Research Article Marcinkiewicz Integral Operators and...
10
Research Article Marcinkiewicz Integral Operators and Commutators on Herz Spaces with Variable Exponents Liwei Wang School of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, China Correspondence should be addressed to Liwei Wang; [email protected] Received 26 July 2014; Accepted 21 September 2014; Published 15 October 2014 Academic Editor: Dashan Fan Copyright © 2014 Liwei Wang. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Our aim in this paper is to give the boundedness of the Marcinkiewicz integral Ω on Herz spaces ̇ (⋅), (⋅) (R ) and (⋅), (⋅) (R ), where the two main indices are variable. Meanwhile, we consider the boundedness of the higher order commutator Ω, generated by Ω and a function in BMO(R ) on these spaces. 1. Introduction Let S −1 be the unit sphere in R ( ≥ 2) equipped with the normalized Lebesgue measure ( ). Suppose that Ω is homogeneous of degree zero on R and has mean zero on S −1 , that is, ∫ S −1 Ω ( ) ( ) = 0. (1) en the Marcinkiewicz integral Ω in higher dimension is defined by Ω () () = (∫ ∞ 0 Ω, () () 2 3 ) 1/2 , (2) where Ω, () () = ∫ |−|≤ Ω ( − ) − −1 () . (3) Denote by N the set of all positive integer numbers. Let ∈ N and ∈ BMO(R ); the higher order commutator Ω, is defined by Ω, () () = (∫ ∞ 0 Ω,, () () 2 3 ) 1/2 , (4) where Ω,, () () = ∫ |−|≤ Ω ( − ) − −1 ( () − ()) () . (5) Stein [1] defined the operator Ω and proved that if Ω∈ Lip (S −1 ) (0 < ≤ 1), then Ω is of type (,) (1 < ≤ 2) and of weak type (1, 1). Benedek et al. [2] showed that Ω is of type (, ) (1 < < ∞) with Ω∈ 1 (S −1 ). Ding et al. [3] improved the previous results to the case of Ω∈ 1 (S −1 ), where 1 (S −1 ) denotes the Hardy space on S −1 . Obviously, 1 Ω, = [, Ω ], which was defined by Torchinsky and Wang in [4]; moreover, they proved that if Ω∈ Lip (S −1 ) (0 < ≤ 1), then [, Ω ] is bounded on (R ) (1 < < ∞). Ding et al. [5] weakened the smoothness of the kernel to a rough kernel and showed that if Ω∈ (S −1 ) (1 < ≤ ∞), then [, Ω ] is of type (,) (1 < < ∞). Ding et al. [6] established the weighted weak log type estimates for Ω, when Ω∈ Lip (S −1 ) (0 < ≤ 1). Recently, Zhang [7] improved the previous result and proved that Ω, enjoys the same weighted weak log type estimates when the kernel Ω satisfies a kind of Dini’s conditions. For further details on recent developments on this field, we refer the readers to [8, 9] and references therein. Function spaces with variable exponents were intensively studied during the past 20 years, due to their applications to PDE with nonstandard growth conditions and so on; we mention [10, 11], for instance. Since the fundamental paper [12] by Kov´ aˇ cik and R´ akosn´ ık appeared in 1991, the Lebesgue spaces with variable exponent (⋅) (R ) have attracted a great attention and many interesting results have been obtained; Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 430365, 9 pages http://dx.doi.org/10.1155/2014/430365
Transcript of Research Article Marcinkiewicz Integral Operators and...
Research ArticleMarcinkiewicz Integral Operators and Commutators onHerz Spaces with Variable Exponents
Liwei Wang
School of Mathematics and Physics Anhui Polytechnic University Wuhu 241000 China
Correspondence should be addressed to Liwei Wang wangliwei8013163com
Received 26 July 2014 Accepted 21 September 2014 Published 15 October 2014
Academic Editor Dashan Fan
Copyright copy 2014 Liwei Wang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Our aim in this paper is to give the boundedness of theMarcinkiewicz integral 120583ΩonHerz spaces 120572(sdot)119902
119901(sdot)(R119899
) and119870120572(sdot)119902
119901(sdot)(R119899
) wherethe two main indices are variable Meanwhile we consider the boundedness of the higher order commutator 120583119898
Ω119887generated by 120583
Ω
and a function 119887 in BMO(R119899) on these spaces
1 Introduction
Let S119899minus1 be the unit sphere in R119899(119899 ge 2) equipped with
the normalized Lebesgue measure 119889120590(1199091015840) Suppose that Ω is
homogeneous of degree zero on R119899 and has mean zero onS119899minus1 that is
intS119899minus1
Ω(1199091015840) 119889120590 (119909
1015840) = 0 (1)
Then the Marcinkiewicz integral 120583Ωin higher dimension is
defined by
120583Ω(119891) (119909) = (int
infin
0
1003816100381610038161003816119865Ω119905(119891) (119909)
10038161003816100381610038162 119889119905
1199053)
12
(2)
where
119865Ω119905
(119891) (119909) = int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891 (119910) 119889119910 (3)
Denote by N the set of all positive integer numbers Let119898 isin N and 119887 isin BMO(R119899
) the higher order commutator 120583119898
Ω119887
is defined by
120583119898
Ω119887(119891) (119909) = (int
infin
0
10038161003816100381610038161003816119865
119898
Ω119887119905(119891) (119909)
10038161003816100381610038161003816
2 119889119905
1199053)
12
(4)
where
119865119898
Ω119887119905(119891) (119909) = int
|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(119887 (119909) minus 119887 (119910))119898119891 (119910) 119889119910
(5)
Stein [1] defined the operator 120583Ωand proved that if Ω isin
Lip120574(S119899minus1
) (0 lt 120574 le 1) then 120583Ωis of type (119901 119901) (1 lt 119901 le 2)
and of weak type (1 1) Benedek et al [2] showed that 120583Ωis
of type (119901 119901) (1 lt 119901 lt infin) withΩ isin 1198621(S119899minus1
) Ding et al [3]improved the previous results to the case of Ω isin 119867
1(S119899minus1
)where1198671
(S119899minus1) denotes the Hardy space on S119899minus1 Obviously
1205831
Ω119887= [119887 120583
Ω] which was defined by Torchinsky and Wang
in [4] moreover they proved that if Ω isin Lip120574(S119899minus1
) (0 lt
120574 le 1) then [119887 120583Ω] is bounded on 119871
119901(R119899
) (1 lt 119901 lt infin)Ding et al [5] weakened the smoothness of the kernel to arough kernel and showed that if Ω isin 119871
119902(S119899minus1
) (1 lt 119902 le infin)then [119887 120583
Ω] is of type (119901 119901) (1 lt 119901 lt infin) Ding et al [6]
established the weighted weak 119871log119871 type estimates for 120583119898
Ω119887
when Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) Recently Zhang [7]improved the previous result and proved that 120583119898
Ω119887enjoys the
same weighted weak 119871log119871 type estimates when the kernelΩ satisfies a kind of Dinirsquos conditions For further details onrecent developments on this field we refer the readers to [8 9]and references therein
Function spaces with variable exponents were intensivelystudied during the past 20 years due to their applicationsto PDE with nonstandard growth conditions and so on wemention [10 11] for instance Since the fundamental paper[12] by Kovacik and Rakosnık appeared in 1991 the Lebesguespaces with variable exponent 119871119901(sdot)
(R119899) have attracted a great
attention and many interesting results have been obtained
Hindawi Publishing CorporationJournal of Function SpacesVolume 2014 Article ID 430365 9 pageshttpdxdoiorg1011552014430365
2 Journal of Function Spaces
see [13ndash15] Izuki [16 17] defined the Herz spaces 120572119902
119901(sdot)(R119899
)
and 119870120572119902
119901(sdot)(R119899
) with variable exponent 119901 but fixed 120572 isin R and119902 isin (0infin] Wang et al [18 19] obtained the boundednessof 120583
Ωand [119887 120583
Ω] on
120572119902
119901(sdot)(R119899
) and 119870120572119902
119901(sdot)(R119899
) Almeida andDrihem [20] established the boundedness of a wide class ofsublinear operators which includes maximal potential andCalderon-Zygmund operators on Herz spaces
120572(sdot)119902
119901(sdot)(R119899
)
and 119870120572(sdot)119902
119901(sdot)(R119899
) where the two main exponents 120572 and 119901 areboth variable In this paper we will give boundedness resultsfor 120583
Ωand 120583119898
Ω119887on Herz spaces 120572(sdot)119902
119901(sdot)(R119899
) and119870120572(sdot)119902
119901(sdot)(R119899
)For brevity |119864| denotes the Lebesgue measure for a
measurable set 119864 sub R119899 119891119864denotes the integral average of
119891 on 119864 that is 119891119864= |119864|
minus1int
119864119891(119909)119889119909 1199011015840
(sdot) stands for theconjugate exponent 1119901(sdot) + 1119901
1015840(sdot) = 1 119861(119909 119903) = 119910 isin
R119899 |119909 minus 119910| lt 119903 119862 denotes a positive constant which may
have different values even in the same line 119891 ≲ 119892means that119891 le 119862119892 and 119891 asymp 119892means that 119891 ≲ 119892 ≲ 119891
2 Preliminaries and Main Results
Let 119864 sub R119899 with |119864| gt 0 and let 119901(sdot) 119864 rarr [1infin) be ameasurable function Let us first recall some definitions andnotations
Definition 1 The Lebesgue space with variable exponent119871
119901(sdot)(119864) is defined by
119871119901(sdot)
(119864)
= 119891 is measurable int119864
(
1003816100381610038161003816119891 (119909)1003816100381610038161003816
120582)
119901(119909)
119889119909 lt infin
for some constant 120582 gt 0
(6)
This is a Banach space with the Luxemburg norm
10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(119864)= inf 120582 gt 0 int
119864
(
1003816100381610038161003816119891 (119909)1003816100381610038161003816
120582)
119901(119909)
119889119909 le 1 (7)
Let 119891 isin 1198711
loc(119864) the Hardy-Littlewood maximal operator119872 is defined by
119872119891(119909) = sup119903gt0
119903minus119899int
119861(119909119903)cap119864
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (8)
Denote
119901minus= ess inf 119901 (119909) 119909 isin 119864
119901+= ess sup 119901 (119909) 119909 isin 119864
P (119864) = 119901 (sdot) 119901minusgt 1 119901
+lt infin
B (119864) = 119901 (sdot) isin P (119864) 119872 is bounded on 119871119901(sdot)
(119864)
(9)
Let 119861119896= 119909 isin R119899
|119909| le 2119896 119877
119896= 119861
119896119861
119896minus1 and 120594
119896= 120594
119877119896
be the characteristic function of the set 119877119896for 119896 isin Z For119898 isin
N one denotes 120594119898= 120594
119877119898
if 119898 ge 1 and 1205940= 120594
1198610
By ℓ119902(0 lt
119902 le infin) we denote the discrete Lebesgue space equippedby the usual quasinorm
Definition 2 Let 0 lt 119902 le infin 119901(sdot) isin P(R119899) and 120572(sdot) R119899
rarr
R with 120572 isin 119871infin(R119899
)
(1) The homogeneous Herz space 120572(sdot)119902
119901(sdot)(R119899
) is definedby
120572(sdot)119902
119901(sdot)(R
119899) = 119891 isin 119871
119901(sdot)
loc (R119899 0)
10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
lt infin
(10)
where10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
=100381710038171003817100381710038171003817100381710038171003817100381710038172
120572(sdot)119896119891120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
infin
119896=minusinfin
100381710038171003817100381710038171003817ℓ119902(Z)
(11)
(2) The inhomogeneous Herz space119870120572(sdot)119902
119901(sdot)(R119899
) is definedby
119870120572(sdot)119902
119901(sdot)(R
119899) = 119891 isin 119871
119901(sdot)
loc (R119899)
10038171003817100381710038171198911003817100381710038171003817119870120572(sdot)119902
119901(sdot)(R119899)
lt infin (12)
where10038171003817100381710038171198911003817100381710038171003817119870120572(sdot)119902
119901(sdot)(R119899)
=100381710038171003817100381710038171003817100381710038171003817100381710038172
120572(sdot)119898119891120594
119898
10038171003817100381710038171003817119871119901(sdot)
(R119899)
infin
119898=0
100381710038171003817100381710038171003817ℓ119902(N)
(13)
with the usual modification when 119902 = infin
Remark 3 It is obvious that if 0 lt 1199021le 119902
2le infin then
120572(sdot)1199021
119901(sdot)(R119899
) sub 120572(sdot)1199022
119901(sdot)(R119899
) and 119870120572(sdot)1199021
119901(sdot)(R119899
) sub 119870120572(sdot)1199022
119901(sdot)(R119899
) Ifboth 120572(sdot) and 119901(sdot) are constants then 120572(sdot)119902
119901(sdot)(R119899
) = 120572119902
119901 (R119899)
and 119870120572(sdot)119902
119901(sdot)(R119899
) = 119870120572119902
119901 (R119899) are classical Herz spaces see
[21 22]
Definition 4 A function 120572(sdot) R119899rarr R is called log-Holder
continuous at the origin if there exists a constant 119862log gt 0
such that
|120572 (119909) minus 120572 (0)| le119862log
log (119890 + 1 |119909|) (14)
for all 119909 isin R119899 If for some 120572infinisin R and 119862log gt 0 there holds
1003816100381610038161003816120572 (119909) minus 120572infin
1003816100381610038161003816 le119862log
log (119890 + |119909|)(15)
for all 119909 isin R119899 then 120572(sdot) is called log-Holder continuous atinfinity
Let one denote
1003817100381710038171003817ℎ1198961003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)= (sum
119896⩾0
1003817100381710038171003817ℎ119896
1003817100381710038171003817119902
119871119901(sdot))
1119902
1003817100381710038171003817ℎ1198961003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)= (sum
119896lt0
1003817100381710038171003817ℎ119896
1003817100381710038171003817119902
119871119901(sdot))
1119902
(16)
Journal of Function Spaces 3
for sequences ℎ119896119896isinZ ofmeasurable functions (with the usual
modification when 119902 = infin)
Proposition 5 (see [20]) Let 0 lt 119902 le infin 119901(sdot) isin P(R119899)
and 120572(sdot) isin 119871infin(R119899
) If 120572(sdot) is log-Holder continuous both at theorigin and at infinity then
10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
(17)
Before stating themain results of this paper we introducesome key lemmas that will be used later
Lemma 6 (generalized Holderrsquos inequality [12]) Let 119901(sdot) isin
P(R119899) if 119891 isin 119871
119901(sdot)(R119899
) and 119892 isin 1198711199011015840
(sdot)(R119899
) then
intR119899
1003816100381610038161003816119891 (119909) 119892 (119909)1003816100381610038161003816 119889119909 le 119903
119901
10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(R119899)
100381710038171003817100381711989210038171003817100381710038171198711199011015840(sdot)
(R119899) (18)
where 119903119901= 1 + 1119901
minusminus 1119901
+
We remark that the following Lemmas 7ndash9 were shown inIzuki [17 23] and Lemma 10 was considered by Wang et alin [18]
Lemma 7 Let 119901(sdot) isin B(R119899) then one has for all balls 119861 in
R119899
1
|119861|
1003817100381710038171003817120594119861
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119861
10038171003817100381710038171198711199011015840(sdot)
(R119899)≲ 1 (19)
Lemma 8 Let 119901(sdot) isin B(R119899) then one has for all balls 119861 in
R119899 and all measurable subsets 119878 sub 1198611003817100381710038171003817120594119878
1003817100381710038171003817119871119901(sdot)
(R119899)1003817100381710038171003817120594119861
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ (|119878|
|119861|)
1205751
1003817100381710038171003817120594119878
10038171003817100381710038171198711199011015840(sdot)
(R119899)1003817100381710038171003817120594119861
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (|119878|
|119861|)
1205752
(20)
where 1205751and 120575
2are constants with 0 lt 120575
1 120575
2lt 1
Lemma 9 Let 119898 isin N 119887 isin BMO(R119899) and 119896 gt 119894 (119896 119894 isin N)
then one has
sup119861subR119899
11003817100381710038171003817120594119861
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887 minus 119887
119861)119898120594
119861
10038171003817100381710038171003817119871119901(sdot)
(R119899)asymp 119887
119898
BMO
100381710038171003817100381710038171003817(119887 minus 119887
119861119894
)119898
120594119861119896
100381710038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119896 minus 119894)
119898119887
119898
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
(21)
Lemma 10 Let Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) 119887 isin BMO(R119899)
and 119901(sdot) isin B(R119899) then one has1003817100381710038171003817120583Ω
(119891)1003817100381710038171003817119871119901(sdot)
(R119899)≲10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ 119887
119898
BMO10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(R119899)
(22)
Our results in this paper can be stated as follows
Theorem 11 Let Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) 0 lt 119902 le infinand 119901(sdot) isin B(R119899
) And let 120572(sdot) isin 119871infin(R119899
) be log-Holder
continuous both at the origin and at infinity such that minus1198991205751lt
120572(0) le 120572infin
lt 1198991205752 where 0 lt 120575
1 120575
2lt 1 are the constants
appearing in Lemma 8 then the operator 120583Ωis bounded on
120572(sdot)119902
119901(sdot)(R119899
) and 119870120572(sdot)119902
119901(sdot)(R119899
)
Theorem 12 Let Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) 119887 isin
BMO(R119899) 0 lt 119902 le infin and 119901(sdot) isin B(R119899
) And let 120572(sdot) isin119871
infin(R119899
) be log-Holder continuous both at the origin and atinfinity such that minus119899120575
1lt 120572(0) le 120572
infinlt 119899120575
2 where 0 lt 120575
1
1205752lt 1 are the constants appearing in Lemma 8 then the
higher order commutator 120583119898
Ω119887is bounded on
120572(sdot)119902
119901(sdot)(R119899
) and119870
120572(sdot)119902
119901(sdot)(R119899
)
Remark 13 If 120572(sdot) equiv 120572 is constant then the statementscorresponding toTheorems 11 and 12 can be found in [19 24]We consider only 0 lt 119902 lt infin in Section 3 The arguments aresimilar in the case 119902 = infin
3 Proofs of the Theorems
