Research Article Multilinear Singular and Fractional Integral...
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Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 735795, 11 pages http://dx.doi.org/10.1155/2013/735795 Research Article Multilinear Singular and Fractional Integral Operators on Weighted Morrey Spaces Hua Wang 1 and Wentan Yi 2 1 College of Mathematics and Econometrics, Hunan University, Changsha 410082, China 2 Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou 450002, China Correspondence should be addressed to Hua Wang; [email protected] Received 2 May 2013; Accepted 12 September 2013 Academic Editor: John R. Akeroyd Copyright © 2013 H. Wang and W. Yi. 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. We will study the boundedness properties of multilinear Calder´ on-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights. 1. Introduction and Main Results Multilinear Calder´ on-Zygmund theory is a natural gener- alization of the linear case. e initial work on the class of multilinear Calder´ on-Zygmund operators was done by Coifman and Meyer in [1] and was later systematically studied by Grafakos and Torres in [2–4]. Let R be the -dimensional Euclidean space, and let (R ) = R ×⋅⋅⋅× R be the -fold product space (∈ N). We denote by S(R ) the space of all Schwartz functions on R and by S (R ) its dual space, the set of all tempered distributions on R . Let ≥2 and be an -linear operator initially defined on the -fold product of Schwartz spaces, and taking values into the space of tempered distributions, : S (R )×⋅⋅⋅× S (R ) → S (R ). (1) Following [2], for given ⃗ = ( 1 ,..., ), we say that is an -linear Calder´ on-Zygmund operator if, for some 1 ,..., ∈ [1, ∞) and ∈ (0, ∞) with 1/ = ∑ =1 1/ , it extends to a bounded multilinear operator from 1 (R )× ⋅⋅⋅× (R ) into (R ) and if there exists a kernel function (, 1 ,..., ) in the class -(, ), defined away from the diagonal = 1 =⋅⋅⋅= in (R ) +1 such that ( ⃗ ) () = ( 1 ,..., ) () =∫ (R ) (, 1 ,..., ) 1 ( 1 )⋅⋅⋅ ( ) 1 ⋅ ⋅ ⋅ , (2) whenever 1 ,..., ∈ S(R ) and ∉⋂ =1 supp . We say that (, 1 ,..., ) is a kernel in the class -(, ) if it satisfies the size condition (, 1 ,..., ) ≤ ( − 1 +⋅⋅⋅+ − ) , (3) for some >0 and all (, 1 ,..., )∈(R ) +1 with ̸ = for some 1≤≤. Moreover, for some >0, it satisfies the regularity condition that (, 1 ,..., ) − ( , 1 ,..., ) ≤ ⋅ − ( − 1 +⋅⋅⋅+ − ) + (4) whenever | − | ≤ (1/2)max 1≤≤ | − | and also that, for each fixed with 1≤≤, (, 1 ,..., ,..., ) − (, 1 ,..., ,..., ) ≤ ⋅ − ( − 1 +⋅⋅⋅+ − ) + (5) whenever | − | ≤ (1/2)max 1≤≤ | − |. In recent years, many authors have been interested in studying the boundedness of these operators on function spaces; see, for example, [5–8]. In 2009, the weighted strong and weak type estimates of multilinear Calder´ on-Zygmund singular integral operators were established in [9] by Lerner et al. New more
Transcript of Research Article Multilinear Singular and Fractional Integral...
Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 735795 11 pageshttpdxdoiorg1011552013735795
Research ArticleMultilinear Singular and Fractional IntegralOperators on Weighted Morrey Spaces
Hua Wang1 and Wentan Yi2
1 College of Mathematics and Econometrics Hunan University Changsha 410082 China2Department of Applied Mathematics Zhengzhou Information Science and Technology Institute Zhengzhou 450002 China
Correspondence should be addressed to Hua Wang wanghuapkueducn
Received 2 May 2013 Accepted 12 September 2013
Academic Editor John R Akeroyd
Copyright copy 2013 H Wang and W Yi This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We will study the boundedness properties of multilinear Calderon-Zygmund operators and multilinear fractional integrals onproducts of weighted Morrey spaces with multiple weights
1 Introduction and Main Results
Multilinear Calderon-Zygmund theory is a natural gener-alization of the linear case The initial work on the classof multilinear Calderon-Zygmund operators was done byCoifman andMeyer in [1] andwas later systematically studiedby Grafakos and Torres in [2ndash4] LetR119899 be the 119899-dimensionalEuclidean space and let (R119899)119898 = R119899 times sdot sdot sdot timesR119899 be the119898-foldproduct space (119898 isin N) We denote by S(R119899) the space of allSchwartz functions on R119899 and by S1015840(R119899) its dual space theset of all tempered distributions onR119899 Let119898 ge 2 and119879 be an119898-linear operator initially defined on the 119898-fold product ofSchwartz spaces and taking values into the space of tempereddistributions
119879 S (R119899
) times sdot sdot sdot timesS (R119899
) 997888rarr S1015840
(R119899
) (1)
Following [2] for given 119891 = (119891
1 119891
119898) we say that
119879 is an 119898-linear Calderon-Zygmund operator if for some1199021 119902
119898isin [1infin) and 119902 isin (0infin) with 1119902 = sum
119898
119896=11119902119896
it extends to a bounded multilinear operator from 1198711199021(R119899) times
sdot sdot sdot times 119871119902119898(R119899) into 119871119902(R119899) and if there exists a kernel function
119870(119909 1199101 119910
119898) in the class119898-119862119885119870(119860 120576) defined away from
the diagonal 119909 = 1199101= sdot sdot sdot = 119910
119898in (R119899)119898+1 such that
119879 (
119891) (119909)
= 119879 (1198911 119891
119898) (119909)
= int
(R119899)119898
119870(119909 1199101 119910
119898) 1198911(1199101) sdot sdot sdot 119891
119898(119910119898) 1198891199101sdot sdot sdot 119889119910119898
(2)
whenever 1198911 119891
119898isin S(R119899) and 119909 notin ⋂119898
119896=1supp 119891
119896 We say
that119870(119909 1199101 119910
119898) is a kernel in the class119898-119862119885119870(119860 120576) if it
satisfies the size condition1003816100381610038161003816119870 (119909 119910
1 119910
119898)1003816100381610038161003816le
119860
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899 (3)
for some 119860 gt 0 and all (119909 1199101 119910
119898) isin (R119899)
119898+1 with 119909 = 119910119896
for some 1 le 119896 le 119898 Moreover for some 120576 gt 0 it satisfies theregularity condition that
10038161003816100381610038161003816119870 (119909 119910
1 119910
119898) minus 119870 (119909
1015840
1199101 119910
119898)
10038161003816100381610038161003816
le
119860 sdot
10038161003816100381610038161003816119909 minus 119909101584010038161003816100381610038161003816
120576
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899+120576
(4)
whenever |119909 minus 1199091015840| le (12)max1le119896le119898
|119909 minus 119910119896| and also that for
each fixed 119896 with 1 le 119896 le 11989810038161003816100381610038161003816119870 (119909 119910
1 119910
119896 119910
119898) minus 119870 (119909 119910
1 119910
1015840
119896 119910
119898)
10038161003816100381610038161003816
le
119860 sdot
10038161003816100381610038161003816119910119896minus 1199101015840
119896
10038161003816100381610038161003816
120576
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899+120576
(5)
whenever |119910119896minus 1199101015840
119896| le (12)max
1le119894le119898|119909 minus 119910
119894| In recent
years many authors have been interested in studying theboundedness of these operators on function spaces see forexample [5ndash8] In 2009 the weighted strong and weak typeestimates ofmultilinear Calderon-Zygmund singular integraloperators were established in [9] by Lerner et al New more
2 Journal of Function Spaces and Applications
refined multilinear maximal function was defined and usedin [9] to characterize the class of multiple 119860
weights
Theorem A (see [9]) Let 119898 ge 2 and 119879 be an 119898-linearCalderon-Zygmund operator If 119901
1 119901
119898isin (1infin) and
119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898)
satisfies the 119860condition then there exists a constant 119862 gt 0
independent of 119891 = (119891
1 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119871119901(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894 (119908119894) (6)
where ]= prod119898
119894=1119908119901119901119894
119894
Theorem B (see [9]) Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)
min1199011 119901
119898 = 1 and 119901 isin (0infin) with 1119901 = sum
119898
119896=11119901119896
