B120572rarr119867infinV120573+ 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
(63)
Similarly
119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593
+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)
Journal of Function Spaces 9
1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)
119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572
+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)
119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573
(69)
Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)
Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572
into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume
to be identically zero Then it follows from Theorem 6 andLemma 1 that
lim119896rarrinfin
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin
sup119911isin119863
10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)
which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816
= lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(74)
For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin
1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin
1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
(75)
Now we use (14) and Lemma 4 to obtain that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1
10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
= 120572119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(76)
Hence (70) holds
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Stochastic AnalysisInternational Journal of
Journal of Function Spaces 7
every 119911 isin 119863with |120593(119911)| gt 12 by the same way as (ii) we canobtain that (40) holds
The converse implication can be shown as in the proof of(i)
(iv) Case 120572 = 2 We have proved that (41) holds aboveTo prove (42) we consider another test function 119905119886(119911) =log(2(1 minus 119886119911)) Clearly 119905119886 isin Z2 and sup12lt|119886|lt1119905119886Z2 lt infinFor every 119911 isin 119863 with |120593(119911)| gt 12 it follows that
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816
(50)
Applying (41) we get
sup|120593(119911)|gt12
log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12
(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin
(51)
Noting ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and usingLemma 4 and (15) we conclude that (42) holds
(v) Case 120572 gt 2 We have proved that (40) holds aboveApplying test function 119891120593(119911) in (23) for every 119911 isin 119863 with|120593(119911)| gt 12 we have
1198781 = sup|120593(119911)|gt12
(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12
(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup
|120593(119911)|gt12
10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)
With the same calculation for test function 119905120593(119911)(120593(119911)) =(1 minus |120593(119911)|2)2(1 minus 120593(119911)119911)120572 with |120593(119911)| gt 12 thensup|120593(119911)|gt(12)119905120593(119911)Z120572 le 119862 and we have that
1198782 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1100381610038161003816100381610038161003816100381610038161003816100381610038161003816 = sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot 100381610038161003816100381610038161003816(119862120593119879119892 (119905120593(119911)))10158401015840 (120593 (119911))100381610038161003816100381610038161003816 le 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817Z120572rarrZ120573sdot sup|120593(119911)|gt12
10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin
(53)
Therefore
sup|120593(119911)|gt12
(1 minus |119911|2)120573
sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)
(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782
+ 120572 sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572
sdot sup|120593(119911)|gt12
100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))10038161003816100381610038161003816100381610038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 1205721198781lt infin
(54)
Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 we conclude that(43) holds
Theorem9 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded
operator if and only if 11989210158401205931015840 isin 119867infinV120573 and 11989210158401015840 isin 119867infinV120573 (ii) If 120572 = 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
if and only if 11989210158401015840 isin 119867infinV120573 andsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin
(55)
8 Journal of Function Spaces
(iii) If 1 lt 120572 lt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a boundedoperator if and only if 11989210158401015840 isin 119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)
(iv) If 120572 = 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin
(57)
(v) If 120572 gt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if (56) holds and
sup119899ge0
(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)
The proof is similar to that of Theorem 8 and the detailsare omitted
3 Essential Norms
In this section we estimate the essential norms of theseintegral type operators on 119885119910119892119898119906119899119889 type spaces in terms of119892 120593 their derivatives and the nth power 120593119899 of 120593
Let Z120572 = 119891 isin Z120572 119891(0) = 1198911015840(0) = 0 and B120572 =119891 isin B120572 119891(0) = 0 We note that every compact operator119879 isin K(Z120572Z120573) can be extended to a compact operator119870 isinK(Z120572Z120573) In fact for every 119891 isin Z120572 119891 minus 119891(0) minus 1198911015840(0)119911 isinZ120572 and we can define119870(119891) = 119879(119891minus119891(0) minus1198911015840(0)119911) +119891(0) +1198911015840(0)119911
For 119903 isin (0 1) we consider the compact operator 119870119903 Z120572 rarr Z120573 defined by 119870119903119891(119911) = 119891(119903119911)Lemma 10 If 119883(119868119892119862120593 119862120593119868119892 119862120593119879119892 119879119892119862120593) is a bounded oper-ator fromZ120572 toZ120573 space then
119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)
Proof Clearly 119883119890Z120572rarrZ120573 ge 119883119890Z120572rarrZ120573 Then we prove119883119890Z120572rarrZ120573 le 119883119890Z120572rarrZ120573 Let 119879 isin 119870(Z120572Z120573) be given Let 119903119899 be an increasing
sequence in (0 1) converging to 1 andA = ℎ | ℎ = 119886 + 119887119911the closed subspace ofZ120572 Then
119883 minus 119879Z120572rarrZ120573 = sup119891isinZ120572119891Z120572le1
1003817100381710038171003817119883 (119891) minus 119879 (119891)1003817100381710038171003817Z120573le sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)minus 119879 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573+ sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup
119892isinZ120572119892Z120572le1
1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup
ℎisinAℎZ120572le1
1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573
(60)
Hence
inf119879isin119870(Z120572 Z120573)
119883 minus 119879Z120572rarrZ120573le inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|Z1205721003817100381710038171003817Z120572rarrZ120573+ inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|1205721003817100381710038171003817Z120572rarrZ120573le 119883119890Z120572rarrZ120573 + lim
119899rarrinfin
10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573
(61)
Since lim119899rarrinfin119883(119868minus119870119903119899)Z120572rarrZ120573 = 0 we have 119883119890Z120572rarrZ120573 le119883119890Z120572rarrZ120573 and the proof is finished
Let119863120572 Z120572 rarr B120572 and 119878120572 B120572 rarr 119867infinV120572 be the derivativeoperatorsThen clearly119863120572 and 119878120572 are linear isometries on Z120572and B120572 respectively and
119878120573119863120573119868119892119862120593119863minus1120572 119878minus1120572 = 119892 (1205931015840) 119862120593 + 1198921015840119862120593119878minus1120572 (62)
Therefore
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890B120572rarr119867infinV120573+ 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
(63)
Similarly
119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593