In this section we prove the boundedness of 120583Ωand 120583119898
Ω119887on
120572(sdot)119902
119901(sdot)(R119899
) (the same arguments can be used in 119870120572(sdot)119902
119901(sdot)(R119899
))some of our decomposition techniques are similar to thoseused by Dong and Xu in [25]
Proof of Theorem 11 In view of Proposition 5 we have
1003817100381710038171003817120583Ω(119891)
1003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896120583
Ω(119891) 120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896120583
Ω(119891) 120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
= 119868lt+ 119868
gt
(23)
Let 119891 isin 120572(sdot)119902
119901(sdot)(R119899
) write
119891 (119909) =
infin
sum
119894=minusinfin
119891 (119909) 120594119894 (119909) =
infin
sum
119894=minusinfin
119891119894 (119909) (24)
Minkowskirsquos inequality implies that
119868lt=
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817120583Ω
(119891) 120594119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896minus2
sum
119894=minusinfin
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896+1
sum
119894=119896minus1
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
infin
sum
119894=119896+2
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119864lt+ 119865
lt+ 119866
lt
(25)
4 Journal of Function Spaces
Similarly we obtain
119868gt=
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817120583Ω(119891) 120594
119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119864gt+ 119865
gt+ 119866
gt
(26)
Thus we get
1003817100381710038171003817120583Ω(119891)
1003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119864 + 119865 + 119866 (27)
where 119864 = 119864lt+ 119864
gt 119865 = 119865
lt+ 119865
gt and 119866 = 119866
lt+ 119866
gt
For 119865 Lemma 10 yields
119865 = 119865lt+ 119865
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(28)
Nowwe turn to estimate119864 Observe that if 119909 isin 119877119896 119910 isin 119877
119894
and 119894 le 119896 minus 2 then |119909 minus 119910| asymp |119909| asymp 2119896 and
1003816100381610038161003816100381610038161003816100381610038161003816
1
1003816100381610038161003816119909 minus 11991010038161003816100381610038162minus
1
|119909|2
1003816100381610038161003816100381610038161003816100381610038161003816
≲
10038161003816100381610038161199101003816100381610038161003816
1003816100381610038161003816119909 minus 11991010038161003816100381610038163 (29)
Since Ω isin Lip120574(S119899minus1
) sub 119871infin(S119899minus1
) by Minkowskirsquosinequality and Lemma 6 we have
1003816100381610038161003816120583Ω(119891
119894) (119909)
1003816100381610038161003816
≲ (int
|119909|
0
100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
|119909 minus 119910|119899minus1119891
119894(119910)119889119910
100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
|119909 minus 119910|119899minus1119891
119894(119910)119889119910
100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910 + int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(30)
Lemmas 7 and 8 lead to
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(31)
Thus we get
119864lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
119896minus2
sum
119894=minusinfin
2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))
)
119902
1119902
(32)
If 1 lt 119902 lt infin since 1198991205752minus 120572(0) gt 0 Holderrsquos inequality
implies that
119864lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))1199022
)
times(
119896minus2
sum
119894=minusinfin
2(119894minus119896)(119899120575
2minus120572(0))119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572(0))1199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(33)
Journal of Function Spaces 5
If 0 lt 119902 le 1 by the well-known inequality
(
infin
sum
119894=1
119886119894)
119902
le
infin
sum
119894=1
119886119902
119894(119886
119894gt 0 119894 = 1 2 ) (34)
we obtain
119864lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572(0))119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(35)
Similarly we have
119864gt≲
infin
sum
119896=0
2119896120572infin
119902(
119896minus2
sum
119894=minusinfin
2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
))
119902
1119902
(36)
If 1 lt 119902 lt infin since 120572infin+119899120575
2gt 2120572
infingt 2120572(0) then we get
119864gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
times (
119896minus2
sum
119894=minusinfin
2(119894minus119896)(119899120575
2minus120572infin
)1199021015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)1199022
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)1199022
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)119894119902
2(120572infin
+1198991205752minus2120572(0))11989411990221003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(37)
If 0 lt 119902 le 1 since 120572(0) le 120572infin we obtain
119864gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(38)
Thus we arrive at
119864 = 119864lt+ 119864
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(39)
For 119866 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896 + 2 then
|119909 minus 119910| asymp |119910| asymp 2119894 and
1003816100381610038161003816100381610038161003816100381610038161003816
1
1003816100381610038161003816119909 minus 11991010038161003816100381610038162minus
1
100381610038161003816100381611991010038161003816100381610038162
1003816100381610038161003816100381610038161003816100381610038161003816
≲|119909|
1003816100381610038161003816119909 minus 11991010038161003816100381610038163 (40)
From Minkowskirsquos inequality and Lemma 6 it followsthat1003816100381610038161003816120583Ω
(119891119894) (119909)
1003816100381610038161003816
≲ (int
|119910|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119910|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119910|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int
infin
|119910|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot|119909|
12
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910 + int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot110038161003816100381610038161199101003816100381610038161003816
119889119910
6 Journal of Function Spaces
≲ 2(119896minus119894)2
2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 1198622minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(41)
By Lemmas 7 and 8 we have
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)10038171003817100381710038171003817120594
119861119894
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(42)
Thus we get
119866lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119866gt≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(43)
Using the same arguments as that for 119864ltand 119864
gt we get
119866 = 119866lt+ 119866
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(44)
Hence the proof of Theorem 11 is completed
Proof of Theorem 12 We apply Proposition 5 again and get
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
= 119869lt+ 119869
gt
(45)
Let 119891 isin 120572(sdot)119902
119901(sdot)(R119899
) and write
119891 (119909) =
infin
sum
119894=minusinfin
119891 (119909) 120594119894 (119909) =
infin
sum
119894=minusinfin
119891119894 (119909) (46)
By Minkowskirsquos inequality we have
119869lt=
minus1
sum
119896=minusinfin
2120572(0)11989611990210038171003817100381710038171003817
120583119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880lt+ 119881
lt+119882
lt
(47)
By the same way we obtain
119869gt=
infin
sum
119896=0
2120572infin
11989611990210038171003817100381710038171003817120583
119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880gt+ 119881
gt+119882
gt
(48)
Thus we have
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119880 + 119881 +119882 (49)
where 119880 = 119880lt+ 119880
gt 119881 = 119881
lt+ 119881
gt and119882 = 119882
lt+119882
gt
For 119881 by Lemma 10 we have
119881 = 119881lt+ 119881
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(50)
Journal of Function Spaces 7
For 119880 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 le 119896 minus 2 then
10038161003816100381610038161003816120583
119898
Ω119887(119891
119894) (119909)
10038161003816100381610038161003816
≲ (int
|119909|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
+ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus119895
int119877119894
10038161003816100381610038161003816119887119861119894
minus 119887 (119910)10038161003816100381610038161003816
119895 1003816100381610038161003816119891119894(119910)
1003816100381610038161003816 119889119910
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus11989510038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(51)
An application of Lemmas 7 8 and 10 gives
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times
119898
sum
119895=0
119862119895
119898
10038171003817100381710038171003817(119887(119909) minus 119887
119861119894
)119898minus119895
120594119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(119877119899)
times
119898
sum
119895=0
119862119895
119898(119896 minus 119894)
119898minus119895119887
119898minus119895
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)119887
119895
BMO10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198981003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
(119894minus119896)11989912057521003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(52)For convenience below we put 120590 = 119899120575
2minus 120572(0) if 1 lt 119902 lt
infin then we use Holderrsquos inequality and obtain
119880lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120590)
119902
1119902
≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205901199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120590119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)1205901199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(53)If 0 lt 119902 le 1 then we get
119880lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120590119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120590119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(54)Similarly we put 120578 = 119899120575
2minus 120572
infin if 1 lt 119902 lt infin by Holderrsquos
inequality we obtain
119880gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120578)
119902
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120578119902
1015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
1119902
(55)
8 Journal of Function Spaces
By the same arguments as 119864gt we get
119880gt≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(56)
If 0 lt 119902 le 1 we obtain
119880gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120578119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(57)
Thus we have119880 = 119880
lt+ 119880
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(58)
For119882 if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896+2 as in the arguments
for 119866 and 119880 we obtain10038171003817100381710038171003817120583
119898
Ω119887(119891
119895)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119894 minus 119896 + 1)
1198982
(119896minus119894)1198991205751
10038171003817100381710038171003817119891
119895
10038171003817100381710038171003817119871119901(sdot)
(R119899) (59)
Thus we get
119882lt
≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
(119894 minus 119896 + 1)1198982
(119896minus119894)11989912057511003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times (119894 minus 119896 + 1)1198982
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119882gt
≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119894 minus 119896 + 1)
1198982
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(60)
Similar to the estimates of 119880ltand 119880
gt we get
119882 = 119882lt+119882
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(61)
Hence the proof of Theorem 12 is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the referees for their time andvaluable comments This work was supported by the NSF ofChina (Grant no 11201003) and University NSR Project ofAnhui Province (Grant no KJ2014A087)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 pp 430ndash466 1958
[2] A Benedek A-P Calderon and R Panzone ldquoConvolutionoperators on Banach space valued functionsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 48 pp 356ndash365 1962
[3] Y Ding D Fan and Y Pan ldquoWeighted boundedness for aclass of rough Marcinkiewicz integralsrdquo Indiana UniversityMathematics Journal vol 48 no 3 pp 1037ndash1055 1999
[4] A Torchinsky and S L Wang ldquoA note on the Marcinkiewiczintegralrdquo Colloquium Mathematicum vol 60-61 no 1 pp 235ndash243 1990
[5] Y Ding S Lu and K Yabuta ldquoOn commutators of Marcink-iewicz integrals with rough kernelrdquo Journal of MathematicalAnalysis and Applications vol 275 no 1 pp 60ndash68 2002
[6] Y Ding S Lu and P Zhang ldquoWeighted weak type estimates forcommutators of the Marcinkiewicz integralsrdquo Science in ChinaA vol 47 no 1 pp 83ndash95 2004
[7] P Zhang ldquoWeighted endpoint estimates for commutators ofMarcinkiewicz integralsrdquo Acta Mathematica Sinica vol 26 no9 pp 1709ndash1722 2010
[8] S Lu ldquoMarcinkiewicz integral with rough kernelsrdquo Frontiers ofMathematics in China vol 3 no 1 pp 1ndash14 2008
[9] Y P Chen and Y Ding ldquo119871119901 boundedness of the commutatorsof Marcinkiewicz integrals with rough kernelsrdquo Forum Mathe-maticum 2013
[10] Y M Chen S Levine and M Rao ldquoVariable exponent lineargrowth functionals in image restorationrdquo SIAM Journal onApplied Mathematics vol 66 no 4 pp 1383ndash1406 2006
[11] P Harjulehto P Hasto U V Le and M Nuortio ldquoOverviewof differential equations with non-standard growthrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 12 pp4551ndash4574 2010
[12] O Kovacik and J Rakosnık ldquoOn spaces 119871119901(119909) and 119882119896119901(119909)rdquo
Czechoslovak Mathematical Journal vol 41 no 4 pp 592ndash6181991
[13] D Cruz-Uribe A Fiorenza J M Martell and C Perez ldquoTheboundedness of classical operators on variable 119871
119901 spacesrdquoAnnales Academiae Scientiarum Fennicae Mathematica vol 31no 1 pp 239ndash264 2006
[14] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011
[15] D V Cruz-Uribe and A Fiorenza Variable Lebesgue SpacesFoundations and Harmonic Analysis Applied and NumericalHarmonic Analysis Birkhauser Basel Switzerland 2013
Journal of Function Spaces 9
[16] M Izuki ldquoHerz and amalgam spaces with variable exponentthe Haar wavelets and greediness of the wavelet systemrdquo EastJournal on Approximations vol 15 no 1 pp 87ndash109 2009
[17] M Izuki ldquoCommutators of fractional integrals on Lebesgueand Herz spaces with variable exponentrdquo Rendiconti del CircoloMatematico di Palermo Second Series vol 59 no 3 pp 461ndash4722010
[18] H B Wang Z W Fu and Z G Liu ldquoHigher-order commu-tators of Marcinkiewicz integrals on variable Lebesgue spacesrdquoActa Mathematica Scientia A vol 32 no 6 pp 1092ndash1101 2012
[19] Z G Liu and H B Wang ldquoBoundedness of Marcinkiewiczintegrals on Herz spaces with variable exponentrdquoThe JordanianJournal of Mathematics and Statistics vol 5 no 4 pp 223ndash2392012
[20] A Almeida and D Drihem ldquoMaximal potential and singulartype operators on Herz spaces with variable exponentsrdquo Journalof Mathematical Analysis and Applications vol 394 no 2 pp781ndash795 2012
[21] S Z Lu D C Yang and G E Hu Herz Type Spaces and TheirApplications Science Press Beijing China 2008
[22] X W Li and D C Yang ldquoBoundedness of some sublinearoperators on Herz spacesrdquo Illinois Journal of Mathematics vol40 no 3 pp 484ndash501 1996
[23] M Izuki ldquoVector-valued inequalities onHERz spaces and char-acterizations of HERz-Sobolev spaces with variable exponentrdquoGlasnik Matematicki vol 45 no 65 pp 475ndash503 2010
[24] L Wang and L Shu ldquoHigher order commutators of Marcink-iewicz integral operator on Herz-Morrey spaces with variableexponentrdquo Journal of Mathematical Research with Applicationsvol 34 no 2 pp 175ndash186 2014
[25] B Dong and J Xu ldquoNew Herz type Besov and Triebel-Lizorkinspaces with variable exponentsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 384593 27 pages 2012
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces
see [13ndash15] Izuki [16 17] defined the Herz spaces 120572119902
119901(sdot)(R119899
)
and 119870120572119902
119901(sdot)(R119899
) with variable exponent 119901 but fixed 120572 isin R and119902 isin (0infin] Wang et al [18 19] obtained the boundednessof 120583
Ωand [119887 120583
Ω] on
120572119902
119901(sdot)(R119899
) and 119870120572119902
119901(sdot)(R119899
) Almeida andDrihem [20] established the boundedness of a wide class ofsublinear operators which includes maximal potential andCalderon-Zygmund operators on Herz spaces
120572(sdot)119902
119901(sdot)(R119899
)
and 119870120572(sdot)119902
119901(sdot)(R119899
) where the two main exponents 120572 and 119901 areboth variable In this paper we will give boundedness resultsfor 120583
Ωand 120583119898
Ω119887on Herz spaces 120572(sdot)119902
119901(sdot)(R119899
) and119870120572(sdot)119902
119901(sdot)(R119899
)For brevity |119864| denotes the Lebesgue measure for a
measurable set 119864 sub R119899 119891119864denotes the integral average of
119891 on 119864 that is 119891119864= |119864|
minus1int
119864119891(119909)119889119909 1199011015840
(sdot) stands for theconjugate exponent 1119901(sdot) + 1119901
1015840(sdot) = 1 119861(119909 119903) = 119910 isin
R119899 |119909 minus 119910| lt 119903 119862 denotes a positive constant which may
have different values even in the same line 119891 ≲ 119892means that119891 le 119862119892 and 119891 asymp 119892means that 119891 ≲ 119892 ≲ 119891