and = (1199081 119908
119898) satisfies the 119860
condition then there
exists a constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such
that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894 (119908119894) (7)
where ]= prod119898
119894=1119908119901119901119894
119894
Let119898 ge 2 and let 0 lt 120572 lt 119898119899 For given 119891 = (119891
1 119891
119898)
the119898-linear fractional integral operator is defined by
119868120572(
119891) (119909) = 119868120572(1198911 119891
119898) (119909)
= int
(R119899)119898
1198911(1199101) sdot sdot sdot 119891
119898(119910119898)
1003816100381610038161003816(119909 minus 119910
1 119909 minus 119910
119898)1003816100381610038161003816
119898119899minus1205721198891199101sdot sdot sdot 119889119910119898
(8)
For the boundedness properties of multilinear fractionalintegrals on various function spaces we refer the reader to[10ndash16] In 2009 Moen [17] considered the weighted norminequalities for multilinear fractional integral operators andconstructed the class of multiple 119860
119902weights (see also [18])
TheoremC (see [17 18]) Let119898 ge 2 0 lt 120572 lt 119898119899 and let 119868120572be
an119898-linear fractional integral operator If1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896and 1119902 = 1119901 minus 120572119899 and = (119908
1 119908
119898)
satisfies the 119860119902
condition then there exists a constant 119862 gt 0
independent of 119891 = (119891
1 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894 (119908119901119894
119894) (9)
where ]= prod119898
119894=1119908119894
TheoremD (see [17 18]) Let119898 ge 2 0 lt 120572 lt 119898119899 and let 119868120572be
an119898-linear fractional integral operator If1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896and 1119902 = 1119901 minus 120572119899
and = (1199081 119908
119898) satisfies the 119860
119902condition then there
exists a constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such
that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894 (119908119901119894
119894) (10)
where ]= prod119898
119894=1119908119894
On the other hand the classical Morrey spaces L119901120582
were originally introduced by Morrey in [19] to study thelocal behavior of solutions to second-order elliptic partialdifferential equations For the boundedness of the Hardy-Littlewood maximal operator the fractional integral opera-tor and the Calderon-Zygmund singular integral operator onthese spaces we refer the reader to [20ndash22] For the propertiesand applications of classical Morrey spaces one can see [23ndash25] and the references therein
In 2009 Komori and Shirai [26] first defined the weightedMorrey spaces 119871119901120581(119908)which could be viewed as an extensionof weighted Lebesgue spaces and studied the boundedness ofthe above classical operators in Harmonic Analysis on theseweighted spaces Recently in [27ndash34] we have establishedthe continuity properties of some other operators and theircommutators on the weighted Morrey spaces 119871119901120581(119908)
The main purpose of this paper is to establish theboundedness properties of multilinear Calderon-Zygmundoperators and multilinear fractional integrals on productsof weighted Morrey spaces with multiple weights We nowformulate our main results as follows
Theorem 1 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin (1infin) and 119901 isin (0infin)
with 1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898) isin 119860
with
1199081 119908
119898isin 119860infin then for any 0 lt 120581 lt 1 there exists a
constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such that
10038171003817100381710038171003817119879(
119891)
10038171003817100381710038171003817119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (11)
where ]= prod119898
119894=1119908119901119901119894
119894
Theorem 2 Let 119898 ge 2 and 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)min119901
1 119901
119898 =
1 and 119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896 and =
(1199081 119908
119898) isin 119860
with 119908
1 119908
119898isin 119860infin then for any
0 lt 120581 lt 1 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (12)
where ]= prod119898
119894=1119908119901119901119894
119894
Theorem 3 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572
be an 119898-linear fractional integral operator If 1199011 119901
119898isin
(1infin) 1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899 and 1119902 =
sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin 119860
119902with
Journal of Function Spaces and Applications 3
1199081199021
1 119908
119902119898
119898isin 119860infin then for any 0 lt 120581 lt 119901119902 there exists a
constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (13)
where ]= prod119898
119894=1119908119894
Theorem 4 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
119860119902
with11990811990211 119908
119902119898
119898isin 119860infin then for any 0 lt 120581 lt 119901119902 there
exists a constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such
that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (14)
where ]= prod119898
119894=1119908119894
2 Notations and Definitions
The classical 119860119901
weight theory was first introduced byMuckenhoupt in the study of weighted 119871
119901 boundedness ofHardy-Littlewood maximal functions in [35] A weight119908 is anonnegative locally integrable function onR119899 119861 = 119861(119909
0 119903119861)
denotes the ball with the center 1199090and radius 119903
119861 For 1 lt 119901 lt
infin a weight function 119908 is said to belong to 119860119901if there is a
constant 119862 gt 0 such that for every ball 119861 sube R119899
(
1
|119861|
int
119861
119908 (119909) 119889119909)(
1
|119861|
int
119861
119908(119909)minus1(119901minus1)
119889119909)
119901minus1
le 119862 (15)
where |119861| denotes the Lebesgue measure of 119861 For the case119901 = 1 119908 isin 119860
1 if there is a constant 119862 gt 0 such that for every
ball 119861 sube R119899
1
|119861|
int
119861
119908 (119909) 119889119909 le 119862 sdot ess inf119909isin119861
119908 (119909) (16)
A weight function 119908 isin 119860infinif it satisfies the 119860
119901condition for
some 1 lt 119901 lt infin We also need another weight class 119860119901119902
introduced by Muckenhoupt andWheeden in [36] A weightfunction 119908 belongs to 119860
119901119902for 1 lt 119901 lt 119902 lt infin if there is a
constant 119862 gt 0 such that for every ball 119861 sube R119899
(
1
|119861|
int
119861
119908(119909)119902
119889119909)
1119902
(
1
|119861|
int
119861
119908(119909)minus1199011015840
119889119909)
11199011015840
le 119862 (17)
When 119901 = 1 119908 is in the class 1198601119902
with 1 lt 119902 lt infin if there isa constant 119862 gt 0 such that for every ball 119861 sube R119899
(
1
|119861|
int
119861
119908(119909)119902
119889119909)
1119902
(ess sup119909isin119861
1
119908 (119909)
) le 119862 (18)
Now let us recall the definitions of multiple weightsFor 119898 exponents 119901
1 119901
119898 we will write for the vector
= (1199011 119901
119898) Let 119901
1 119901
119898isin [1infin) and let 119901 isin
(0infin) with 1119901 = sum119898
119896=11119901119896 Given = (119908
1 119908
119898) set
]= prod119898
119894=1119908119901119901119894
119894 We say that satisfies the 119860
condition if it
satisfies
sup119861
(
1
|119861|
int
119861
](119909) 119889119909)
1119901 119898
prod
119894=1
(
1
|119861|
int
119861
119908119894(119909)1minus1199011015840
119894119889119909)
11199011015840
119894
lt infin
(19)
When 119901119894= 1 ((1|119861|) int
119861
119908119894(119909)1minus1199011015840
119894119889119909)11199011015840
119894 is understood as(inf119909isin119861
119908119894(119909))minus1
Let 1199011 119901
119898isin [1infin) let 1119901 = sum
119898
119896=11119901119896 and let 119902 gt
0 Given = (1199081 119908
119898) set ]
= prod119898
119894=1119908119894 We say that
satisfies the 119860119902
condition if it satisfies
sup119861
(
1
|119861|
int
119861
](119909)119902
119889119909)
1119902
times
119898
prod
119894=1
(
1
|119861|
int
119861
119908119894(119909)minus1199011015840
119894119889119909)
11199011015840
119894
lt infin
(20)
When 119901119894= 1 ((1|119861|) int
119861