+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)
Journal of Function Spaces 9
1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)
119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572
+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)
119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573
(69)
Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)
Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572
into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume
to be identically zero Then it follows from Theorem 6 andLemma 1 that
lim119896rarrinfin
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin
sup119911isin119863
10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)
which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816
= lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(74)
For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin
1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin
1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
(75)
Now we use (14) and Lemma 4 to obtain that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1
10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
= 120572119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(76)
Hence (70) holds
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Function Spaces
(iii) If 1 lt 120572 lt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a boundedoperator if and only if 11989210158401015840 isin 119867infinV120573 and
sup119899ge0
(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)
(iv) If 120572 = 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if
sup119899ge0
(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin
(57)
(v) If 120572 gt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if (56) holds and
sup119899ge0
(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2
1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)
The proof is similar to that of Theorem 8 and the detailsare omitted
3 Essential Norms
In this section we estimate the essential norms of theseintegral type operators on 119885119910119892119898119906119899119889 type spaces in terms of119892 120593 their derivatives and the nth power 120593119899 of 120593
Let Z120572 = 119891 isin Z120572 119891(0) = 1198911015840(0) = 0 and B120572 =119891 isin B120572 119891(0) = 0 We note that every compact operator119879 isin K(Z120572Z120573) can be extended to a compact operator119870 isinK(Z120572Z120573) In fact for every 119891 isin Z120572 119891 minus 119891(0) minus 1198911015840(0)119911 isinZ120572 and we can define119870(119891) = 119879(119891minus119891(0) minus1198911015840(0)119911) +119891(0) +1198911015840(0)119911
For 119903 isin (0 1) we consider the compact operator 119870119903 Z120572 rarr Z120573 defined by 119870119903119891(119911) = 119891(119903119911)Lemma 10 If 119883(119868119892119862120593 119862120593119868119892 119862120593119879119892 119879119892119862120593) is a bounded oper-ator fromZ120572 toZ120573 space then
119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)
Proof Clearly 119883119890Z120572rarrZ120573 ge 119883119890Z120572rarrZ120573 Then we prove119883119890Z120572rarrZ120573 le 119883119890Z120572rarrZ120573 Let 119879 isin 119870(Z120572Z120573) be given Let 119903119899 be an increasing
sequence in (0 1) converging to 1 andA = ℎ | ℎ = 119886 + 119887119911the closed subspace ofZ120572 Then
119883 minus 119879Z120572rarrZ120573 = sup119891isinZ120572119891Z120572le1
1003817100381710038171003817119883 (119891) minus 119879 (119891)1003817100381710038171003817Z120573le sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)minus 119879 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573+ sup119891isinZ120572119891Z120572le1
10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup
119892isinZ120572119892Z120572le1
1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup
ℎisinAℎZ120572le1
1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573
(60)
Hence
inf119879isin119870(Z120572 Z120573)
119883 minus 119879Z120572rarrZ120573le inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|Z1205721003817100381710038171003817Z120572rarrZ120573+ inf119879isin119870(Z120572 Z120573)
1003817100381710038171003817119883 minus 119879|1205721003817100381710038171003817Z120572rarrZ120573le 119883119890Z120572rarrZ120573 + lim
119899rarrinfin
10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573
(61)
Since lim119899rarrinfin119883(119868minus119870119903119899)Z120572rarrZ120573 = 0 we have 119883119890Z120572rarrZ120573 le119883119890Z120572rarrZ120573 and the proof is finished
Let119863120572 Z120572 rarr B120572 and 119878120572 B120572 rarr 119867infinV120572 be the derivativeoperatorsThen clearly119863120572 and 119878120572 are linear isometries on Z120572and B120572 respectively and
119878120573119863120573119868119892119862120593119863minus1120572 119878minus1120572 = 119892 (1205931015840) 119862120593 + 1198921015840119862120593119878minus1120572 (62)
Therefore
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890B120572rarr119867infinV120573+ 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
(63)
Similarly
119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593
+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)
Journal of Function Spaces 9
1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)
119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572
+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)
119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573
(69)
Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)
Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572
into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume
to be identically zero Then it follows from Theorem 6 andLemma 1 that
lim119896rarrinfin
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin
sup119911isin119863
10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)
which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816
= lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(74)
For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin
1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin
1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
(75)
Now we use (14) and Lemma 4 to obtain that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1
10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
= 120572119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(76)
Hence (70) holds
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 9
1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)
119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572
+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573
+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)
119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573
(69)
Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)
Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572
into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume
to be identically zero Then it follows from Theorem 6 andLemma 1 that
lim119896rarrinfin
sup119911isin119863
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin
sup119911isin119863
10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)
which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1
(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816
= lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(74)
For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin
1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin
1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
(75)
Now we use (14) and Lemma 4 to obtain that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1
10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572
= 120572119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)
120572
lim sup119899rarrinfin
(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(76)
Hence (70) holds
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
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Stochastic AnalysisInternational Journal of
10 Journal of Function Spaces
(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that