2 Preliminaries and Main Results
Let 119864 sub R119899 with |119864| gt 0 and let 119901(sdot) 119864 rarr [1infin) be ameasurable function Let us first recall some definitions andnotations
Definition 1 The Lebesgue space with variable exponent119871
119901(sdot)(119864) is defined by
119871119901(sdot)
(119864)
= 119891 is measurable int119864
(
1003816100381610038161003816119891 (119909)1003816100381610038161003816
120582)
119901(119909)
119889119909 lt infin
for some constant 120582 gt 0
(6)
This is a Banach space with the Luxemburg norm
10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(119864)= inf 120582 gt 0 int
119864
(
1003816100381610038161003816119891 (119909)1003816100381610038161003816
120582)
119901(119909)
119889119909 le 1 (7)
Let 119891 isin 1198711
loc(119864) the Hardy-Littlewood maximal operator119872 is defined by
119872119891(119909) = sup119903gt0
119903minus119899int
119861(119909119903)cap119864
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (8)
Denote
119901minus= ess inf 119901 (119909) 119909 isin 119864
119901+= ess sup 119901 (119909) 119909 isin 119864
P (119864) = 119901 (sdot) 119901minusgt 1 119901
+lt infin
B (119864) = 119901 (sdot) isin P (119864) 119872 is bounded on 119871119901(sdot)
(119864)
(9)
Let 119861119896= 119909 isin R119899
|119909| le 2119896 119877
119896= 119861
119896119861
119896minus1 and 120594
119896= 120594
119877119896
be the characteristic function of the set 119877119896for 119896 isin Z For119898 isin
N one denotes 120594119898= 120594
119877119898
if 119898 ge 1 and 1205940= 120594
1198610
By ℓ119902(0 lt
119902 le infin) we denote the discrete Lebesgue space equippedby the usual quasinorm
Definition 2 Let 0 lt 119902 le infin 119901(sdot) isin P(R119899) and 120572(sdot) R119899
rarr
R with 120572 isin 119871infin(R119899
)
(1) The homogeneous Herz space 120572(sdot)119902
119901(sdot)(R119899
) is definedby
120572(sdot)119902
119901(sdot)(R
119899) = 119891 isin 119871
119901(sdot)
loc (R119899 0)
10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
lt infin
(10)
where10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
=100381710038171003817100381710038171003817100381710038171003817100381710038172
120572(sdot)119896119891120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
infin
119896=minusinfin
100381710038171003817100381710038171003817ℓ119902(Z)
(11)
(2) The inhomogeneous Herz space119870120572(sdot)119902
119901(sdot)(R119899
) is definedby
119870120572(sdot)119902
119901(sdot)(R
119899) = 119891 isin 119871
119901(sdot)
loc (R119899)
10038171003817100381710038171198911003817100381710038171003817119870120572(sdot)119902
119901(sdot)(R119899)
lt infin (12)
where10038171003817100381710038171198911003817100381710038171003817119870120572(sdot)119902
119901(sdot)(R119899)
=100381710038171003817100381710038171003817100381710038171003817100381710038172
120572(sdot)119898119891120594
119898
10038171003817100381710038171003817119871119901(sdot)
(R119899)
infin
119898=0
100381710038171003817100381710038171003817ℓ119902(N)
(13)
with the usual modification when 119902 = infin
Remark 3 It is obvious that if 0 lt 1199021le 119902
2le infin then
120572(sdot)1199021
119901(sdot)(R119899
) sub 120572(sdot)1199022
119901(sdot)(R119899
) and 119870120572(sdot)1199021
119901(sdot)(R119899
) sub 119870120572(sdot)1199022
119901(sdot)(R119899
) Ifboth 120572(sdot) and 119901(sdot) are constants then 120572(sdot)119902
119901(sdot)(R119899
) = 120572119902
119901 (R119899)
and 119870120572(sdot)119902
119901(sdot)(R119899
) = 119870120572119902
119901 (R119899) are classical Herz spaces see
[21 22]
Definition 4 A function 120572(sdot) R119899rarr R is called log-Holder
continuous at the origin if there exists a constant 119862log gt 0
such that
|120572 (119909) minus 120572 (0)| le119862log
log (119890 + 1 |119909|) (14)
for all 119909 isin R119899 If for some 120572infinisin R and 119862log gt 0 there holds
1003816100381610038161003816120572 (119909) minus 120572infin
1003816100381610038161003816 le119862log
log (119890 + |119909|)(15)
for all 119909 isin R119899 then 120572(sdot) is called log-Holder continuous atinfinity
Let one denote
1003817100381710038171003817ℎ1198961003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)= (sum
119896⩾0
1003817100381710038171003817ℎ119896
1003817100381710038171003817119902
119871119901(sdot))
1119902
1003817100381710038171003817ℎ1198961003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)= (sum
119896lt0
1003817100381710038171003817ℎ119896
1003817100381710038171003817119902
119871119901(sdot))
1119902
(16)
Journal of Function Spaces 3
for sequences ℎ119896119896isinZ ofmeasurable functions (with the usual
modification when 119902 = infin)
Proposition 5 (see [20]) Let 0 lt 119902 le infin 119901(sdot) isin P(R119899)
and 120572(sdot) isin 119871infin(R119899
) If 120572(sdot) is log-Holder continuous both at theorigin and at infinity then
10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
(17)
Before stating themain results of this paper we introducesome key lemmas that will be used later
Lemma 6 (generalized Holderrsquos inequality [12]) Let 119901(sdot) isin
P(R119899) if 119891 isin 119871
119901(sdot)(R119899
) and 119892 isin 1198711199011015840
(sdot)(R119899
) then
intR119899
1003816100381610038161003816119891 (119909) 119892 (119909)1003816100381610038161003816 119889119909 le 119903
119901
10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(R119899)
100381710038171003817100381711989210038171003817100381710038171198711199011015840(sdot)
(R119899) (18)
where 119903119901= 1 + 1119901
minusminus 1119901
+
We remark that the following Lemmas 7ndash9 were shown inIzuki [17 23] and Lemma 10 was considered by Wang et alin [18]
Lemma 7 Let 119901(sdot) isin B(R119899) then one has for all balls 119861 in
R119899
1
|119861|
1003817100381710038171003817120594119861
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119861
10038171003817100381710038171198711199011015840(sdot)
(R119899)≲ 1 (19)
Lemma 8 Let 119901(sdot) isin B(R119899) then one has for all balls 119861 in
R119899 and all measurable subsets 119878 sub 1198611003817100381710038171003817120594119878
1003817100381710038171003817119871119901(sdot)
(R119899)1003817100381710038171003817120594119861
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ (|119878|
|119861|)
1205751
1003817100381710038171003817120594119878
10038171003817100381710038171198711199011015840(sdot)
(R119899)1003817100381710038171003817120594119861
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (|119878|
|119861|)
1205752
(20)
where 1205751and 120575
2are constants with 0 lt 120575
1 120575
2lt 1
Lemma 9 Let 119898 isin N 119887 isin BMO(R119899) and 119896 gt 119894 (119896 119894 isin N)
then one has
sup119861subR119899
11003817100381710038171003817120594119861
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887 minus 119887
119861)119898120594
119861
10038171003817100381710038171003817119871119901(sdot)
(R119899)asymp 119887
119898
BMO
100381710038171003817100381710038171003817(119887 minus 119887
119861119894
)119898
120594119861119896
100381710038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119896 minus 119894)
119898119887
119898
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
(21)
Lemma 10 Let Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) 119887 isin BMO(R119899)
and 119901(sdot) isin B(R119899) then one has1003817100381710038171003817120583Ω
(119891)1003817100381710038171003817119871119901(sdot)
(R119899)≲10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ 119887
119898
BMO10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(R119899)
(22)
Our results in this paper can be stated as follows
Theorem 11 Let Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) 0 lt 119902 le infinand 119901(sdot) isin B(R119899
) And let 120572(sdot) isin 119871infin(R119899
) be log-Holder
continuous both at the origin and at infinity such that minus1198991205751lt
120572(0) le 120572infin
lt 1198991205752 where 0 lt 120575
1 120575
2lt 1 are the constants
appearing in Lemma 8 then the operator 120583Ωis bounded on
120572(sdot)119902
119901(sdot)(R119899
) and 119870120572(sdot)119902
119901(sdot)(R119899
)
Theorem 12 Let Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) 119887 isin
BMO(R119899) 0 lt 119902 le infin and 119901(sdot) isin B(R119899
) And let 120572(sdot) isin119871
infin(R119899
) be log-Holder continuous both at the origin and atinfinity such that minus119899120575
1lt 120572(0) le 120572
infinlt 119899120575
2 where 0 lt 120575
1
1205752lt 1 are the constants appearing in Lemma 8 then the
higher order commutator 120583119898
Ω119887is bounded on
120572(sdot)119902
119901(sdot)(R119899
) and119870
120572(sdot)119902
119901(sdot)(R119899
)
Remark 13 If 120572(sdot) equiv 120572 is constant then the statementscorresponding toTheorems 11 and 12 can be found in [19 24]We consider only 0 lt 119902 lt infin in Section 3 The arguments aresimilar in the case 119902 = infin
3 Proofs of the Theorems
In this section we prove the boundedness of 120583Ωand 120583119898
Ω119887on
120572(sdot)119902
119901(sdot)(R119899
) (the same arguments can be used in 119870120572(sdot)119902
119901(sdot)(R119899
))some of our decomposition techniques are similar to thoseused by Dong and Xu in [25]
Proof of Theorem 11 In view of Proposition 5 we have
1003817100381710038171003817120583Ω(119891)
1003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896120583
Ω(119891) 120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896120583
Ω(119891) 120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
= 119868lt+ 119868
gt
(23)
Let 119891 isin 120572(sdot)119902
119901(sdot)(R119899
) write
119891 (119909) =
infin
sum
119894=minusinfin
119891 (119909) 120594119894 (119909) =
infin
sum
119894=minusinfin
119891119894 (119909) (24)
Minkowskirsquos inequality implies that
119868lt=
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817120583Ω
(119891) 120594119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896minus2
sum
119894=minusinfin
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896+1
sum
119894=119896minus1
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
infin
sum
119894=119896+2
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119864lt+ 119865
lt+ 119866
lt
(25)
4 Journal of Function Spaces
Similarly we obtain
119868gt=
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817120583Ω(119891) 120594
119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119864gt+ 119865
gt+ 119866
gt
(26)
Thus we get
1003817100381710038171003817120583Ω(119891)
1003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119864 + 119865 + 119866 (27)
where 119864 = 119864lt+ 119864
gt 119865 = 119865
lt+ 119865
gt and 119866 = 119866
lt+ 119866
gt
For 119865 Lemma 10 yields
119865 = 119865lt+ 119865
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(28)
Nowwe turn to estimate119864 Observe that if 119909 isin 119877119896 119910 isin 119877
119894
and 119894 le 119896 minus 2 then |119909 minus 119910| asymp |119909| asymp 2119896 and
1003816100381610038161003816100381610038161003816100381610038161003816
1
1003816100381610038161003816119909 minus 11991010038161003816100381610038162minus
1
|119909|2
1003816100381610038161003816100381610038161003816100381610038161003816
≲
10038161003816100381610038161199101003816100381610038161003816
1003816100381610038161003816119909 minus 11991010038161003816100381610038163 (29)
Since Ω isin Lip120574(S119899minus1
) sub 119871infin(S119899minus1
) by Minkowskirsquosinequality and Lemma 6 we have
1003816100381610038161003816120583Ω(119891
119894) (119909)
1003816100381610038161003816
≲ (int
|119909|
0
100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
|119909 minus 119910|119899minus1119891
119894(119910)119889119910
100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
|119909 minus 119910|119899minus1119891
119894(119910)119889119910
100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910 + int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(30)
Lemmas 7 and 8 lead to
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(31)
Thus we get
119864lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
119896minus2
sum
119894=minusinfin
2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))
)
119902
1119902
(32)
If 1 lt 119902 lt infin since 1198991205752minus 120572(0) gt 0 Holderrsquos inequality
implies that
119864lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))1199022
)
times(
119896minus2
sum
119894=minusinfin
2(119894minus119896)(119899120575
2minus120572(0))119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572(0))1199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(33)
Journal of Function Spaces 5
If 0 lt 119902 le 1 by the well-known inequality
(
infin
sum
119894=1
119886119894)
119902
le
infin
sum
119894=1
119886119902
119894(119886
119894gt 0 119894 = 1 2 ) (34)
we obtain
119864lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572(0))119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(35)
Similarly we have
119864gt≲
infin
sum
119896=0
2119896120572infin
119902(
119896minus2
sum
119894=minusinfin
2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
))
119902
1119902
(36)
If 1 lt 119902 lt infin since 120572infin+119899120575
2gt 2120572
infingt 2120572(0) then we get
119864gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
times (
119896minus2
sum
119894=minusinfin
2(119894minus119896)(119899120575
2minus120572infin
)1199021015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)1199022
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)1199022
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)119894119902
2(120572infin
+1198991205752minus2120572(0))11989411990221003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(37)
If 0 lt 119902 le 1 since 120572(0) le 120572infin we obtain
119864gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(38)
Thus we arrive at
119864 = 119864lt+ 119864
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(39)
For 119866 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896 + 2 then
|119909 minus 119910| asymp |119910| asymp 2119894 and
1003816100381610038161003816100381610038161003816100381610038161003816
1
1003816100381610038161003816119909 minus 11991010038161003816100381610038162minus
1
100381610038161003816100381611991010038161003816100381610038162
1003816100381610038161003816100381610038161003816100381610038161003816
≲|119909|
1003816100381610038161003816119909 minus 11991010038161003816100381610038163 (40)
From Minkowskirsquos inequality and Lemma 6 it followsthat1003816100381610038161003816120583Ω
(119891119894) (119909)
1003816100381610038161003816
≲ (int
|119910|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119910|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119910|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int
infin
|119910|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot|119909|
12
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910 + int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot110038161003816100381610038161199101003816100381610038161003816
119889119910
6 Journal of Function Spaces
≲ 2(119896minus119894)2
2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 1198622minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(41)
By Lemmas 7 and 8 we have
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)10038171003817100381710038171003817120594
119861119894
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(42)
Thus we get
119866lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119866gt≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(43)
Using the same arguments as that for 119864ltand 119864
gt we get
119866 = 119866lt+ 119866
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(44)
Hence the proof of Theorem 11 is completed
Proof of Theorem 12 We apply Proposition 5 again and get
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
= 119869lt+ 119869
gt
(45)
Let 119891 isin 120572(sdot)119902
119901(sdot)(R119899
) and write
119891 (119909) =
infin
sum
119894=minusinfin
119891 (119909) 120594119894 (119909) =
infin
sum
119894=minusinfin
119891119894 (119909) (46)
By Minkowskirsquos inequality we have
119869lt=
minus1
sum
119896=minusinfin
2120572(0)11989611990210038171003817100381710038171003817
120583119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880lt+ 119881
lt+119882
lt
(47)
By the same way we obtain
119869gt=
infin
sum
119896=0
2120572infin
11989611990210038171003817100381710038171003817120583
119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880gt+ 119881
gt+119882
gt
(48)
Thus we have
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119880 + 119881 +119882 (49)
where 119880 = 119880lt+ 119880
gt 119881 = 119881
lt+ 119881
gt and119882 = 119882
lt+119882
gt
For 119881 by Lemma 10 we have
119881 = 119881lt+ 119881
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(50)
Journal of Function Spaces 7
For 119880 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 le 119896 minus 2 then
10038161003816100381610038161003816120583
119898
Ω119887(119891
119894) (119909)
10038161003816100381610038161003816
≲ (int
|119909|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
+ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus119895
int119877119894
10038161003816100381610038161003816119887119861119894
minus 119887 (119910)10038161003816100381610038161003816
119895 1003816100381610038161003816119891119894(119910)
1003816100381610038161003816 119889119910
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus11989510038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(51)