119908119894(119909)minus1199011015840
119894119889119909)11199011015840
119894 is understood as(inf119909isin119861
119908119894(119909))minus1
Given a ball 119861 and 120582 gt 0 120582119861 denotes the ball with thesame center as 119861whose radius is 120582 times that of 119861 For a givenweight function119908 and a measurable set 119864 we also denote theLebesgue measure of 119864 by |119864| and the weighted measure of 119864by 119908(119864) where 119908(119864) = int
119864
119908(119909)119889119909
Lemma 5 (see [37]) Let 119908 isin 119860119901with 1 le 119901 lt infin Then for
any ball 119861 there exists an absolute constant 119862 gt 0 such that
119908 (2119861) le 119862119908 (119861) (21)
Lemma 6 (see [38]) Let 119908 isin 119860infin Then for all balls 119861 sube R119899
the following reverse Jensen inequality holds
int
119861
119908 (119909) 119889119909 le 119862 |119861| sdot exp( 1
|119861|
int
119861
log 119908 (119909) 119889119909) (22)
Lemma 7 (see [37]) Let119908 isin 119860infin Then for all balls 119861 and all
measurable subsets 119864 of 119861 there exists 120575 gt 0 such that
119908 (119864)
119908 (119861)
le 119862(
|119864|
|119861|
)
120575
(23)
Lemma 8 (see [9]) Let 1199011 119901
119898isin [1infin) and let 1119901 =
sum119898
119896=11119901119896 Then = (119908
1 119908
119898) isin 119860if and only if
]isin 119860119898119901
1199081minus1199011015840
119894
119894isin 1198601198981199011015840
119894
119894 = 1 119898
(24)
where ]= prod119898
119894=1119908119901119901119894
119894and the condition 1199081minus119901
1015840
119894
119894isin 1198601198981199011015840
119894
in thecase 119901
119894= 1 is understood as 1199081119898
119894isin 1198601
4 Journal of Function Spaces and Applications
Lemma 9 (see [17 18]) Let 0 lt 120572 lt 119898119899 and 1199011 119901
119898isin
[1infin) let 1119901 = sum119898
119896=11119901119896 and let 1119902 = 1119901 minus 120572119899 If =
(1199081 119908
119898) isin 119860119902
then
(])119902
isin 119860119898119902
119908minus1199011015840
119894
119894isin 1198601198981199011015840
119894
119894 = 1 119898
(25)
where ]= prod119898
119894=1119908119894
Given a weight function 119908 on R119899 for 0 lt 119901 lt infin theweighted Lebesgue space 119871119901(119908) is defined as the set of allfunctions 119891 such that
10038171003817100381710038171198911003817100381710038171003817119871119901(119908)
= (int
R119899
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119908 (119909) 119889119909)
1119901
lt infin (26)
We also denote by119882119871119901
(119908) the weighted weak space consist-ing of all measurable functions 119891 such that
10038171003817100381710038171198911003817100381710038171003817119882119871119901(119908)
= sup120582gt0
120582 sdot 119908(119909 isin R119899
1003816100381610038161003816119891 (119909)
1003816100381610038161003816gt 120582)1119901
lt infin
(27)
In 2009 Komori and Shirai [26] first defined the weightedMorrey spaces 119871119901120581(119908) for 1 le 119901 lt infin In order to deal withthe multilinear case 119898 ge 2 we will define 119871119901120581(119908) for all 0 lt119901 lt infin
Definition 10 Let 0 lt 119901 lt infin let 0 lt 120581 lt 1 and let 119908 bea weight function on R119899 Then the weighted Morrey space isdefined by
119871119901120581
(119908) = 119891 isin 119871119901
loc (119908) 10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
lt infin (28)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
= sup119861
(
1
119908(119861)120581int
119861
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119908 (119909) 119889119909)
1119901
(29)
and the supremum is taken over all balls 119861 in R119899
Definition 11 Let 0 lt 119901 lt infin let 0 lt 120581 lt 1 and let 119908 be aweight function onR119899Then theweightedweakMorrey spaceis defined by
119882119871119901120581
(119908) = 119891 measurable 10038171003817100381710038171198911003817100381710038171003817119882119871119901120581(119908)
lt infin (30)
where
10038171003817100381710038171198911003817100381710038171003817119882119871119901120581(119908)
= sup119861
sup120582gt0
1
119908(119861)120581119901
120582 sdot 119908(119909 isin 119861 1003816100381610038161003816119891 (119909)
1003816100381610038161003816gt 120582)1119901
(31)
Furthermore in order to deal with the fractional ordercase we need to consider the weighted Morrey spaces withtwo weights
Definition 12 Let 0 lt 119901 lt infin and 0 lt 120581 lt 1 Then for twoweights 119906 and V the weighted Morrey space is defined by
119871119901120581
(119906 V) = 119891 isin 119871119901
loc (119906) 10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V)
lt infin (32)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V)
= sup119861
(
1
V(119861)120581int
119861
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119906 (119909) 119889119909)
1119901
(33)
Throughout this paper we will use 119862 to denote a positiveconstant which is independent of the main parameters andnot necessarily the same at each occurrence Moreover wewill denote the conjugate exponent of119901 gt 1 by1199011015840 = 119901(119901minus1)
3 Proofs of Theorems 1 and 2
Before proving the main theorems of this section we need toestablish the following lemma
Lemma 13 Let 119898 ge 2 let 1199011 119901
119898isin [1infin) and let 119901 isin
(0infin) with 1119901 = sum119898
119896=11119901119896 Assume that 119908
1 119908
119898isin 119860infin
and ]= prod119898
119894=1119908119901119901119894
119894 then for any ball119861 there exists a constant
119862 gt 0 such that
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862int
119861
](119909) 119889119909 (34)
Proof Since 1199081 119908
119898isin 119860infin then by using Lemma 6 we
have
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862
119898
prod
119894=1
(|119861| sdot exp( 1
|119861|
int
119861
log119908119894(119909) 119889119909))
119901119901119894
= 119862
119898
prod
119894=1
(|119861|119901119901119894sdot exp( 1
|119861|
int
119861
log119908119894(119909)119901119901119894119889119909))
= 119862 sdot (|119861|)sum119898
119894=1119901119901119894sdot exp(
119898
sum
119894=1
1
|119861|
int
119861
log119908119894(119909)119901119901119894119889119909)
(35)
Note that sum119898119894=1
119901119901119894= 1 and ]
(119909) = prod
119898
119894=1119908119894(119909)119901119901119894 Then by
Jensen inequality we obtain
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862 sdot |119861| sdot exp( 1
|119861|
int
119861
log ](119909) 119889119909)
le 119862int
119861
](119909) 119889119909
(36)
We are done
Journal of Function Spaces and Applications 5
Proof of Theorem 1 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
letting 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898 and
1205942119861
denotes the characteristic function of 2119861 then we write
119898
prod
119894=1
119891119894(119910119894) =
119898
prod
119894=1
(1198910
119894(119910119894) + 119891infin
119894(119910119894))
= sum
1205721120572119898isin0infin
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
=
119898
prod
119894=1
1198910
119894(119910119894) +sum
1015840
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
(37)
where each term of sum1015840 contains at least one 120572119894= 0 Since 119879 is
an119898-linear operator then we have
1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911 119891
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
le
1
](119861)120581119901
(int
119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119901
](119909)119889119909)
1119901
+sum
1015840 1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
= 1198680
+sum
1015840
1198681205721120572119898
(38)
In view of Lemma 8 we have that ]
isin 119860119898119901 Applying
Theorem A and Lemmas 13 and 5 we get
1198680
le 119862 sdot
1
](119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
prod119898
119894=1119908119894(2119861)120581119901119894
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
](2119861)120581119901
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(39)
For the other terms let us first consider the case when 1205721=
sdot sdot sdot = 120572119898= infin By the size condition for any 119909 isin 119861 we obtain
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot+