10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1
(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(77)
Now we use Lemmas 4 5 and (63) to conclude that
1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)
sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862
sdot lim sup119899rarrinfin
10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862
sdotmaxlim sup119899rarrinfin
(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573
(78)
On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given
ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1
minus (int1199110log3 2
1 minus 120593 (119911119899)120596119889120596)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus2
(79)
where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and
ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899
1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat
119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup
119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573
(81)
Now we take another function
119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)
From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin
sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
(83)
Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573
ge lim sup|120593(119911119899)|rarr1
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
= lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)
Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar
to that of (ii) From the proof of (i) we get that for someconstant 119862
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1
10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)
Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in
Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
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Journal of Function Spaces 11
Z1205720 converging to zero uniformly on compact subsets of 119863therefore
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin
10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup
119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816
minus lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)
By (85) we have
119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1
100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)
and the rest of the proof is similar to that of the previous andwe omit it
Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin
sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin
sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)
(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(91)
(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim
119899rarrinfinsup (119899 + 1)120572minus1
sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin
sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(92)
Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0
Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator
Theorem 34 in [28] ensures that
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1
(1 minus |119911|2)120573
sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162
(93)
From (67) and Lemmas 4 and 5 we have
1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup
119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(94)
In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1
120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)
2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 minus 119886119899(95)
where
119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)
sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (96)
and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have
119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Function Spaces
For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that
11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162
ge lim sup119899rarrinfin
100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin
(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573
(98)
This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we
do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in
[29] we get that
100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin
(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573
(99)
which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8
and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
ge lim sup119899rarrinfin
(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)
On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function
119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)
2)
sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)
minus1 (101)
From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816
sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)
ApplyingTheorem 8 we get
119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816
minus lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim
119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573
sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573
(103)
Hence
lim119899rarrinfin
(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2
1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)
On the other hand the lower bound can be easily proved byLemmas 4 and 5
If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12
Using the same methods of Theorems 11 and 12 we canhave the following results
Theorem 13 Let 119862120593119868119892 be a bounded operator fromZ120572 toZ120573space
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin
(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 13
(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(106)
(iii) If 120572 gt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572
sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573
(107)
Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator
(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)
(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup
119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)
(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup
119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)
(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)
(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup
119899rarrinfin(119899 + 1)120572minus1
sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin
(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573
(112)
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)
References
[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002
[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008
[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008
[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008
[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007
[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006
[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008
[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013
[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008
[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008
[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008
[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012
[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010
[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009
[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009
[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Journal of Function Spaces
[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013
[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016
[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015
[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977
[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010
[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013
[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009
[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012
[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000
[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012
[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003
[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008
[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of