An application of Lemmas 7 8 and 10 gives
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times
119898
sum
119895=0
119862119895
119898
10038171003817100381710038171003817(119887(119909) minus 119887
119861119894
)119898minus119895
120594119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(119877119899)
times
119898
sum
119895=0
119862119895
119898(119896 minus 119894)
119898minus119895119887
119898minus119895
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)119887
119895
BMO10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198981003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
(119894minus119896)11989912057521003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(52)For convenience below we put 120590 = 119899120575
2minus 120572(0) if 1 lt 119902 lt
infin then we use Holderrsquos inequality and obtain
119880lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120590)
119902
1119902
≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205901199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120590119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)1205901199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(53)If 0 lt 119902 le 1 then we get
119880lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120590119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120590119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(54)Similarly we put 120578 = 119899120575
2minus 120572
infin if 1 lt 119902 lt infin by Holderrsquos
inequality we obtain
119880gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120578)
119902
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120578119902
1015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
1119902
(55)
8 Journal of Function Spaces
By the same arguments as 119864gt we get
119880gt≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(56)
If 0 lt 119902 le 1 we obtain
119880gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120578119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(57)
Thus we have119880 = 119880
lt+ 119880
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(58)
For119882 if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896+2 as in the arguments
for 119866 and 119880 we obtain10038171003817100381710038171003817120583
119898
Ω119887(119891
119895)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119894 minus 119896 + 1)
1198982
(119896minus119894)1198991205751
10038171003817100381710038171003817119891
119895
10038171003817100381710038171003817119871119901(sdot)
(R119899) (59)
Thus we get
119882lt
≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
(119894 minus 119896 + 1)1198982
(119896minus119894)11989912057511003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times (119894 minus 119896 + 1)1198982
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119882gt
≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119894 minus 119896 + 1)
1198982
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(60)
Similar to the estimates of 119880ltand 119880
gt we get
119882 = 119882lt+119882
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(61)
Hence the proof of Theorem 12 is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the referees for their time andvaluable comments This work was supported by the NSF ofChina (Grant no 11201003) and University NSR Project ofAnhui Province (Grant no KJ2014A087)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 pp 430ndash466 1958
[2] A Benedek A-P Calderon and R Panzone ldquoConvolutionoperators on Banach space valued functionsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 48 pp 356ndash365 1962
[3] Y Ding D Fan and Y Pan ldquoWeighted boundedness for aclass of rough Marcinkiewicz integralsrdquo Indiana UniversityMathematics Journal vol 48 no 3 pp 1037ndash1055 1999
[4] A Torchinsky and S L Wang ldquoA note on the Marcinkiewiczintegralrdquo Colloquium Mathematicum vol 60-61 no 1 pp 235ndash243 1990
[5] Y Ding S Lu and K Yabuta ldquoOn commutators of Marcink-iewicz integrals with rough kernelrdquo Journal of MathematicalAnalysis and Applications vol 275 no 1 pp 60ndash68 2002
[6] Y Ding S Lu and P Zhang ldquoWeighted weak type estimates forcommutators of the Marcinkiewicz integralsrdquo Science in ChinaA vol 47 no 1 pp 83ndash95 2004
[7] P Zhang ldquoWeighted endpoint estimates for commutators ofMarcinkiewicz integralsrdquo Acta Mathematica Sinica vol 26 no9 pp 1709ndash1722 2010
[8] S Lu ldquoMarcinkiewicz integral with rough kernelsrdquo Frontiers ofMathematics in China vol 3 no 1 pp 1ndash14 2008
[9] Y P Chen and Y Ding ldquo119871119901 boundedness of the commutatorsof Marcinkiewicz integrals with rough kernelsrdquo Forum Mathe-maticum 2013
[10] Y M Chen S Levine and M Rao ldquoVariable exponent lineargrowth functionals in image restorationrdquo SIAM Journal onApplied Mathematics vol 66 no 4 pp 1383ndash1406 2006
[11] P Harjulehto P Hasto U V Le and M Nuortio ldquoOverviewof differential equations with non-standard growthrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 12 pp4551ndash4574 2010
[12] O Kovacik and J Rakosnık ldquoOn spaces 119871119901(119909) and 119882119896119901(119909)rdquo
Czechoslovak Mathematical Journal vol 41 no 4 pp 592ndash6181991
[13] D Cruz-Uribe A Fiorenza J M Martell and C Perez ldquoTheboundedness of classical operators on variable 119871
119901 spacesrdquoAnnales Academiae Scientiarum Fennicae Mathematica vol 31no 1 pp 239ndash264 2006
[14] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011
[15] D V Cruz-Uribe and A Fiorenza Variable Lebesgue SpacesFoundations and Harmonic Analysis Applied and NumericalHarmonic Analysis Birkhauser Basel Switzerland 2013
Journal of Function Spaces 9
[16] M Izuki ldquoHerz and amalgam spaces with variable exponentthe Haar wavelets and greediness of the wavelet systemrdquo EastJournal on Approximations vol 15 no 1 pp 87ndash109 2009
[17] M Izuki ldquoCommutators of fractional integrals on Lebesgueand Herz spaces with variable exponentrdquo Rendiconti del CircoloMatematico di Palermo Second Series vol 59 no 3 pp 461ndash4722010
[18] H B Wang Z W Fu and Z G Liu ldquoHigher-order commu-tators of Marcinkiewicz integrals on variable Lebesgue spacesrdquoActa Mathematica Scientia A vol 32 no 6 pp 1092ndash1101 2012
[19] Z G Liu and H B Wang ldquoBoundedness of Marcinkiewiczintegrals on Herz spaces with variable exponentrdquoThe JordanianJournal of Mathematics and Statistics vol 5 no 4 pp 223ndash2392012
[20] A Almeida and D Drihem ldquoMaximal potential and singulartype operators on Herz spaces with variable exponentsrdquo Journalof Mathematical Analysis and Applications vol 394 no 2 pp781ndash795 2012
[21] S Z Lu D C Yang and G E Hu Herz Type Spaces and TheirApplications Science Press Beijing China 2008
[22] X W Li and D C Yang ldquoBoundedness of some sublinearoperators on Herz spacesrdquo Illinois Journal of Mathematics vol40 no 3 pp 484ndash501 1996
[23] M Izuki ldquoVector-valued inequalities onHERz spaces and char-acterizations of HERz-Sobolev spaces with variable exponentrdquoGlasnik Matematicki vol 45 no 65 pp 475ndash503 2010
[24] L Wang and L Shu ldquoHigher order commutators of Marcink-iewicz integral operator on Herz-Morrey spaces with variableexponentrdquo Journal of Mathematical Research with Applicationsvol 34 no 2 pp 175ndash186 2014
[25] B Dong and J Xu ldquoNew Herz type Besov and Triebel-Lizorkinspaces with variable exponentsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 384593 27 pages 2012
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 3
for sequences ℎ119896119896isinZ ofmeasurable functions (with the usual
modification when 119902 = infin)
Proposition 5 (see [20]) Let 0 lt 119902 le infin 119901(sdot) isin P(R119899)
and 120572(sdot) isin 119871infin(R119899
) If 120572(sdot) is log-Holder continuous both at theorigin and at infinity then
10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
(17)
Before stating themain results of this paper we introducesome key lemmas that will be used later
Lemma 6 (generalized Holderrsquos inequality [12]) Let 119901(sdot) isin
P(R119899) if 119891 isin 119871
119901(sdot)(R119899
) and 119892 isin 1198711199011015840
(sdot)(R119899
) then
intR119899
1003816100381610038161003816119891 (119909) 119892 (119909)1003816100381610038161003816 119889119909 le 119903
119901
10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(R119899)
100381710038171003817100381711989210038171003817100381710038171198711199011015840(sdot)
(R119899) (18)
where 119903119901= 1 + 1119901
minusminus 1119901
+
We remark that the following Lemmas 7ndash9 were shown inIzuki [17 23] and Lemma 10 was considered by Wang et alin [18]
Lemma 7 Let 119901(sdot) isin B(R119899) then one has for all balls 119861 in
R119899
1
|119861|
1003817100381710038171003817120594119861
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119861
10038171003817100381710038171198711199011015840(sdot)
(R119899)≲ 1 (19)
Lemma 8 Let 119901(sdot) isin B(R119899) then one has for all balls 119861 in
R119899 and all measurable subsets 119878 sub 1198611003817100381710038171003817120594119878
1003817100381710038171003817119871119901(sdot)
(R119899)1003817100381710038171003817120594119861
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ (|119878|
|119861|)
1205751
1003817100381710038171003817120594119878
10038171003817100381710038171198711199011015840(sdot)
(R119899)1003817100381710038171003817120594119861
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (|119878|
|119861|)
1205752
(20)
where 1205751and 120575
2are constants with 0 lt 120575
1 120575
2lt 1
Lemma 9 Let 119898 isin N 119887 isin BMO(R119899) and 119896 gt 119894 (119896 119894 isin N)
then one has
sup119861subR119899
11003817100381710038171003817120594119861
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887 minus 119887
119861)119898120594
119861
10038171003817100381710038171003817119871119901(sdot)
(R119899)asymp 119887
119898
BMO
100381710038171003817100381710038171003817(119887 minus 119887
119861119894
)119898
120594119861119896
100381710038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119896 minus 119894)
119898119887
119898
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
(21)
Lemma 10 Let Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) 119887 isin BMO(R119899)
and 119901(sdot) isin B(R119899) then one has1003817100381710038171003817120583Ω
(119891)1003817100381710038171003817119871119901(sdot)
(R119899)≲10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ 119887
119898
BMO10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)
(R119899)
(22)
Our results in this paper can be stated as follows
Theorem 11 Let Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) 0 lt 119902 le infinand 119901(sdot) isin B(R119899
) And let 120572(sdot) isin 119871infin(R119899
) be log-Holder
continuous both at the origin and at infinity such that minus1198991205751lt
120572(0) le 120572infin
lt 1198991205752 where 0 lt 120575
1 120575
2lt 1 are the constants
appearing in Lemma 8 then the operator 120583Ωis bounded on
120572(sdot)119902
119901(sdot)(R119899
) and 119870120572(sdot)119902
119901(sdot)(R119899
)
Theorem 12 Let Ω isin Lip120574(S119899minus1
) (0 lt 120574 le 1) 119887 isin
BMO(R119899) 0 lt 119902 le infin and 119901(sdot) isin B(R119899
) And let 120572(sdot) isin119871
infin(R119899
) be log-Holder continuous both at the origin and atinfinity such that minus119899120575
1lt 120572(0) le 120572
infinlt 119899120575
2 where 0 lt 120575
1
1205752lt 1 are the constants appearing in Lemma 8 then the
higher order commutator 120583119898
Ω119887is bounded on
120572(sdot)119902
119901(sdot)(R119899
) and119870
120572(sdot)119902
119901(sdot)(R119899
)
Remark 13 If 120572(sdot) equiv 120572 is constant then the statementscorresponding toTheorems 11 and 12 can be found in [19 24]We consider only 0 lt 119902 lt infin in Section 3 The arguments aresimilar in the case 119902 = infin
3 Proofs of the Theorems
In this section we prove the boundedness of 120583Ωand 120583119898
Ω119887on
120572(sdot)119902
119901(sdot)(R119899
) (the same arguments can be used in 119870120572(sdot)119902
119901(sdot)(R119899
))some of our decomposition techniques are similar to thoseused by Dong and Xu in [25]
Proof of Theorem 11 In view of Proposition 5 we have
1003817100381710038171003817120583Ω(119891)
1003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896120583
Ω(119891) 120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896120583
Ω(119891) 120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
= 119868lt+ 119868
gt
(23)
Let 119891 isin 120572(sdot)119902
119901(sdot)(R119899
) write
119891 (119909) =
infin
sum
119894=minusinfin
119891 (119909) 120594119894 (119909) =
infin
sum
119894=minusinfin
119891119894 (119909) (24)
Minkowskirsquos inequality implies that
119868lt=
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817120583Ω
(119891) 120594119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896minus2
sum
119894=minusinfin
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896+1
sum
119894=119896minus1
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
infin
sum
119894=119896+2
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119864lt+ 119865
lt+ 119866
lt
(25)
4 Journal of Function Spaces
Similarly we obtain
119868gt=
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817120583Ω(119891) 120594
119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119864gt+ 119865
gt+ 119866
gt
(26)
Thus we get
1003817100381710038171003817120583Ω(119891)
1003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119864 + 119865 + 119866 (27)
where 119864 = 119864lt+ 119864
gt 119865 = 119865
lt+ 119865
gt and 119866 = 119866
lt+ 119866
gt
For 119865 Lemma 10 yields
119865 = 119865lt+ 119865
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(28)
Nowwe turn to estimate119864 Observe that if 119909 isin 119877119896 119910 isin 119877
119894
and 119894 le 119896 minus 2 then |119909 minus 119910| asymp |119909| asymp 2119896 and
1003816100381610038161003816100381610038161003816100381610038161003816
1
1003816100381610038161003816119909 minus 11991010038161003816100381610038162minus
1
|119909|2
1003816100381610038161003816100381610038161003816100381610038161003816
≲
10038161003816100381610038161199101003816100381610038161003816
1003816100381610038161003816119909 minus 11991010038161003816100381610038163 (29)
Since Ω isin Lip120574(S119899minus1
) sub 119871infin(S119899minus1
) by Minkowskirsquosinequality and Lemma 6 we have
1003816100381610038161003816120583Ω(119891
119894) (119909)
1003816100381610038161003816
≲ (int
|119909|
0
100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
|119909 minus 119910|119899minus1119891
119894(119910)119889119910
100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
|119909 minus 119910|119899minus1119891
119894(119910)119889119910
100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910 + int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(30)
Lemmas 7 and 8 lead to
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(31)
Thus we get
119864lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
119896minus2
sum
119894=minusinfin
2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))
)
119902
1119902
(32)
If 1 lt 119902 lt infin since 1198991205752minus 120572(0) gt 0 Holderrsquos inequality
implies that
119864lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))1199022
)
times(
119896minus2
sum
119894=minusinfin
2(119894minus119896)(119899120575
2minus120572(0))119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572(0))1199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(33)
Journal of Function Spaces 5
If 0 lt 119902 le 1 by the well-known inequality
(
infin
sum
119894=1
119886119894)
119902
le
infin
sum
119894=1
119886119902
119894(119886
119894gt 0 119894 = 1 2 ) (34)
we obtain
119864lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572(0))119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(35)
Similarly we have
119864gt≲
infin
sum
119896=0
2119896120572infin
119902(
119896minus2
sum
119894=minusinfin
2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
))
119902
1119902
(36)
If 1 lt 119902 lt infin since 120572infin+119899120575
2gt 2120572
infingt 2120572(0) then we get
119864gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
times (
119896minus2
sum
119894=minusinfin
2(119894minus119896)(119899120575
2minus120572infin
)1199021015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)1199022
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)1199022
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)119894119902
2(120572infin
+1198991205752minus2120572(0))11989411990221003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(37)
If 0 lt 119902 le 1 since 120572(0) le 120572infin we obtain
119864gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(38)
Thus we arrive at
119864 = 119864lt+ 119864
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(39)
For 119866 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896 + 2 then
|119909 minus 119910| asymp |119910| asymp 2119894 and
1003816100381610038161003816100381610038161003816100381610038161003816
1
1003816100381610038161003816119909 minus 11991010038161003816100381610038162minus
1
100381610038161003816100381611991010038161003816100381610038162
1003816100381610038161003816100381610038161003816100381610038161003816
≲|119909|
1003816100381610038161003816119909 minus 11991010038161003816100381610038163 (40)
From Minkowskirsquos inequality and Lemma 6 it followsthat1003816100381610038161003816120583Ω