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(40)
where we have used the notation 119864119898
= 119864 times sdot sdot sdot times 119864Furthermore by using Holderrsquos inequality the multiple 119860
condition and Lemma 13 we deduce that
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119901+sum119898
119894=1(1minus1119901
119894)
](2119895+1119861)1119901
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)119908119894(2119895+1
119861)
120581119901119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
prod119898
119894=1119908119894(2119895+1
119861)
120581119901119894
](2119895+1119861)1119901
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(41)
Since ]isin 119860119898119901
sub 119860infin then it follows directly fromLemma 7
that
](119861)
](2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575
(42)
Hence
119868infininfin
le ](119861)(1minus120581)119901 1003816
100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
|2119895+1119861|
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(43)
where the last inequality holds since 0 lt 120581 lt 1 and 120575 gt 0We now consider the case where exactly ℓ of the 120572
119894are infin
for some 1 le ℓ lt 119898 We only give the arguments for oneof these cases The rest are similar and can easily be obtainedfrom the arguments below by permuting the indices Usingthe size condition again we deduce that for any 119909 isin 119861
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899
times 1198891199101sdot sdot sdot 119889119910119898
6 Journal of Function Spaces and Applications
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(44)
and we arrive at the expression considered in the previouscase So for any 119909 isin 119861 we also have
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(45)
Therefore by the inequality (42) and the above pointwiseinequality we have
1198681205721120572119898
le ](119861)(1minus120581)119901
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(46)
Combining the above estimates and then taking the supre-mum over all balls 119861 sube R119899 we complete the proof ofTheorem 1
Proof of Theorem 2 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
decomposing 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898
then for any given 120582 gt 0 we can write
](119909 isin 119861
1003816100381610038161003816119879 (1198911 119891
119898)1003816100381610038161003816gt 120582)1119901
le ](119909 isin 119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119901
+sum
1015840
](119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119901
= 1198680
lowast+sum
1015840
1198681205721120572119898
lowast
(47)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 8
again we know that ]isin 119860119898119901
with 1 le 119898119901 lt infin ApplyingTheorem B and Lemmas 13 and 5 we have
1198680
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119894(2119861)120581119901119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(48)
In the proof of Theorem 1 we have already showed thefollowing pointwise estimate (see (40) and (44)) Consider
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(49)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1
Using Holderrsquos inequality the multiple 119860condition and
Lemma 13 we obtain
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894
times ( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(50)
Observe that ]isin 119860119898119901
with 1 le 119898119901 lt infin Thus it followsfrom the inequality (42) that for any 119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
Journal of Function Spaces and Applications 7
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(51)
If 119909 isin 119861 |119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198681205721120572119898
lowastle
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(52)
holds trivially Now if instead we suppose that 119909 isin 119861
|119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (51) we have
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(53)
which is equivalent to
](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (54)
Therefore
1198681205721120572119898
lowastle ](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (55)
Summing up all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we completethe proof of Theorem 2
By usingHolderrsquos inequality it is easy to check that if each119908119894is in 119860
119901119894
then
119898
prod
119894=1
119860119901119894
sub 119860 (56)
and this inclusion is strict (see [9]) Thus as direct conse-quences of Theorems 1 and 2 we immediately obtain thefollowing
Corollary 14 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If119901
1 119901
119898isin (1infin) and119901 isin (0infin)with
1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898) isin prod
119898
119894=1119860119901119894
then forany 0 lt 120581 lt 1 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (57)
where ]= prod119898
119894=1119908119901119901119894
119894
Corollary 15 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)min119901
1 119901
119898 =
1 and 119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896 and =
(1199081 119908
119898) isin prod
119898
119894=1119860119901119894
then for any 0 lt 120581 lt 1 there existsa constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (58)
where ]= prod119898
119894=1119908119901119901119894
119894
4 Proofs of Theorems 3 and 4
Following along the same lines as those of Lemma 13 we canalso show the following result which plays an important rolein our proofs of Theorems 3 and 4
Lemma 16 Let 119898 ge 2 1199021 119902
119898isin [1infin) and 119902 isin (0infin)
with 1119902 = sum119898
119896=11119902119896 Assume that 1199081199021
1 119908
119902119898
119898isin 119860infin
and]= prod119898
119894=1119908119894 then for any ball 119861 there exists a constant119862 gt 0
such that119898
prod
119894=1
(int
119861
119908119902119894
119894(119909) 119889119909)
119902119902119894
le 119862int
119861
](119909)119902
119889119909 (59)
Proof of Theorem 3 Arguing as in the proof ofTheorem 1 fixa ball119861 = 119861(119909
0 119903119861) sube R119899 and decompose119891
119894= 1198910
119894+119891infin
119894 where
1198910
119894= 1198911198941205942119861 119894 = 1 119898 Since 119868
120572is an119898-linear operator then
we have1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911 119891
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
le
1
]119902
(119861)120581119901
(int
119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
+sum
1015840 1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
= 1198690
+sum
1015840
1198691205721120572119898
(60)
where each term of sum1015840 contains at least one 120572119894= 0 In view of
Lemma 9 we can see that (])119902
isin 119860119898119902 UsingTheoremC and
Lemmas 16 and 5 we get
1198690
le 119862 sdot
1
]119902
(119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
]119902
(119861)120581119901
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
(prod119898
119894=1119908119902119894
119894(2119861)119902119902119894)
120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
]119902
(2119861)120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(61)
8 Journal of Function Spaces and Applications
For the other terms let us first deal with the case when 1205721=
sdot sdot sdot = 120572119898= infin By the definition of 119868
120572 for any 119909 isin 119861 we
obtain
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
= int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
1198891199101sdot sdot sdot119889119910119898
=
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899minus120572119898
119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(62)
Moreover by using Holderrsquos inequality the multiple 119860119902
condition and Lemma 16 we deduce that
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times (int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119902+sum119898
119894=1(1minus1119901
119894)
]119902
(2119895+1119861)1119902
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)119908119902119894
119894(2119895+1
119861)
120581119902119901119902119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
[
[
[
[
(prod119898
119894=1119908119902119894
119894(2119895+1