(119891119894) (119909)
1003816100381610038161003816
≲ (int
|119910|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119910|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119910|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int
infin
|119910|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot|119909|
12
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910 + int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot110038161003816100381610038161199101003816100381610038161003816
119889119910
6 Journal of Function Spaces
≲ 2(119896minus119894)2
2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 1198622minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(41)
By Lemmas 7 and 8 we have
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)10038171003817100381710038171003817120594
119861119894
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(42)
Thus we get
119866lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119866gt≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(43)
Using the same arguments as that for 119864ltand 119864
gt we get
119866 = 119866lt+ 119866
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(44)
Hence the proof of Theorem 11 is completed
Proof of Theorem 12 We apply Proposition 5 again and get
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
= 119869lt+ 119869
gt
(45)
Let 119891 isin 120572(sdot)119902
119901(sdot)(R119899
) and write
119891 (119909) =
infin
sum
119894=minusinfin
119891 (119909) 120594119894 (119909) =
infin
sum
119894=minusinfin
119891119894 (119909) (46)
By Minkowskirsquos inequality we have
119869lt=
minus1
sum
119896=minusinfin
2120572(0)11989611990210038171003817100381710038171003817
120583119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880lt+ 119881
lt+119882
lt
(47)
By the same way we obtain
119869gt=
infin
sum
119896=0
2120572infin
11989611990210038171003817100381710038171003817120583
119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880gt+ 119881
gt+119882
gt
(48)
Thus we have
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119880 + 119881 +119882 (49)
where 119880 = 119880lt+ 119880
gt 119881 = 119881
lt+ 119881
gt and119882 = 119882
lt+119882
gt
For 119881 by Lemma 10 we have
119881 = 119881lt+ 119881
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(50)
Journal of Function Spaces 7
For 119880 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 le 119896 minus 2 then
10038161003816100381610038161003816120583
119898
Ω119887(119891
119894) (119909)
10038161003816100381610038161003816
≲ (int
|119909|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
+ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus119895
int119877119894
10038161003816100381610038161003816119887119861119894
minus 119887 (119910)10038161003816100381610038161003816
119895 1003816100381610038161003816119891119894(119910)
1003816100381610038161003816 119889119910
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus11989510038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(51)
An application of Lemmas 7 8 and 10 gives
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times
119898
sum
119895=0
119862119895
119898
10038171003817100381710038171003817(119887(119909) minus 119887
119861119894
)119898minus119895
120594119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(119877119899)
times
119898
sum
119895=0
119862119895
119898(119896 minus 119894)
119898minus119895119887
119898minus119895
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)119887
119895
BMO10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198981003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
(119894minus119896)11989912057521003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(52)For convenience below we put 120590 = 119899120575
2minus 120572(0) if 1 lt 119902 lt
infin then we use Holderrsquos inequality and obtain
119880lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120590)
119902
1119902
≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205901199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120590119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)1205901199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(53)If 0 lt 119902 le 1 then we get
119880lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120590119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120590119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(54)Similarly we put 120578 = 119899120575
2minus 120572
infin if 1 lt 119902 lt infin by Holderrsquos
inequality we obtain
119880gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120578)
119902
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120578119902
1015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
1119902
(55)
8 Journal of Function Spaces
By the same arguments as 119864gt we get
119880gt≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(56)
If 0 lt 119902 le 1 we obtain
119880gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120578119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(57)
Thus we have119880 = 119880
lt+ 119880
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(58)
For119882 if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896+2 as in the arguments
for 119866 and 119880 we obtain10038171003817100381710038171003817120583
119898
Ω119887(119891
119895)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119894 minus 119896 + 1)
1198982
(119896minus119894)1198991205751
10038171003817100381710038171003817119891
119895
10038171003817100381710038171003817119871119901(sdot)
(R119899) (59)
Thus we get
119882lt
≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
(119894 minus 119896 + 1)1198982
(119896minus119894)11989912057511003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times (119894 minus 119896 + 1)1198982
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119882gt
≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119894 minus 119896 + 1)
1198982
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(60)
Similar to the estimates of 119880ltand 119880
gt we get
119882 = 119882lt+119882
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(61)
Hence the proof of Theorem 12 is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the referees for their time andvaluable comments This work was supported by the NSF ofChina (Grant no 11201003) and University NSR Project ofAnhui Province (Grant no KJ2014A087)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 pp 430ndash466 1958
[2] A Benedek A-P Calderon and R Panzone ldquoConvolutionoperators on Banach space valued functionsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 48 pp 356ndash365 1962
[3] Y Ding D Fan and Y Pan ldquoWeighted boundedness for aclass of rough Marcinkiewicz integralsrdquo Indiana UniversityMathematics Journal vol 48 no 3 pp 1037ndash1055 1999
[4] A Torchinsky and S L Wang ldquoA note on the Marcinkiewiczintegralrdquo Colloquium Mathematicum vol 60-61 no 1 pp 235ndash243 1990
[5] Y Ding S Lu and K Yabuta ldquoOn commutators of Marcink-iewicz integrals with rough kernelrdquo Journal of MathematicalAnalysis and Applications vol 275 no 1 pp 60ndash68 2002
[6] Y Ding S Lu and P Zhang ldquoWeighted weak type estimates forcommutators of the Marcinkiewicz integralsrdquo Science in ChinaA vol 47 no 1 pp 83ndash95 2004
[7] P Zhang ldquoWeighted endpoint estimates for commutators ofMarcinkiewicz integralsrdquo Acta Mathematica Sinica vol 26 no9 pp 1709ndash1722 2010
[8] S Lu ldquoMarcinkiewicz integral with rough kernelsrdquo Frontiers ofMathematics in China vol 3 no 1 pp 1ndash14 2008
[9] Y P Chen and Y Ding ldquo119871119901 boundedness of the commutatorsof Marcinkiewicz integrals with rough kernelsrdquo Forum Mathe-maticum 2013
[10] Y M Chen S Levine and M Rao ldquoVariable exponent lineargrowth functionals in image restorationrdquo SIAM Journal onApplied Mathematics vol 66 no 4 pp 1383ndash1406 2006
[11] P Harjulehto P Hasto U V Le and M Nuortio ldquoOverviewof differential equations with non-standard growthrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 12 pp4551ndash4574 2010
[12] O Kovacik and J Rakosnık ldquoOn spaces 119871119901(119909) and 119882119896119901(119909)rdquo
Czechoslovak Mathematical Journal vol 41 no 4 pp 592ndash6181991
[13] D Cruz-Uribe A Fiorenza J M Martell and C Perez ldquoTheboundedness of classical operators on variable 119871
119901 spacesrdquoAnnales Academiae Scientiarum Fennicae Mathematica vol 31no 1 pp 239ndash264 2006
[14] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011
[15] D V Cruz-Uribe and A Fiorenza Variable Lebesgue SpacesFoundations and Harmonic Analysis Applied and NumericalHarmonic Analysis Birkhauser Basel Switzerland 2013
Journal of Function Spaces 9
[16] M Izuki ldquoHerz and amalgam spaces with variable exponentthe Haar wavelets and greediness of the wavelet systemrdquo EastJournal on Approximations vol 15 no 1 pp 87ndash109 2009
[17] M Izuki ldquoCommutators of fractional integrals on Lebesgueand Herz spaces with variable exponentrdquo Rendiconti del CircoloMatematico di Palermo Second Series vol 59 no 3 pp 461ndash4722010
[18] H B Wang Z W Fu and Z G Liu ldquoHigher-order commu-tators of Marcinkiewicz integrals on variable Lebesgue spacesrdquoActa Mathematica Scientia A vol 32 no 6 pp 1092ndash1101 2012
[19] Z G Liu and H B Wang ldquoBoundedness of Marcinkiewiczintegrals on Herz spaces with variable exponentrdquoThe JordanianJournal of Mathematics and Statistics vol 5 no 4 pp 223ndash2392012
[20] A Almeida and D Drihem ldquoMaximal potential and singulartype operators on Herz spaces with variable exponentsrdquo Journalof Mathematical Analysis and Applications vol 394 no 2 pp781ndash795 2012
[21] S Z Lu D C Yang and G E Hu Herz Type Spaces and TheirApplications Science Press Beijing China 2008
[22] X W Li and D C Yang ldquoBoundedness of some sublinearoperators on Herz spacesrdquo Illinois Journal of Mathematics vol40 no 3 pp 484ndash501 1996
[23] M Izuki ldquoVector-valued inequalities onHERz spaces and char-acterizations of HERz-Sobolev spaces with variable exponentrdquoGlasnik Matematicki vol 45 no 65 pp 475ndash503 2010
[24] L Wang and L Shu ldquoHigher order commutators of Marcink-iewicz integral operator on Herz-Morrey spaces with variableexponentrdquo Journal of Mathematical Research with Applicationsvol 34 no 2 pp 175ndash186 2014
[25] B Dong and J Xu ldquoNew Herz type Besov and Triebel-Lizorkinspaces with variable exponentsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 384593 27 pages 2012
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Function Spaces
Similarly we obtain
119868gt=
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817120583Ω(119891) 120594
119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
1003817100381710038171003817120583Ω(119891
119894) 120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119864gt+ 119865
gt+ 119866
gt
(26)
Thus we get
1003817100381710038171003817120583Ω(119891)
1003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119864 + 119865 + 119866 (27)
where 119864 = 119864lt+ 119864
gt 119865 = 119865
lt+ 119865
gt and 119866 = 119866
lt+ 119866
gt
For 119865 Lemma 10 yields
119865 = 119865lt+ 119865
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(28)
Nowwe turn to estimate119864 Observe that if 119909 isin 119877119896 119910 isin 119877
119894
and 119894 le 119896 minus 2 then |119909 minus 119910| asymp |119909| asymp 2119896 and
1003816100381610038161003816100381610038161003816100381610038161003816
1
1003816100381610038161003816119909 minus 11991010038161003816100381610038162minus
1
|119909|2
1003816100381610038161003816100381610038161003816100381610038161003816
≲
10038161003816100381610038161199101003816100381610038161003816
1003816100381610038161003816119909 minus 11991010038161003816100381610038163 (29)
Since Ω isin Lip120574(S119899minus1
) sub 119871infin(S119899minus1
) by Minkowskirsquosinequality and Lemma 6 we have
1003816100381610038161003816120583Ω(119891
119894) (119909)
1003816100381610038161003816
≲ (int
|119909|
0
100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
|119909 minus 119910|119899minus1119891
119894(119910)119889119910
100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
|119909 minus 119910|119899minus1119891
119894(119910)119889119910
100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910 + int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(30)
Lemmas 7 and 8 lead to
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(31)
Thus we get
119864lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
119896minus2
sum
119894=minusinfin
2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))
)
119902
1119902
(32)
If 1 lt 119902 lt infin since 1198991205752minus 120572(0) gt 0 Holderrsquos inequality
implies that
119864lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))1199022
)
times(
119896minus2
sum
119894=minusinfin
2(119894minus119896)(119899120575
2minus120572(0))119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572(0))1199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(33)
Journal of Function Spaces 5
If 0 lt 119902 le 1 by the well-known inequality
(
infin
sum
119894=1
119886119894)
119902
le
infin
sum
119894=1
119886119902
119894(119886
119894gt 0 119894 = 1 2 ) (34)
we obtain
119864lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572(0))119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(35)
Similarly we have
119864gt≲
infin
sum
119896=0
2119896120572infin
119902(
119896minus2
sum
119894=minusinfin
2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
))
119902
1119902
(36)
If 1 lt 119902 lt infin since 120572infin+119899120575
2gt 2120572
infingt 2120572(0) then we get
119864gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
times (
119896minus2
sum
119894=minusinfin
2(119894minus119896)(119899120575
2minus120572infin
)1199021015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)1199022
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)1199022
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)119894119902
2(120572infin
+1198991205752minus2120572(0))11989411990221003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(37)
If 0 lt 119902 le 1 since 120572(0) le 120572infin we obtain
119864gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(38)
Thus we arrive at
119864 = 119864lt+ 119864
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(39)
For 119866 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896 + 2 then
|119909 minus 119910| asymp |119910| asymp 2119894 and
1003816100381610038161003816100381610038161003816100381610038161003816
1
1003816100381610038161003816119909 minus 11991010038161003816100381610038162minus
1
100381610038161003816100381611991010038161003816100381610038162
1003816100381610038161003816100381610038161003816100381610038161003816
≲|119909|
1003816100381610038161003816119909 minus 11991010038161003816100381610038163 (40)
From Minkowskirsquos inequality and Lemma 6 it followsthat1003816100381610038161003816120583Ω
(119891119894) (119909)
1003816100381610038161003816
≲ (int
|119910|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119910|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119910|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int
infin
|119910|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot|119909|
12
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910 + int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot110038161003816100381610038161199101003816100381610038161003816
119889119910
6 Journal of Function Spaces
≲ 2(119896minus119894)2
2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 1198622minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(41)
By Lemmas 7 and 8 we have
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)10038171003817100381710038171003817120594
119861119894
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(42)
Thus we get
119866lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119866gt≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(43)
Using the same arguments as that for 119864ltand 119864
gt we get
119866 = 119866lt+ 119866
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(44)
Hence the proof of Theorem 11 is completed
Proof of Theorem 12 We apply Proposition 5 again and get
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
= 119869lt+ 119869
gt
(45)
Let 119891 isin 120572(sdot)119902
119901(sdot)(R119899
) and write
119891 (119909) =
infin
sum
119894=minusinfin
119891 (119909) 120594119894 (119909) =
infin
sum
119894=minusinfin
119891119894 (119909) (46)
By Minkowskirsquos inequality we have
119869lt=
minus1
sum
119896=minusinfin
2120572(0)11989611990210038171003817100381710038171003817
120583119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880lt+ 119881
lt+119882
lt
(47)
By the same way we obtain
119869gt=
infin
sum
119896=0
2120572infin
11989611990210038171003817100381710038171003817120583
119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880gt+ 119881
gt+119882
gt
(48)
Thus we have
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119880 + 119881 +119882 (49)
where 119880 = 119880lt+ 119880
gt 119881 = 119881
lt+ 119881
gt and119882 = 119882
lt+119882
gt
For 119881 by Lemma 10 we have
119881 = 119881lt+ 119881
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(50)
Journal of Function Spaces 7
For 119880 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 le 119896 minus 2 then
10038161003816100381610038161003816120583
119898
Ω119887(119891
119894) (119909)
10038161003816100381610038161003816
≲ (int
|119909|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
+ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus119895