119861)
119902119902119894
)
120581119901
]119902
(2119895+1119861)1119902
]
]
]
]
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(63)
Since (])119902
isin 119860119898119902
sub 119860infin then it follows immediately from
Lemma 7 that
]119902
(119861)
]119902
(2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840
(64)
Hence
119869infininfin
le ]119902
(119861)1119902minus120581119901 1003816
100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(65)
where in the last inequality we have used the fact that 0 lt 120581 lt119901119902 and 1205751015840 gt 0 We now consider the case where exactly ℓ ofthe 120572119894areinfin for some 1 le ℓ lt 119898 We only give the arguments
for one of these cases The rest are similar and can easily beobtained from the arguments belowby permuting the indicesUsing the definition of 119868
120572again we can see that for any 119909 isin 119861
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
= int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(66)
and we arrive at the expression considered in the previouscase Thus for any 119909 isin 119861 we also have
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(67)
Journal of Function Spaces and Applications 9
Therefore by the inequality (64) and the above pointwiseinequality we obtain
1198691205721120572119898
le ]119902
(119861)1119902minus120581119901
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(68)
Summarizing the estimates derived above and then takingthe supremum over all balls 119861 sube R119899 we finish the proof ofTheorem 3
Proof of Theorem 4 As before fix a ball 119861 = 119861(1199090 119903119861) sube R119899
and split119891119894into119891
119894= 1198910
119894+119891infin
119894 where1198910
119894= 1198911198941205942119861 119894 = 1 119898
Then for each fixed 120582 gt 0 we can write
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911 119891
119898)1003816100381610038161003816gt 120582)1119902
le ]119902
(119909 isin 119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119902
+sum
1015840
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119902
= 1198690
lowast+sum
1015840
1198691205721120572119898
lowast
(69)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 9
again we know that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin UsingTheorem D and Lemmas 16 and 5 we have
1198690
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(70)
In the proof of Theorem 3 we have already proved thefollowing pointwise estimate (see (62) and (66)) Consider
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(71)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1 By
using Holderrsquos inequality the multiple 119860119902
condition andLemma 16 we obtain1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(72)
Note that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin Hence it followsfrom the inequality (64) that for any 119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(73)
If 119909 isin 119861 |119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198691205721120572119898
lowastle
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(74)
holds trivially Now if instead we assume that 119909 isin 119861
|119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (73) we get
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(75)
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces and Applications
refined multilinear maximal function was defined and usedin [9] to characterize the class of multiple 119860
weights
Theorem A (see [9]) Let 119898 ge 2 and 119879 be an 119898-linearCalderon-Zygmund operator If 119901
1 119901
119898isin (1infin) and
119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898)
satisfies the 119860condition then there exists a constant 119862 gt 0
independent of 119891 = (119891
1 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119871119901(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894 (119908119894) (6)
where ]= prod119898
119894=1119908119901119901119894
119894
Theorem B (see [9]) Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)
min1199011 119901
119898 = 1 and 119901 isin (0infin) with 1119901 = sum
119898
119896=11119901119896
and = (1199081 119908
119898) satisfies the 119860
condition then there
exists a constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such
that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894 (119908119894) (7)
where ]= prod119898
119894=1119908119901119901119894
119894
Let119898 ge 2 and let 0 lt 120572 lt 119898119899 For given 119891 = (119891
1 119891
119898)
the119898-linear fractional integral operator is defined by
119868120572(
119891) (119909) = 119868120572(1198911 119891
119898) (119909)
= int
(R119899)119898
1198911(1199101) sdot sdot sdot 119891
119898(119910119898)
1003816100381610038161003816(119909 minus 119910
1 119909 minus 119910
119898)1003816100381610038161003816
119898119899minus1205721198891199101sdot sdot sdot 119889119910119898
(8)
For the boundedness properties of multilinear fractionalintegrals on various function spaces we refer the reader to[10ndash16] In 2009 Moen [17] considered the weighted norminequalities for multilinear fractional integral operators andconstructed the class of multiple 119860
119902weights (see also [18])
TheoremC (see [17 18]) Let119898 ge 2 0 lt 120572 lt 119898119899 and let 119868120572be
an119898-linear fractional integral operator If1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896and 1119902 = 1119901 minus 120572119899 and = (119908
1 119908
119898)
satisfies the 119860119902
condition then there exists a constant 119862 gt 0
independent of 119891 = (119891
1 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894 (119908119901119894
119894) (9)
where ]= prod119898
119894=1119908119894
TheoremD (see [17 18]) Let119898 ge 2 0 lt 120572 lt 119898119899 and let 119868120572be
an119898-linear fractional integral operator If1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896and 1119902 = 1119901 minus 120572119899
and = (1199081 119908
119898) satisfies the 119860
119902condition then there
exists a constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such
that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894 (119908119901119894
119894) (10)
where ]= prod119898
119894=1119908119894
On the other hand the classical Morrey spaces L119901120582
were originally introduced by Morrey in [19] to study thelocal behavior of solutions to second-order elliptic partialdifferential equations For the boundedness of the Hardy-Littlewood maximal operator the fractional integral opera-tor and the Calderon-Zygmund singular integral operator onthese spaces we refer the reader to [20ndash22] For the propertiesand applications of classical Morrey spaces one can see [23ndash25] and the references therein
In 2009 Komori and Shirai [26] first defined the weightedMorrey spaces 119871119901120581(119908)which could be viewed as an extensionof weighted Lebesgue spaces and studied the boundedness ofthe above classical operators in Harmonic Analysis on theseweighted spaces Recently in [27ndash34] we have establishedthe continuity properties of some other operators and theircommutators on the weighted Morrey spaces 119871119901120581(119908)
The main purpose of this paper is to establish theboundedness properties of multilinear Calderon-Zygmundoperators and multilinear fractional integrals on productsof weighted Morrey spaces with multiple weights We nowformulate our main results as follows
Theorem 1 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin (1infin) and 119901 isin (0infin)
with 1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898) isin 119860
with
1199081 119908
119898isin 119860infin then for any 0 lt 120581 lt 1 there exists a
constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such that
10038171003817100381710038171003817119879(
119891)
10038171003817100381710038171003817119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (11)
where ]= prod119898
119894=1119908119901119901119894
119894
Theorem 2 Let 119898 ge 2 and 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)min119901
1 119901
119898 =
1 and 119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896 and =
(1199081 119908
119898) isin 119860
with 119908
1 119908
119898isin 119860infin then for any