int119877119894
10038161003816100381610038161003816119887119861119894
minus 119887 (119910)10038161003816100381610038161003816
119895 1003816100381610038161003816119891119894(119910)
1003816100381610038161003816 119889119910
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus11989510038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(51)
An application of Lemmas 7 8 and 10 gives
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times
119898
sum
119895=0
119862119895
119898
10038171003817100381710038171003817(119887(119909) minus 119887
119861119894
)119898minus119895
120594119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(119877119899)
times
119898
sum
119895=0
119862119895
119898(119896 minus 119894)
119898minus119895119887
119898minus119895
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)119887
119895
BMO10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198981003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
(119894minus119896)11989912057521003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(52)For convenience below we put 120590 = 119899120575
2minus 120572(0) if 1 lt 119902 lt
infin then we use Holderrsquos inequality and obtain
119880lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120590)
119902
1119902
≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205901199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120590119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)1205901199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(53)If 0 lt 119902 le 1 then we get
119880lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120590119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120590119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(54)Similarly we put 120578 = 119899120575
2minus 120572
infin if 1 lt 119902 lt infin by Holderrsquos
inequality we obtain
119880gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120578)
119902
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120578119902
1015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
1119902
(55)
8 Journal of Function Spaces
By the same arguments as 119864gt we get
119880gt≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(56)
If 0 lt 119902 le 1 we obtain
119880gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120578119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(57)
Thus we have119880 = 119880
lt+ 119880
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(58)
For119882 if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896+2 as in the arguments
for 119866 and 119880 we obtain10038171003817100381710038171003817120583
119898
Ω119887(119891
119895)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119894 minus 119896 + 1)
1198982
(119896minus119894)1198991205751
10038171003817100381710038171003817119891
119895
10038171003817100381710038171003817119871119901(sdot)
(R119899) (59)
Thus we get
119882lt
≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
(119894 minus 119896 + 1)1198982
(119896minus119894)11989912057511003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times (119894 minus 119896 + 1)1198982
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119882gt
≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119894 minus 119896 + 1)
1198982
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(60)
Similar to the estimates of 119880ltand 119880
gt we get
119882 = 119882lt+119882
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(61)
Hence the proof of Theorem 12 is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the referees for their time andvaluable comments This work was supported by the NSF ofChina (Grant no 11201003) and University NSR Project ofAnhui Province (Grant no KJ2014A087)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 pp 430ndash466 1958
[2] A Benedek A-P Calderon and R Panzone ldquoConvolutionoperators on Banach space valued functionsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 48 pp 356ndash365 1962
[3] Y Ding D Fan and Y Pan ldquoWeighted boundedness for aclass of rough Marcinkiewicz integralsrdquo Indiana UniversityMathematics Journal vol 48 no 3 pp 1037ndash1055 1999
[4] A Torchinsky and S L Wang ldquoA note on the Marcinkiewiczintegralrdquo Colloquium Mathematicum vol 60-61 no 1 pp 235ndash243 1990
[5] Y Ding S Lu and K Yabuta ldquoOn commutators of Marcink-iewicz integrals with rough kernelrdquo Journal of MathematicalAnalysis and Applications vol 275 no 1 pp 60ndash68 2002
[6] Y Ding S Lu and P Zhang ldquoWeighted weak type estimates forcommutators of the Marcinkiewicz integralsrdquo Science in ChinaA vol 47 no 1 pp 83ndash95 2004
[7] P Zhang ldquoWeighted endpoint estimates for commutators ofMarcinkiewicz integralsrdquo Acta Mathematica Sinica vol 26 no9 pp 1709ndash1722 2010
[8] S Lu ldquoMarcinkiewicz integral with rough kernelsrdquo Frontiers ofMathematics in China vol 3 no 1 pp 1ndash14 2008
[9] Y P Chen and Y Ding ldquo119871119901 boundedness of the commutatorsof Marcinkiewicz integrals with rough kernelsrdquo Forum Mathe-maticum 2013
[10] Y M Chen S Levine and M Rao ldquoVariable exponent lineargrowth functionals in image restorationrdquo SIAM Journal onApplied Mathematics vol 66 no 4 pp 1383ndash1406 2006
[11] P Harjulehto P Hasto U V Le and M Nuortio ldquoOverviewof differential equations with non-standard growthrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 12 pp4551ndash4574 2010
[12] O Kovacik and J Rakosnık ldquoOn spaces 119871119901(119909) and 119882119896119901(119909)rdquo
Czechoslovak Mathematical Journal vol 41 no 4 pp 592ndash6181991
[13] D Cruz-Uribe A Fiorenza J M Martell and C Perez ldquoTheboundedness of classical operators on variable 119871
119901 spacesrdquoAnnales Academiae Scientiarum Fennicae Mathematica vol 31no 1 pp 239ndash264 2006
[14] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011
[15] D V Cruz-Uribe and A Fiorenza Variable Lebesgue SpacesFoundations and Harmonic Analysis Applied and NumericalHarmonic Analysis Birkhauser Basel Switzerland 2013
Journal of Function Spaces 9
[16] M Izuki ldquoHerz and amalgam spaces with variable exponentthe Haar wavelets and greediness of the wavelet systemrdquo EastJournal on Approximations vol 15 no 1 pp 87ndash109 2009
[17] M Izuki ldquoCommutators of fractional integrals on Lebesgueand Herz spaces with variable exponentrdquo Rendiconti del CircoloMatematico di Palermo Second Series vol 59 no 3 pp 461ndash4722010
[18] H B Wang Z W Fu and Z G Liu ldquoHigher-order commu-tators of Marcinkiewicz integrals on variable Lebesgue spacesrdquoActa Mathematica Scientia A vol 32 no 6 pp 1092ndash1101 2012
[19] Z G Liu and H B Wang ldquoBoundedness of Marcinkiewiczintegrals on Herz spaces with variable exponentrdquoThe JordanianJournal of Mathematics and Statistics vol 5 no 4 pp 223ndash2392012
[20] A Almeida and D Drihem ldquoMaximal potential and singulartype operators on Herz spaces with variable exponentsrdquo Journalof Mathematical Analysis and Applications vol 394 no 2 pp781ndash795 2012
[21] S Z Lu D C Yang and G E Hu Herz Type Spaces and TheirApplications Science Press Beijing China 2008
[22] X W Li and D C Yang ldquoBoundedness of some sublinearoperators on Herz spacesrdquo Illinois Journal of Mathematics vol40 no 3 pp 484ndash501 1996
[23] M Izuki ldquoVector-valued inequalities onHERz spaces and char-acterizations of HERz-Sobolev spaces with variable exponentrdquoGlasnik Matematicki vol 45 no 65 pp 475ndash503 2010
[24] L Wang and L Shu ldquoHigher order commutators of Marcink-iewicz integral operator on Herz-Morrey spaces with variableexponentrdquo Journal of Mathematical Research with Applicationsvol 34 no 2 pp 175ndash186 2014
[25] B Dong and J Xu ldquoNew Herz type Besov and Triebel-Lizorkinspaces with variable exponentsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 384593 27 pages 2012
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 5
If 0 lt 119902 le 1 by the well-known inequality
(
infin
sum
119894=1
119886119894)
119902
le
infin
sum
119894=1
119886119902
119894(119886
119894gt 0 119894 = 1 2 ) (34)
we obtain
119864lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572(0))119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572(0))119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(35)
Similarly we have
119864gt≲
infin
sum
119896=0
2119896120572infin
119902(
119896minus2
sum
119894=minusinfin
2(119894minus119896)119899120575
21003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
))
119902
1119902
(36)
If 1 lt 119902 lt infin since 120572infin+119899120575
2gt 2120572
infingt 2120572(0) then we get
119864gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
times (
119896minus2
sum
119894=minusinfin
2(119894minus119896)(119899120575
2minus120572infin
)1199021015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)1199022)
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)1199022
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)1199022
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)119894119902
2(120572infin
+1198991205752minus2120572(0))11989411990221003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(37)
If 0 lt 119902 le 1 since 120572(0) le 120572infin we obtain
119864gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)(1198991205752minus120572infin
)119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
2(119894minus119896)(119899120575
2minus120572infin
)119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
2(119894minus119896)(119899120575
2minus120572infin
)119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(38)
Thus we arrive at
119864 = 119864lt+ 119864
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(39)
For 119866 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896 + 2 then
|119909 minus 119910| asymp |119910| asymp 2119894 and
1003816100381610038161003816100381610038161003816100381610038161003816
1
1003816100381610038161003816119909 minus 11991010038161003816100381610038162minus
1
100381610038161003816100381611991010038161003816100381610038162
1003816100381610038161003816100381610038161003816100381610038161003816
≲|119909|
1003816100381610038161003816119909 minus 11991010038161003816100381610038163 (40)
From Minkowskirsquos inequality and Lemma 6 it followsthat1003816100381610038161003816120583Ω
(119891119894) (119909)
1003816100381610038161003816
≲ (int
|119910|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119910|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
119891119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119910|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
(int
infin
|119910|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot|119909|
12
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910 + int119877119894
1003816100381610038161003816119891119894(119910)
10038161003816100381610038161003816100381610038161003816119909 minus 119910
1003816100381610038161003816119899minus1
sdot110038161003816100381610038161199101003816100381610038161003816
119889119910
6 Journal of Function Spaces
≲ 2(119896minus119894)2
2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 1198622minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(41)
By Lemmas 7 and 8 we have
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)10038171003817100381710038171003817120594
119861119894
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(42)
Thus we get
119866lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119866gt≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(43)
Using the same arguments as that for 119864ltand 119864
gt we get
119866 = 119866lt+ 119866
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(44)
Hence the proof of Theorem 11 is completed
Proof of Theorem 12 We apply Proposition 5 again and get
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
= 119869lt+ 119869
gt
(45)
Let 119891 isin 120572(sdot)119902
119901(sdot)(R119899
) and write
119891 (119909) =
infin
sum
119894=minusinfin
119891 (119909) 120594119894 (119909) =
infin
sum
119894=minusinfin
119891119894 (119909) (46)
By Minkowskirsquos inequality we have
119869lt=
minus1
sum
119896=minusinfin
2120572(0)11989611990210038171003817100381710038171003817
120583119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880lt+ 119881
lt+119882
lt
(47)
By the same way we obtain
119869gt=
infin
sum
119896=0
2120572infin
11989611990210038171003817100381710038171003817120583
119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880gt+ 119881
gt+119882
gt
(48)
Thus we have
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119880 + 119881 +119882 (49)
where 119880 = 119880lt+ 119880
gt 119881 = 119881
lt+ 119881
gt and119882 = 119882
lt+119882
gt
For 119881 by Lemma 10 we have
119881 = 119881lt+ 119881
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(50)
Journal of Function Spaces 7
For 119880 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 le 119896 minus 2 then
10038161003816100381610038161003816120583
119898
Ω119887(119891
119894) (119909)
10038161003816100381610038161003816
≲ (int
|119909|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
+ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus119895
int119877119894
10038161003816100381610038161003816119887119861119894
minus 119887 (119910)10038161003816100381610038161003816
119895 1003816100381610038161003816119891119894(119910)
1003816100381610038161003816 119889119910
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus11989510038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(51)
An application of Lemmas 7 8 and 10 gives
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times
119898
sum
119895=0
119862119895
119898
10038171003817100381710038171003817(119887(119909) minus 119887
119861119894
)119898minus119895
120594119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(119877119899)
times
119898
sum
119895=0
119862119895
119898(119896 minus 119894)
119898minus119895119887
119898minus119895
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)119887
119895
BMO10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198981003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
(119894minus119896)11989912057521003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(52)For convenience below we put 120590 = 119899120575
2minus 120572(0) if 1 lt 119902 lt
infin then we use Holderrsquos inequality and obtain
119880lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120590)
119902
1119902
≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205901199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120590119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)1205901199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(53)If 0 lt 119902 le 1 then we get
119880lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120590119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120590119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(54)Similarly we put 120578 = 119899120575
2minus 120572
infin if 1 lt 119902 lt infin by Holderrsquos
inequality we obtain
119880gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120578)
119902
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120578119902
1015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
1119902
(55)
8 Journal of Function Spaces
By the same arguments as 119864gt we get
119880gt≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(56)
If 0 lt 119902 le 1 we obtain
119880gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120578119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(57)
Thus we have119880 = 119880
lt+ 119880
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(58)
For119882 if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896+2 as in the arguments
for 119866 and 119880 we obtain10038171003817100381710038171003817120583
119898
Ω119887(119891
119895)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119894 minus 119896 + 1)
1198982
(119896minus119894)1198991205751
10038171003817100381710038171003817119891
119895
10038171003817100381710038171003817119871119901(sdot)
(R119899) (59)
Thus we get
119882lt
≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
(119894 minus 119896 + 1)1198982
(119896minus119894)11989912057511003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times (119894 minus 119896 + 1)1198982
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119882gt
≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119894 minus 119896 + 1)
1198982
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(60)
Similar to the estimates of 119880ltand 119880
gt we get
119882 = 119882lt+119882
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(61)
Hence the proof of Theorem 12 is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the referees for their time andvaluable comments This work was supported by the NSF ofChina (Grant no 11201003) and University NSR Project ofAnhui Province (Grant no KJ2014A087)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 pp 430ndash466 1958
[2] A Benedek A-P Calderon and R Panzone ldquoConvolutionoperators on Banach space valued functionsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 48 pp 356ndash365 1962
[3] Y Ding D Fan and Y Pan ldquoWeighted boundedness for aclass of rough Marcinkiewicz integralsrdquo Indiana UniversityMathematics Journal vol 48 no 3 pp 1037ndash1055 1999
[4] A Torchinsky and S L Wang ldquoA note on the Marcinkiewiczintegralrdquo Colloquium Mathematicum vol 60-61 no 1 pp 235ndash243 1990
[5] Y Ding S Lu and K Yabuta ldquoOn commutators of Marcink-iewicz integrals with rough kernelrdquo Journal of MathematicalAnalysis and Applications vol 275 no 1 pp 60ndash68 2002