0 lt 120581 lt 1 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (12)
where ]= prod119898
119894=1119908119901119901119894
119894
Theorem 3 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572
be an 119898-linear fractional integral operator If 1199011 119901
119898isin
(1infin) 1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899 and 1119902 =
sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin 119860
119902with
Journal of Function Spaces and Applications 3
1199081199021
1 119908
119902119898
119898isin 119860infin then for any 0 lt 120581 lt 119901119902 there exists a
constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (13)
where ]= prod119898
119894=1119908119894
Theorem 4 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
119860119902
with11990811990211 119908
119902119898
119898isin 119860infin then for any 0 lt 120581 lt 119901119902 there
exists a constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such
that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (14)
where ]= prod119898
119894=1119908119894
2 Notations and Definitions
The classical 119860119901
weight theory was first introduced byMuckenhoupt in the study of weighted 119871
119901 boundedness ofHardy-Littlewood maximal functions in [35] A weight119908 is anonnegative locally integrable function onR119899 119861 = 119861(119909
0 119903119861)
denotes the ball with the center 1199090and radius 119903
119861 For 1 lt 119901 lt
infin a weight function 119908 is said to belong to 119860119901if there is a
constant 119862 gt 0 such that for every ball 119861 sube R119899
(
1
|119861|
int
119861
119908 (119909) 119889119909)(
1
|119861|
int
119861
119908(119909)minus1(119901minus1)
119889119909)
119901minus1
le 119862 (15)
where |119861| denotes the Lebesgue measure of 119861 For the case119901 = 1 119908 isin 119860
1 if there is a constant 119862 gt 0 such that for every
ball 119861 sube R119899
1
|119861|
int
119861
119908 (119909) 119889119909 le 119862 sdot ess inf119909isin119861
119908 (119909) (16)
A weight function 119908 isin 119860infinif it satisfies the 119860
119901condition for
some 1 lt 119901 lt infin We also need another weight class 119860119901119902
introduced by Muckenhoupt andWheeden in [36] A weightfunction 119908 belongs to 119860
119901119902for 1 lt 119901 lt 119902 lt infin if there is a
constant 119862 gt 0 such that for every ball 119861 sube R119899
(
1
|119861|
int
119861
119908(119909)119902
119889119909)
1119902
(
1
|119861|
int
119861
119908(119909)minus1199011015840
119889119909)
11199011015840
le 119862 (17)
When 119901 = 1 119908 is in the class 1198601119902
with 1 lt 119902 lt infin if there isa constant 119862 gt 0 such that for every ball 119861 sube R119899
(
1
|119861|
int
119861
119908(119909)119902
119889119909)
1119902
(ess sup119909isin119861
1
119908 (119909)
) le 119862 (18)
Now let us recall the definitions of multiple weightsFor 119898 exponents 119901
1 119901
119898 we will write for the vector
= (1199011 119901
119898) Let 119901
1 119901
119898isin [1infin) and let 119901 isin
(0infin) with 1119901 = sum119898
119896=11119901119896 Given = (119908
1 119908
119898) set
]= prod119898
119894=1119908119901119901119894
119894 We say that satisfies the 119860
condition if it
satisfies
sup119861
(
1
|119861|
int
119861
](119909) 119889119909)
1119901 119898
prod
119894=1
(
1
|119861|
int
119861
119908119894(119909)1minus1199011015840
119894119889119909)
11199011015840
119894
lt infin
(19)
When 119901119894= 1 ((1|119861|) int
119861
119908119894(119909)1minus1199011015840
119894119889119909)11199011015840
119894 is understood as(inf119909isin119861
119908119894(119909))minus1
Let 1199011 119901
119898isin [1infin) let 1119901 = sum
119898
119896=11119901119896 and let 119902 gt
0 Given = (1199081 119908
119898) set ]
= prod119898
119894=1119908119894 We say that
satisfies the 119860119902
condition if it satisfies
sup119861
(
1
|119861|
int
119861
](119909)119902
119889119909)
1119902
times
119898
prod
119894=1
(
1
|119861|
int
119861
119908119894(119909)minus1199011015840
119894119889119909)
11199011015840
119894
lt infin
(20)
When 119901119894= 1 ((1|119861|) int
119861
119908119894(119909)minus1199011015840
119894119889119909)11199011015840
119894 is understood as(inf119909isin119861
119908119894(119909))minus1
Given a ball 119861 and 120582 gt 0 120582119861 denotes the ball with thesame center as 119861whose radius is 120582 times that of 119861 For a givenweight function119908 and a measurable set 119864 we also denote theLebesgue measure of 119864 by |119864| and the weighted measure of 119864by 119908(119864) where 119908(119864) = int
119864
119908(119909)119889119909
Lemma 5 (see [37]) Let 119908 isin 119860119901with 1 le 119901 lt infin Then for
any ball 119861 there exists an absolute constant 119862 gt 0 such that
119908 (2119861) le 119862119908 (119861) (21)
Lemma 6 (see [38]) Let 119908 isin 119860infin Then for all balls 119861 sube R119899
the following reverse Jensen inequality holds
int
119861
119908 (119909) 119889119909 le 119862 |119861| sdot exp( 1
|119861|
int
119861
log 119908 (119909) 119889119909) (22)
Lemma 7 (see [37]) Let119908 isin 119860infin Then for all balls 119861 and all
measurable subsets 119864 of 119861 there exists 120575 gt 0 such that
119908 (119864)
119908 (119861)
le 119862(
|119864|
|119861|
)
120575
(23)
Lemma 8 (see [9]) Let 1199011 119901
119898isin [1infin) and let 1119901 =
sum119898
119896=11119901119896 Then = (119908
1 119908
119898) isin 119860if and only if
]isin 119860119898119901
1199081minus1199011015840
119894
119894isin 1198601198981199011015840
119894
119894 = 1 119898
(24)
where ]= prod119898
119894=1119908119901119901119894
119894and the condition 1199081minus119901
1015840
119894
119894isin 1198601198981199011015840
119894
in thecase 119901
119894= 1 is understood as 1199081119898
119894isin 1198601
4 Journal of Function Spaces and Applications
Lemma 9 (see [17 18]) Let 0 lt 120572 lt 119898119899 and 1199011 119901
119898isin
[1infin) let 1119901 = sum119898
119896=11119901119896 and let 1119902 = 1119901 minus 120572119899 If =
(1199081 119908
119898) isin 119860119902
then
(])119902
isin 119860119898119902
119908minus1199011015840
119894
119894isin 1198601198981199011015840
119894
119894 = 1 119898
(25)
where ]= prod119898
119894=1119908119894
Given a weight function 119908 on R119899 for 0 lt 119901 lt infin theweighted Lebesgue space 119871119901(119908) is defined as the set of allfunctions 119891 such that
10038171003817100381710038171198911003817100381710038171003817119871119901(119908)
= (int
R119899
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119908 (119909) 119889119909)
1119901
lt infin (26)
We also denote by119882119871119901
(119908) the weighted weak space consist-ing of all measurable functions 119891 such that
10038171003817100381710038171198911003817100381710038171003817119882119871119901(119908)
= sup120582gt0
120582 sdot 119908(119909 isin R119899
1003816100381610038161003816119891 (119909)
1003816100381610038161003816gt 120582)1119901
lt infin
(27)
In 2009 Komori and Shirai [26] first defined the weightedMorrey spaces 119871119901120581(119908) for 1 le 119901 lt infin In order to deal withthe multilinear case 119898 ge 2 we will define 119871119901120581(119908) for all 0 lt119901 lt infin
Definition 10 Let 0 lt 119901 lt infin let 0 lt 120581 lt 1 and let 119908 bea weight function on R119899 Then the weighted Morrey space isdefined by
119871119901120581
(119908) = 119891 isin 119871119901
loc (119908) 10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
lt infin (28)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
= sup119861
(
1
119908(119861)120581int
119861
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119908 (119909) 119889119909)
1119901
(29)
and the supremum is taken over all balls 119861 in R119899
Definition 11 Let 0 lt 119901 lt infin let 0 lt 120581 lt 1 and let 119908 be aweight function onR119899Then theweightedweakMorrey spaceis defined by
119882119871119901120581
(119908) = 119891 measurable 10038171003817100381710038171198911003817100381710038171003817119882119871119901120581(119908)
lt infin (30)
where
10038171003817100381710038171198911003817100381710038171003817119882119871119901120581(119908)
= sup119861
sup120582gt0
1
119908(119861)120581119901
120582 sdot 119908(119909 isin 119861 1003816100381610038161003816119891 (119909)
1003816100381610038161003816gt 120582)1119901
(31)
Furthermore in order to deal with the fractional ordercase we need to consider the weighted Morrey spaces withtwo weights
Definition 12 Let 0 lt 119901 lt infin and 0 lt 120581 lt 1 Then for twoweights 119906 and V the weighted Morrey space is defined by
119871119901120581
(119906 V) = 119891 isin 119871119901
loc (119906) 10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V)
lt infin (32)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V)
= sup119861
(
1
V(119861)120581int
119861
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119906 (119909) 119889119909)
1119901
(33)
Throughout this paper we will use 119862 to denote a positiveconstant which is independent of the main parameters andnot necessarily the same at each occurrence Moreover wewill denote the conjugate exponent of119901 gt 1 by1199011015840 = 119901(119901minus1)
3 Proofs of Theorems 1 and 2
Before proving the main theorems of this section we need toestablish the following lemma
Lemma 13 Let 119898 ge 2 let 1199011 119901
119898isin [1infin) and let 119901 isin
(0infin) with 1119901 = sum119898
119896=11119901119896 Assume that 119908
1 119908
119898isin 119860infin
and ]= prod119898
119894=1119908119901119901119894
119894 then for any ball119861 there exists a constant
119862 gt 0 such that
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862int
119861
](119909) 119889119909 (34)
Proof Since 1199081 119908
119898isin 119860infin then by using Lemma 6 we
have
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862
119898
prod
119894=1
(|119861| sdot exp( 1
|119861|
int
119861
log119908119894(119909) 119889119909))
119901119901119894
= 119862
119898
prod
119894=1
(|119861|119901119901119894sdot exp( 1
|119861|
int
119861
log119908119894(119909)119901119901119894119889119909))
= 119862 sdot (|119861|)sum119898
119894=1119901119901119894sdot exp(
119898
sum
119894=1
1
|119861|
int
119861
log119908119894(119909)119901119901119894119889119909)
(35)
Note that sum119898119894=1
119901119901119894= 1 and ]
(119909) = prod
119898
119894=1119908119894(119909)119901119901119894 Then by
Jensen inequality we obtain
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862 sdot |119861| sdot exp( 1
|119861|
int
119861
log ](119909) 119889119909)
le 119862int
119861
](119909) 119889119909
(36)
We are done
Journal of Function Spaces and Applications 5
Proof of Theorem 1 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
letting 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898 and
1205942119861
denotes the characteristic function of 2119861 then we write
119898
prod
119894=1
119891119894(119910119894) =
119898
prod
119894=1
(1198910
119894(119910119894) + 119891infin
119894(119910119894))
= sum
1205721120572119898isin0infin
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
=
119898
prod
119894=1
1198910
119894(119910119894) +sum
1015840
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
(37)
where each term of sum1015840 contains at least one 120572119894= 0 Since 119879 is
an119898-linear operator then we have
1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911 119891
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
le
1
](119861)120581119901
(int
119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119901
](119909)119889119909)
1119901
+sum
1015840 1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
= 1198680
+sum
1015840
1198681205721120572119898
(38)
In view of Lemma 8 we have that ]
isin 119860119898119901 Applying
Theorem A and Lemmas 13 and 5 we get
1198680
le 119862 sdot
1
](119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
prod119898
119894=1119908119894(2119861)120581119901119894
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
](2119861)120581119901
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(39)
For the other terms let us first consider the case when 1205721=
sdot sdot sdot = 120572119898= infin By the size condition for any 119909 isin 119861 we obtain
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot+
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(40)
where we have used the notation 119864119898
= 119864 times sdot sdot sdot times 119864Furthermore by using Holderrsquos inequality the multiple 119860
condition and Lemma 13 we deduce that
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119901+sum119898
119894=1(1minus1119901
119894)
](2119895+1119861)1119901
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)119908119894(2119895+1
119861)
120581119901119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
prod119898
119894=1119908119894(2119895+1
119861)
120581119901119894
](2119895+1119861)1119901
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(41)
Since ]isin 119860119898119901
sub 119860infin then it follows directly fromLemma 7
that
](119861)
](2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575
(42)
Hence
119868infininfin
le ](119861)(1minus120581)119901 1003816
100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
|2119895+1119861|
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(43)
where the last inequality holds since 0 lt 120581 lt 1 and 120575 gt 0We now consider the case where exactly ℓ of the 120572
119894are infin
for some 1 le ℓ lt 119898 We only give the arguments for oneof these cases The rest are similar and can easily be obtainedfrom the arguments below by permuting the indices Usingthe size condition again we deduce that for any 119909 isin 119861
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899
times 1198891199101sdot sdot sdot 119889119910119898
6 Journal of Function Spaces and Applications
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(44)
and we arrive at the expression considered in the previouscase So for any 119909 isin 119861 we also have
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(45)
Therefore by the inequality (42) and the above pointwiseinequality we have
1198681205721120572119898
le ](119861)(1minus120581)119901
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(46)
Combining the above estimates and then taking the supre-mum over all balls 119861 sube R119899 we complete the proof ofTheorem 1
Proof of Theorem 2 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
decomposing 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898
then for any given 120582 gt 0 we can write
](119909 isin 119861
1003816100381610038161003816119879 (1198911 119891
119898)1003816100381610038161003816gt 120582)1119901
le ](119909 isin 119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119901
+sum
1015840
](119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119901
= 1198680
lowast+sum
1015840
1198681205721120572119898
lowast
(47)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 8
again we know that ]isin 119860119898119901
with 1 le 119898119901 lt infin ApplyingTheorem B and Lemmas 13 and 5 we have
1198680
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119894(2119861)120581119901119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(48)
In the proof of Theorem 1 we have already showed thefollowing pointwise estimate (see (40) and (44)) Consider
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(49)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1
Using Holderrsquos inequality the multiple 119860condition and
Lemma 13 we obtain
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894
times ( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(50)
Observe that ]isin 119860119898119901
with 1 le 119898119901 lt infin Thus it followsfrom the inequality (42) that for any 119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
Journal of Function Spaces and Applications 7
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(51)
If 119909 isin 119861 |119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198681205721120572119898
lowastle
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(52)
holds trivially Now if instead we suppose that 119909 isin 119861
|119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (51) we have
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(53)
which is equivalent to