[6] Y Ding S Lu and P Zhang ldquoWeighted weak type estimates forcommutators of the Marcinkiewicz integralsrdquo Science in ChinaA vol 47 no 1 pp 83ndash95 2004
[7] P Zhang ldquoWeighted endpoint estimates for commutators ofMarcinkiewicz integralsrdquo Acta Mathematica Sinica vol 26 no9 pp 1709ndash1722 2010
[8] S Lu ldquoMarcinkiewicz integral with rough kernelsrdquo Frontiers ofMathematics in China vol 3 no 1 pp 1ndash14 2008
[9] Y P Chen and Y Ding ldquo119871119901 boundedness of the commutatorsof Marcinkiewicz integrals with rough kernelsrdquo Forum Mathe-maticum 2013
[10] Y M Chen S Levine and M Rao ldquoVariable exponent lineargrowth functionals in image restorationrdquo SIAM Journal onApplied Mathematics vol 66 no 4 pp 1383ndash1406 2006
[11] P Harjulehto P Hasto U V Le and M Nuortio ldquoOverviewof differential equations with non-standard growthrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 12 pp4551ndash4574 2010
[12] O Kovacik and J Rakosnık ldquoOn spaces 119871119901(119909) and 119882119896119901(119909)rdquo
Czechoslovak Mathematical Journal vol 41 no 4 pp 592ndash6181991
[13] D Cruz-Uribe A Fiorenza J M Martell and C Perez ldquoTheboundedness of classical operators on variable 119871
119901 spacesrdquoAnnales Academiae Scientiarum Fennicae Mathematica vol 31no 1 pp 239ndash264 2006
[14] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011
[15] D V Cruz-Uribe and A Fiorenza Variable Lebesgue SpacesFoundations and Harmonic Analysis Applied and NumericalHarmonic Analysis Birkhauser Basel Switzerland 2013
Journal of Function Spaces 9
[16] M Izuki ldquoHerz and amalgam spaces with variable exponentthe Haar wavelets and greediness of the wavelet systemrdquo EastJournal on Approximations vol 15 no 1 pp 87ndash109 2009
[17] M Izuki ldquoCommutators of fractional integrals on Lebesgueand Herz spaces with variable exponentrdquo Rendiconti del CircoloMatematico di Palermo Second Series vol 59 no 3 pp 461ndash4722010
[18] H B Wang Z W Fu and Z G Liu ldquoHigher-order commu-tators of Marcinkiewicz integrals on variable Lebesgue spacesrdquoActa Mathematica Scientia A vol 32 no 6 pp 1092ndash1101 2012
[19] Z G Liu and H B Wang ldquoBoundedness of Marcinkiewiczintegrals on Herz spaces with variable exponentrdquoThe JordanianJournal of Mathematics and Statistics vol 5 no 4 pp 223ndash2392012
[20] A Almeida and D Drihem ldquoMaximal potential and singulartype operators on Herz spaces with variable exponentsrdquo Journalof Mathematical Analysis and Applications vol 394 no 2 pp781ndash795 2012
[21] S Z Lu D C Yang and G E Hu Herz Type Spaces and TheirApplications Science Press Beijing China 2008
[22] X W Li and D C Yang ldquoBoundedness of some sublinearoperators on Herz spacesrdquo Illinois Journal of Mathematics vol40 no 3 pp 484ndash501 1996
[23] M Izuki ldquoVector-valued inequalities onHERz spaces and char-acterizations of HERz-Sobolev spaces with variable exponentrdquoGlasnik Matematicki vol 45 no 65 pp 475ndash503 2010
[24] L Wang and L Shu ldquoHigher order commutators of Marcink-iewicz integral operator on Herz-Morrey spaces with variableexponentrdquo Journal of Mathematical Research with Applicationsvol 34 no 2 pp 175ndash186 2014
[25] B Dong and J Xu ldquoNew Herz type Besov and Triebel-Lizorkinspaces with variable exponentsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 384593 27 pages 2012
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Function Spaces
≲ 2(119896minus119894)2
2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
+ 1198622minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
1003817100381710038171003817120594119894
10038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(41)
By Lemmas 7 and 8 we have
1003817100381710038171003817120583Ω(119891
119894)(119909)120594
119896
1003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198941198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲1003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)10038171003817100381710038171003817120594
119861119894
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(42)
Thus we get
119866lt≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
2(119896minus119894)119899120575
11003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119866gt≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)2
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(43)
Using the same arguments as that for 119864ltand 119864
gt we get
119866 = 119866lt+ 119866
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(44)
Hence the proof of Theorem 11 is completed
Proof of Theorem 12 We apply Proposition 5 again and get
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
asymp100381710038171003817100381710038172
120572(0)119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
+100381710038171003817100381710038172
120572infin
119896120583
119898
Ω119887(119891) 120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
= 119869lt+ 119869
gt
(45)
Let 119891 isin 120572(sdot)119902
119901(sdot)(R119899
) and write
119891 (119909) =
infin
sum
119894=minusinfin
119891 (119909) 120594119894 (119909) =
infin
sum
119894=minusinfin
119891119894 (119909) (46)
By Minkowskirsquos inequality we have
119869lt=
minus1
sum
119896=minusinfin
2120572(0)11989611990210038171003817100381710038171003817
120583119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
minus1
sum
119896=minusinfin
2120572(0)119896119902
(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880lt+ 119881
lt+119882
lt
(47)
By the same way we obtain
119869gt=
infin
sum
119896=0
2120572infin
11989611990210038171003817100381710038171003817120583
119898
Ω119887(119891) 120594
119896
10038171003817100381710038171003817
119902
119871119901(sdot)
(R119899)
1119902
≲
infin
sum
119896=0
2120572infin
119896119902(
119896minus2
sum
119894=minusinfin
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
119896+1
sum
119894=119896minus1
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
+
infin
sum
119896=0
2120572infin
119896119902(
infin
sum
119894=119896+2
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894) 120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
= 119880gt+ 119881
gt+119882
gt
(48)
Thus we have
10038171003817100381710038171003817120583
119898
Ω119887(119891)
10038171003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
≲ 119880 + 119881 +119882 (49)
where 119880 = 119880lt+ 119880
gt 119881 = 119881
lt+ 119881
gt and119882 = 119882
lt+119882
gt
For 119881 by Lemma 10 we have
119881 = 119881lt+ 119881
gt
≲
minus1
sum
119896=minusinfin
2120572(0)1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
+
infin
sum
119896=0
2120572infin
1198961199021003817100381710038171003817119891119896
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(50)
Journal of Function Spaces 7
For 119880 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 le 119896 minus 2 then
10038161003816100381610038161003816120583
119898
Ω119887(119891
119894) (119909)
10038161003816100381610038161003816
≲ (int
|119909|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
+ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus119895
int119877119894
10038161003816100381610038161003816119887119861119894
minus 119887 (119910)10038161003816100381610038161003816
119895 1003816100381610038161003816119891119894(119910)
1003816100381610038161003816 119889119910
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus11989510038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(51)
An application of Lemmas 7 8 and 10 gives
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times
119898
sum
119895=0
119862119895
119898
10038171003817100381710038171003817(119887(119909) minus 119887
119861119894
)119898minus119895
120594119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(119877119899)
times
119898
sum
119895=0
119862119895
119898(119896 minus 119894)
119898minus119895119887
119898minus119895
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)119887
119895
BMO10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198981003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
(119894minus119896)11989912057521003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(52)For convenience below we put 120590 = 119899120575
2minus 120572(0) if 1 lt 119902 lt
infin then we use Holderrsquos inequality and obtain
119880lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120590)
119902
1119902
≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205901199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120590119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)1205901199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(53)If 0 lt 119902 le 1 then we get
119880lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120590119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120590119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(54)Similarly we put 120578 = 119899120575
2minus 120572
infin if 1 lt 119902 lt infin by Holderrsquos
inequality we obtain
119880gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120578)
119902
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120578119902
1015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
1119902
(55)
8 Journal of Function Spaces
By the same arguments as 119864gt we get
119880gt≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(56)
If 0 lt 119902 le 1 we obtain
119880gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120578119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(57)
Thus we have119880 = 119880
lt+ 119880
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(58)
For119882 if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896+2 as in the arguments
for 119866 and 119880 we obtain10038171003817100381710038171003817120583
119898
Ω119887(119891
119895)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119894 minus 119896 + 1)
1198982
(119896minus119894)1198991205751
10038171003817100381710038171003817119891
119895
10038171003817100381710038171003817119871119901(sdot)
(R119899) (59)
Thus we get
119882lt
≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
(119894 minus 119896 + 1)1198982
(119896minus119894)11989912057511003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times (119894 minus 119896 + 1)1198982
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119882gt
≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119894 minus 119896 + 1)
1198982
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(60)
Similar to the estimates of 119880ltand 119880
gt we get
119882 = 119882lt+119882
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(61)
Hence the proof of Theorem 12 is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the referees for their time andvaluable comments This work was supported by the NSF ofChina (Grant no 11201003) and University NSR Project ofAnhui Province (Grant no KJ2014A087)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 pp 430ndash466 1958
[2] A Benedek A-P Calderon and R Panzone ldquoConvolutionoperators on Banach space valued functionsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 48 pp 356ndash365 1962
[3] Y Ding D Fan and Y Pan ldquoWeighted boundedness for aclass of rough Marcinkiewicz integralsrdquo Indiana UniversityMathematics Journal vol 48 no 3 pp 1037ndash1055 1999
[4] A Torchinsky and S L Wang ldquoA note on the Marcinkiewiczintegralrdquo Colloquium Mathematicum vol 60-61 no 1 pp 235ndash243 1990
[5] Y Ding S Lu and K Yabuta ldquoOn commutators of Marcink-iewicz integrals with rough kernelrdquo Journal of MathematicalAnalysis and Applications vol 275 no 1 pp 60ndash68 2002
[6] Y Ding S Lu and P Zhang ldquoWeighted weak type estimates forcommutators of the Marcinkiewicz integralsrdquo Science in ChinaA vol 47 no 1 pp 83ndash95 2004
[7] P Zhang ldquoWeighted endpoint estimates for commutators ofMarcinkiewicz integralsrdquo Acta Mathematica Sinica vol 26 no9 pp 1709ndash1722 2010
[8] S Lu ldquoMarcinkiewicz integral with rough kernelsrdquo Frontiers ofMathematics in China vol 3 no 1 pp 1ndash14 2008
[9] Y P Chen and Y Ding ldquo119871119901 boundedness of the commutatorsof Marcinkiewicz integrals with rough kernelsrdquo Forum Mathe-maticum 2013
[10] Y M Chen S Levine and M Rao ldquoVariable exponent lineargrowth functionals in image restorationrdquo SIAM Journal onApplied Mathematics vol 66 no 4 pp 1383ndash1406 2006
[11] P Harjulehto P Hasto U V Le and M Nuortio ldquoOverviewof differential equations with non-standard growthrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 12 pp4551ndash4574 2010
[12] O Kovacik and J Rakosnık ldquoOn spaces 119871119901(119909) and 119882119896119901(119909)rdquo
Czechoslovak Mathematical Journal vol 41 no 4 pp 592ndash6181991
[13] D Cruz-Uribe A Fiorenza J M Martell and C Perez ldquoTheboundedness of classical operators on variable 119871
119901 spacesrdquoAnnales Academiae Scientiarum Fennicae Mathematica vol 31no 1 pp 239ndash264 2006
[14] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011
[15] D V Cruz-Uribe and A Fiorenza Variable Lebesgue SpacesFoundations and Harmonic Analysis Applied and NumericalHarmonic Analysis Birkhauser Basel Switzerland 2013
Journal of Function Spaces 9
[16] M Izuki ldquoHerz and amalgam spaces with variable exponentthe Haar wavelets and greediness of the wavelet systemrdquo EastJournal on Approximations vol 15 no 1 pp 87ndash109 2009
[17] M Izuki ldquoCommutators of fractional integrals on Lebesgueand Herz spaces with variable exponentrdquo Rendiconti del CircoloMatematico di Palermo Second Series vol 59 no 3 pp 461ndash4722010
[18] H B Wang Z W Fu and Z G Liu ldquoHigher-order commu-tators of Marcinkiewicz integrals on variable Lebesgue spacesrdquoActa Mathematica Scientia A vol 32 no 6 pp 1092ndash1101 2012
[19] Z G Liu and H B Wang ldquoBoundedness of Marcinkiewiczintegrals on Herz spaces with variable exponentrdquoThe JordanianJournal of Mathematics and Statistics vol 5 no 4 pp 223ndash2392012
[20] A Almeida and D Drihem ldquoMaximal potential and singulartype operators on Herz spaces with variable exponentsrdquo Journalof Mathematical Analysis and Applications vol 394 no 2 pp781ndash795 2012
[21] S Z Lu D C Yang and G E Hu Herz Type Spaces and TheirApplications Science Press Beijing China 2008
[22] X W Li and D C Yang ldquoBoundedness of some sublinearoperators on Herz spacesrdquo Illinois Journal of Mathematics vol40 no 3 pp 484ndash501 1996
[23] M Izuki ldquoVector-valued inequalities onHERz spaces and char-acterizations of HERz-Sobolev spaces with variable exponentrdquoGlasnik Matematicki vol 45 no 65 pp 475ndash503 2010
[24] L Wang and L Shu ldquoHigher order commutators of Marcink-iewicz integral operator on Herz-Morrey spaces with variableexponentrdquo Journal of Mathematical Research with Applicationsvol 34 no 2 pp 175ndash186 2014
[25] B Dong and J Xu ldquoNew Herz type Besov and Triebel-Lizorkinspaces with variable exponentsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 384593 27 pages 2012
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 7
For 119880 observe that if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 le 119896 minus 2 then
10038161003816100381610038161003816120583
119898
Ω119887(119891
119894) (119909)
10038161003816100381610038161003816
≲ (int
|119909|
0
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
+ (int
infin
|119909|
1003816100381610038161003816100381610038161003816100381610038161003816
int|119909minus119910|le119905
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
[119887 (119909) minus 119887 (119910)]119898119891
119894(119910)119889119910
1003816100381610038161003816100381610038161003816100381610038161003816
2
119889119905
1199053)
12
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int|119909minus119910|le119905|119909|ge119905
119889119905
1199053)
12
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
(int
infin
|119909|
119889119905
1199053)
12
119889119910
≲ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot
1003816100381610038161003816119910100381610038161003816100381612
1003816100381610038161003816119909 minus 119910100381610038161003816100381632
119889119910
+ int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899minus1
sdot1
|119909|119889119910
≲ 2(119894minus119896)2
2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
+ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
int119877119894
1003816100381610038161003816119887 (119909) minus 119887 (119910)1003816100381610038161003816119898 1003816100381610038161003816119891119894
(119910)1003816100381610038161003816 119889119910
≲ 2minus119896119899
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus119895
int119877119894
10038161003816100381610038161003816119887119861119894
minus 119887 (119910)10038161003816100381610038161003816
119895 1003816100381610038161003816119891119894(119910)
1003816100381610038161003816 119889119910
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
119898
sum
119895=0
119862119895
119898
10038161003816100381610038161003816119887 (119909) minus 119887119861
119894
10038161003816100381610038161003816
119898minus11989510038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
(51)
An application of Lemmas 7 8 and 10 gives
10038171003817100381710038171003817120583
119898
Ω119887(119891
119894)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times
119898
sum
119895=0
119862119895
119898
10038171003817100381710038171003817(119887(119909) minus 119887
119861119894
)119898minus119895
120594119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817(119887
119861119894
minus 119887)119895120594
119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ 2minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(119877119899)
times
119898
sum
119895=0
119862119895
119898(119896 minus 119894)
119898minus119895119887
119898minus119895
BMO10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)119887
119895
BMO10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
minus1198961198991003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198981003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
10038171003817100381710038171003817120594
119861119894
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)10038171003817100381710038171003817120594
119861119896
100381710038171003817100381710038171198711199011015840(sdot)
(R119899)
≲ (119896 minus 119894 + 1)1198982
(119894minus119896)11989912057521003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
(52)For convenience below we put 120590 = 119899120575
2minus 120572(0) if 1 lt 119902 lt
infin then we use Holderrsquos inequality and obtain
119880lt≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120590)
119902
1119902
≲
minus1
sum
119896=minusinfin
(
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205901199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120590119902
1015840
2)
1199021199021015840
1119902
≲
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
2(119894minus119896)1205901199022
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(53)If 0 lt 119902 le 1 then we get
119880lt≲
minus1
sum
119896=minusinfin
119896minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120590119902
1119902
asymp
minus3
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
minus1
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120590119902
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)
≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(54)Similarly we put 120578 = 119899120575
2minus 120572
infin if 1 lt 119902 lt infin by Holderrsquos
inequality we obtain
119880gt≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198982
(119894minus119896)120578)
119902
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
times (
119896minus2
sum
119894=minusinfin
(119896 minus 119894 + 1)1198981199021015840
2(119894minus119896)120578119902
1015840
2)
1199021199021015840
1119902
≲
infin
sum
119896=0
(
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)2
(119894minus119896)1205781199022)
1119902
(55)
8 Journal of Function Spaces
By the same arguments as 119864gt we get
119880gt≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(56)
If 0 lt 119902 le 1 we obtain
119880gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120578119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(57)
Thus we have119880 = 119880
lt+ 119880
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(58)
For119882 if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896+2 as in the arguments
for 119866 and 119880 we obtain10038171003817100381710038171003817120583
119898
Ω119887(119891
119895)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119894 minus 119896 + 1)
1198982
(119896minus119894)1198991205751
10038171003817100381710038171003817119891
119895
10038171003817100381710038171003817119871119901(sdot)
(R119899) (59)
Thus we get
119882lt
≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
(119894 minus 119896 + 1)1198982
(119896minus119894)11989912057511003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times (119894 minus 119896 + 1)1198982
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119882gt
≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119894 minus 119896 + 1)
1198982
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(60)
Similar to the estimates of 119880ltand 119880
gt we get
119882 = 119882lt+119882
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(61)
Hence the proof of Theorem 12 is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the referees for their time andvaluable comments This work was supported by the NSF ofChina (Grant no 11201003) and University NSR Project ofAnhui Province (Grant no KJ2014A087)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 pp 430ndash466 1958
[2] A Benedek A-P Calderon and R Panzone ldquoConvolutionoperators on Banach space valued functionsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 48 pp 356ndash365 1962
[3] Y Ding D Fan and Y Pan ldquoWeighted boundedness for aclass of rough Marcinkiewicz integralsrdquo Indiana UniversityMathematics Journal vol 48 no 3 pp 1037ndash1055 1999
[4] A Torchinsky and S L Wang ldquoA note on the Marcinkiewiczintegralrdquo Colloquium Mathematicum vol 60-61 no 1 pp 235ndash243 1990
[5] Y Ding S Lu and K Yabuta ldquoOn commutators of Marcink-iewicz integrals with rough kernelrdquo Journal of MathematicalAnalysis and Applications vol 275 no 1 pp 60ndash68 2002
[6] Y Ding S Lu and P Zhang ldquoWeighted weak type estimates forcommutators of the Marcinkiewicz integralsrdquo Science in ChinaA vol 47 no 1 pp 83ndash95 2004
[7] P Zhang ldquoWeighted endpoint estimates for commutators ofMarcinkiewicz integralsrdquo Acta Mathematica Sinica vol 26 no9 pp 1709ndash1722 2010
[8] S Lu ldquoMarcinkiewicz integral with rough kernelsrdquo Frontiers ofMathematics in China vol 3 no 1 pp 1ndash14 2008
[9] Y P Chen and Y Ding ldquo119871119901 boundedness of the commutatorsof Marcinkiewicz integrals with rough kernelsrdquo Forum Mathe-maticum 2013
[10] Y M Chen S Levine and M Rao ldquoVariable exponent lineargrowth functionals in image restorationrdquo SIAM Journal onApplied Mathematics vol 66 no 4 pp 1383ndash1406 2006
[11] P Harjulehto P Hasto U V Le and M Nuortio ldquoOverviewof differential equations with non-standard growthrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 12 pp4551ndash4574 2010
[12] O Kovacik and J Rakosnık ldquoOn spaces 119871119901(119909) and 119882119896119901(119909)rdquo
Czechoslovak Mathematical Journal vol 41 no 4 pp 592ndash6181991
[13] D Cruz-Uribe A Fiorenza J M Martell and C Perez ldquoTheboundedness of classical operators on variable 119871
119901 spacesrdquoAnnales Academiae Scientiarum Fennicae Mathematica vol 31no 1 pp 239ndash264 2006
[14] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011
[15] D V Cruz-Uribe and A Fiorenza Variable Lebesgue SpacesFoundations and Harmonic Analysis Applied and NumericalHarmonic Analysis Birkhauser Basel Switzerland 2013
Journal of Function Spaces 9
[16] M Izuki ldquoHerz and amalgam spaces with variable exponentthe Haar wavelets and greediness of the wavelet systemrdquo EastJournal on Approximations vol 15 no 1 pp 87ndash109 2009
[17] M Izuki ldquoCommutators of fractional integrals on Lebesgueand Herz spaces with variable exponentrdquo Rendiconti del CircoloMatematico di Palermo Second Series vol 59 no 3 pp 461ndash4722010
[18] H B Wang Z W Fu and Z G Liu ldquoHigher-order commu-tators of Marcinkiewicz integrals on variable Lebesgue spacesrdquoActa Mathematica Scientia A vol 32 no 6 pp 1092ndash1101 2012
[19] Z G Liu and H B Wang ldquoBoundedness of Marcinkiewiczintegrals on Herz spaces with variable exponentrdquoThe JordanianJournal of Mathematics and Statistics vol 5 no 4 pp 223ndash2392012
[20] A Almeida and D Drihem ldquoMaximal potential and singulartype operators on Herz spaces with variable exponentsrdquo Journalof Mathematical Analysis and Applications vol 394 no 2 pp781ndash795 2012
[21] S Z Lu D C Yang and G E Hu Herz Type Spaces and TheirApplications Science Press Beijing China 2008
[22] X W Li and D C Yang ldquoBoundedness of some sublinearoperators on Herz spacesrdquo Illinois Journal of Mathematics vol40 no 3 pp 484ndash501 1996
[23] M Izuki ldquoVector-valued inequalities onHERz spaces and char-acterizations of HERz-Sobolev spaces with variable exponentrdquoGlasnik Matematicki vol 45 no 65 pp 475ndash503 2010
[24] L Wang and L Shu ldquoHigher order commutators of Marcink-iewicz integral operator on Herz-Morrey spaces with variableexponentrdquo Journal of Mathematical Research with Applicationsvol 34 no 2 pp 175ndash186 2014
[25] B Dong and J Xu ldquoNew Herz type Besov and Triebel-Lizorkinspaces with variable exponentsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 384593 27 pages 2012
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Function Spaces
By the same arguments as 119864gt we get
119880gt≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(56)
If 0 lt 119902 le 1 we obtain
119880gt≲
infin
sum
119896=0
119896minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)(119896 minus 119894 + 1)
1198981199022
(119894minus119896)120578119902
1119902
asymp
minus2
sum
119894=minusinfin
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=0
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
infin
sum
119896=119894+2
(119896 minus 119894 + 1)1198981199022
(119894minus119896)120578119902
1119902
≲
minus2
sum
119894=minusinfin
2120572(0)1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)+
infin
sum
119894=minus1
2120572infin
1198941199021003817100381710038171003817119891119894
1003817100381710038171003817119902
119871119901(sdot)
(R119899)
1119902
≲100381710038171003817100381710038172
120572(0)119896119891120594
11989610038171003817100381710038171003817ℓ119902
lt(119871119901(sdot)
)+100381710038171003817100381710038172
120572infin
119896119891120594
11989610038171003817100381710038171003817ℓ119902
gt(119871119901(sdot)
)
asymp10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(57)
Thus we have119880 = 119880
lt+ 119880
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(58)
For119882 if 119909 isin 119877119896 119910 isin 119877
119894 and 119894 ge 119896+2 as in the arguments
for 119866 and 119880 we obtain10038171003817100381710038171003817120583
119898
Ω119887(119891
119895)120594
119896
10038171003817100381710038171003817119871119901(sdot)
(R119899)≲ (119894 minus 119896 + 1)
1198982
(119896minus119894)1198991205751
10038171003817100381710038171003817119891
119895
10038171003817100381710038171003817119871119901(sdot)
(R119899) (59)
Thus we get
119882lt
≲
minus1
sum
119896=minusinfin
2119896120572(0)119902
(
infin
sum
119894=119896+2
(119894 minus 119896 + 1)1198982
(119896minus119894)11989912057511003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899))
119902
1119902
asymp
minus1
sum
119896=minusinfin
(
infin
sum
119894=119896+2
2120572(0)1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)
times (119894 minus 119896 + 1)1198982
(119896minus119894)(1198991205751+120572(0))
)
119902
1119902
119882gt
≲
infin
sum
119896=0
(
infin
sum
119894=119896+2
2120572infin
1198941003817100381710038171003817119891119894
1003817100381710038171003817119871119901(sdot)
(R119899)(119894 minus 119896 + 1)
1198982
(119896minus119894)(1198991205751+120572infin
))
119902
1119902
(60)
Similar to the estimates of 119880ltand 119880
gt we get
119882 = 119882lt+119882
gt≲10038171003817100381710038171198911003817100381710038171003817120572(sdot)119902
119901(sdot)(R119899)
(61)
Hence the proof of Theorem 12 is completed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the referees for their time andvaluable comments This work was supported by the NSF ofChina (Grant no 11201003) and University NSR Project ofAnhui Province (Grant no KJ2014A087)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 pp 430ndash466 1958
[2] A Benedek A-P Calderon and R Panzone ldquoConvolutionoperators on Banach space valued functionsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 48 pp 356ndash365 1962
[3] Y Ding D Fan and Y Pan ldquoWeighted boundedness for aclass of rough Marcinkiewicz integralsrdquo Indiana UniversityMathematics Journal vol 48 no 3 pp 1037ndash1055 1999
[4] A Torchinsky and S L Wang ldquoA note on the Marcinkiewiczintegralrdquo Colloquium Mathematicum vol 60-61 no 1 pp 235ndash243 1990
[5] Y Ding S Lu and K Yabuta ldquoOn commutators of Marcink-iewicz integrals with rough kernelrdquo Journal of MathematicalAnalysis and Applications vol 275 no 1 pp 60ndash68 2002
[6] Y Ding S Lu and P Zhang ldquoWeighted weak type estimates forcommutators of the Marcinkiewicz integralsrdquo Science in ChinaA vol 47 no 1 pp 83ndash95 2004
[7] P Zhang ldquoWeighted endpoint estimates for commutators ofMarcinkiewicz integralsrdquo Acta Mathematica Sinica vol 26 no9 pp 1709ndash1722 2010
[8] S Lu ldquoMarcinkiewicz integral with rough kernelsrdquo Frontiers ofMathematics in China vol 3 no 1 pp 1ndash14 2008
[9] Y P Chen and Y Ding ldquo119871119901 boundedness of the commutatorsof Marcinkiewicz integrals with rough kernelsrdquo Forum Mathe-maticum 2013
[10] Y M Chen S Levine and M Rao ldquoVariable exponent lineargrowth functionals in image restorationrdquo SIAM Journal onApplied Mathematics vol 66 no 4 pp 1383ndash1406 2006
[11] P Harjulehto P Hasto U V Le and M Nuortio ldquoOverviewof differential equations with non-standard growthrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 12 pp4551ndash4574 2010
[12] O Kovacik and J Rakosnık ldquoOn spaces 119871119901(119909) and 119882119896119901(119909)rdquo
Czechoslovak Mathematical Journal vol 41 no 4 pp 592ndash6181991
[13] D Cruz-Uribe A Fiorenza J M Martell and C Perez ldquoTheboundedness of classical operators on variable 119871
119901 spacesrdquoAnnales Academiae Scientiarum Fennicae Mathematica vol 31no 1 pp 239ndash264 2006
[14] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents vol 2017 of LectureNotes in Mathematics Springer Heidelberg Germany 2011
[15] D V Cruz-Uribe and A Fiorenza Variable Lebesgue SpacesFoundations and Harmonic Analysis Applied and NumericalHarmonic Analysis Birkhauser Basel Switzerland 2013
Journal of Function Spaces 9
[16] M Izuki ldquoHerz and amalgam spaces with variable exponentthe Haar wavelets and greediness of the wavelet systemrdquo EastJournal on Approximations vol 15 no 1 pp 87ndash109 2009
[17] M Izuki ldquoCommutators of fractional integrals on Lebesgueand Herz spaces with variable exponentrdquo Rendiconti del CircoloMatematico di Palermo Second Series vol 59 no 3 pp 461ndash4722010
[18] H B Wang Z W Fu and Z G Liu ldquoHigher-order commu-tators of Marcinkiewicz integrals on variable Lebesgue spacesrdquoActa Mathematica Scientia A vol 32 no 6 pp 1092ndash1101 2012
[19] Z G Liu and H B Wang ldquoBoundedness of Marcinkiewiczintegrals on Herz spaces with variable exponentrdquoThe JordanianJournal of Mathematics and Statistics vol 5 no 4 pp 223ndash2392012
[20] A Almeida and D Drihem ldquoMaximal potential and singulartype operators on Herz spaces with variable exponentsrdquo Journalof Mathematical Analysis and Applications vol 394 no 2 pp781ndash795 2012
[21] S Z Lu D C Yang and G E Hu Herz Type Spaces and TheirApplications Science Press Beijing China 2008
[22] X W Li and D C Yang ldquoBoundedness of some sublinearoperators on Herz spacesrdquo Illinois Journal of Mathematics vol40 no 3 pp 484ndash501 1996
[23] M Izuki ldquoVector-valued inequalities onHERz spaces and char-acterizations of HERz-Sobolev spaces with variable exponentrdquoGlasnik Matematicki vol 45 no 65 pp 475ndash503 2010
[24] L Wang and L Shu ldquoHigher order commutators of Marcink-iewicz integral operator on Herz-Morrey spaces with variableexponentrdquo Journal of Mathematical Research with Applicationsvol 34 no 2 pp 175ndash186 2014
[25] B Dong and J Xu ldquoNew Herz type Besov and Triebel-Lizorkinspaces with variable exponentsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 384593 27 pages 2012
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 9
[16] M Izuki ldquoHerz and amalgam spaces with variable exponentthe Haar wavelets and greediness of the wavelet systemrdquo EastJournal on Approximations vol 15 no 1 pp 87ndash109 2009
[17] M Izuki ldquoCommutators of fractional integrals on Lebesgueand Herz spaces with variable exponentrdquo Rendiconti del CircoloMatematico di Palermo Second Series vol 59 no 3 pp 461ndash4722010
[18] H B Wang Z W Fu and Z G Liu ldquoHigher-order commu-tators of Marcinkiewicz integrals on variable Lebesgue spacesrdquoActa Mathematica Scientia A vol 32 no 6 pp 1092ndash1101 2012
[19] Z G Liu and H B Wang ldquoBoundedness of Marcinkiewiczintegrals on Herz spaces with variable exponentrdquoThe JordanianJournal of Mathematics and Statistics vol 5 no 4 pp 223ndash2392012
[20] A Almeida and D Drihem ldquoMaximal potential and singulartype operators on Herz spaces with variable exponentsrdquo Journalof Mathematical Analysis and Applications vol 394 no 2 pp781ndash795 2012
[21] S Z Lu D C Yang and G E Hu Herz Type Spaces and TheirApplications Science Press Beijing China 2008
[22] X W Li and D C Yang ldquoBoundedness of some sublinearoperators on Herz spacesrdquo Illinois Journal of Mathematics vol40 no 3 pp 484ndash501 1996
[23] M Izuki ldquoVector-valued inequalities onHERz spaces and char-acterizations of HERz-Sobolev spaces with variable exponentrdquoGlasnik Matematicki vol 45 no 65 pp 475ndash503 2010
[24] L Wang and L Shu ldquoHigher order commutators of Marcink-iewicz integral operator on Herz-Morrey spaces with variableexponentrdquo Journal of Mathematical Research with Applicationsvol 34 no 2 pp 175ndash186 2014
[25] B Dong and J Xu ldquoNew Herz type Besov and Triebel-Lizorkinspaces with variable exponentsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 384593 27 pages 2012
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
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 2014
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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![LECTURES ON SINGULAR INTEGRAL OPERATORS · [Ch7] , Inversion in some algebras of singular integral operators, Rev. Mat. Iberoamericana 4(1988), 219-225. [Ch8], A T(b) theorem with](https://static.fdocuments.net/doc/165x107/5f921f442edc1f419a6e88f4/lectures-on-singular-integral-ch7-inversion-in-some-algebras-of-singular-integral.jpg)