](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (54)
Therefore
1198681205721120572119898
lowastle ](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (55)
Summing up all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we completethe proof of Theorem 2
By usingHolderrsquos inequality it is easy to check that if each119908119894is in 119860
119901119894
then
119898
prod
119894=1
119860119901119894
sub 119860 (56)
and this inclusion is strict (see [9]) Thus as direct conse-quences of Theorems 1 and 2 we immediately obtain thefollowing
Corollary 14 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If119901
1 119901
119898isin (1infin) and119901 isin (0infin)with
1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898) isin prod
119898
119894=1119860119901119894
then forany 0 lt 120581 lt 1 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (57)
where ]= prod119898
119894=1119908119901119901119894
119894
Corollary 15 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)min119901
1 119901
119898 =
1 and 119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896 and =
(1199081 119908
119898) isin prod
119898
119894=1119860119901119894
then for any 0 lt 120581 lt 1 there existsa constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (58)
where ]= prod119898
119894=1119908119901119901119894
119894
4 Proofs of Theorems 3 and 4
Following along the same lines as those of Lemma 13 we canalso show the following result which plays an important rolein our proofs of Theorems 3 and 4
Lemma 16 Let 119898 ge 2 1199021 119902
119898isin [1infin) and 119902 isin (0infin)
with 1119902 = sum119898
119896=11119902119896 Assume that 1199081199021
1 119908
119902119898
119898isin 119860infin
and]= prod119898
119894=1119908119894 then for any ball 119861 there exists a constant119862 gt 0
such that119898
prod
119894=1
(int
119861
119908119902119894
119894(119909) 119889119909)
119902119902119894
le 119862int
119861
](119909)119902
119889119909 (59)
Proof of Theorem 3 Arguing as in the proof ofTheorem 1 fixa ball119861 = 119861(119909
0 119903119861) sube R119899 and decompose119891
119894= 1198910
119894+119891infin
119894 where
1198910
119894= 1198911198941205942119861 119894 = 1 119898 Since 119868
120572is an119898-linear operator then
we have1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911 119891
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
le
1
]119902
(119861)120581119901
(int
119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
+sum
1015840 1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
= 1198690
+sum
1015840
1198691205721120572119898
(60)
where each term of sum1015840 contains at least one 120572119894= 0 In view of
Lemma 9 we can see that (])119902
isin 119860119898119902 UsingTheoremC and
Lemmas 16 and 5 we get
1198690
le 119862 sdot
1
]119902
(119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
]119902
(119861)120581119901
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
(prod119898
119894=1119908119902119894
119894(2119861)119902119902119894)
120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
]119902
(2119861)120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(61)
8 Journal of Function Spaces and Applications
For the other terms let us first deal with the case when 1205721=
sdot sdot sdot = 120572119898= infin By the definition of 119868
120572 for any 119909 isin 119861 we
obtain
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
= int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
1198891199101sdot sdot sdot119889119910119898
=
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899minus120572119898
119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(62)
Moreover by using Holderrsquos inequality the multiple 119860119902
condition and Lemma 16 we deduce that
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times (int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119902+sum119898
119894=1(1minus1119901
119894)
]119902
(2119895+1119861)1119902
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)119908119902119894
119894(2119895+1
119861)
120581119902119901119902119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
[
[
[
[
(prod119898
119894=1119908119902119894
119894(2119895+1
119861)
119902119902119894
)
120581119901
]119902
(2119895+1119861)1119902
]
]
]
]
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(63)
Since (])119902
isin 119860119898119902
sub 119860infin then it follows immediately from
Lemma 7 that
]119902
(119861)
]119902
(2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840
(64)
Hence
119869infininfin
le ]119902
(119861)1119902minus120581119901 1003816
100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(65)
where in the last inequality we have used the fact that 0 lt 120581 lt119901119902 and 1205751015840 gt 0 We now consider the case where exactly ℓ ofthe 120572119894areinfin for some 1 le ℓ lt 119898 We only give the arguments
for one of these cases The rest are similar and can easily beobtained from the arguments belowby permuting the indicesUsing the definition of 119868
120572again we can see that for any 119909 isin 119861
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
= int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(66)
and we arrive at the expression considered in the previouscase Thus for any 119909 isin 119861 we also have
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(67)
Journal of Function Spaces and Applications 9
Therefore by the inequality (64) and the above pointwiseinequality we obtain
1198691205721120572119898
le ]119902
(119861)1119902minus120581119901
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(68)
Summarizing the estimates derived above and then takingthe supremum over all balls 119861 sube R119899 we finish the proof ofTheorem 3
Proof of Theorem 4 As before fix a ball 119861 = 119861(1199090 119903119861) sube R119899
and split119891119894into119891
119894= 1198910
119894+119891infin
119894 where1198910
119894= 1198911198941205942119861 119894 = 1 119898
Then for each fixed 120582 gt 0 we can write
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911 119891
119898)1003816100381610038161003816gt 120582)1119902
le ]119902
(119909 isin 119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119902
+sum
1015840
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119902
= 1198690
lowast+sum
1015840
1198691205721120572119898
lowast
(69)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 9
again we know that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin UsingTheorem D and Lemmas 16 and 5 we have
1198690
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(70)
In the proof of Theorem 3 we have already proved thefollowing pointwise estimate (see (62) and (66)) Consider
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(71)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1 By
using Holderrsquos inequality the multiple 119860119902
condition andLemma 16 we obtain1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(72)
Note that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin Hence it followsfrom the inequality (64) that for any 119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(73)
If 119909 isin 119861 |119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198691205721120572119898
lowastle
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(74)
holds trivially Now if instead we assume that 119909 isin 119861
|119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (73) we get
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(75)
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces and Applications 3
1199081199021
1 119908
119902119898
119898isin 119860infin then for any 0 lt 120581 lt 119901119902 there exists a
constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (13)
where ]= prod119898
119894=1119908119894
Theorem 4 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
119860119902
with11990811990211 119908
119902119898
119898isin 119860infin then for any 0 lt 120581 lt 119901119902 there
exists a constant 119862 gt 0 independent of 119891 = (119891
1 119891
119898) such
that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (14)
where ]= prod119898
119894=1119908119894
2 Notations and Definitions
The classical 119860119901
weight theory was first introduced byMuckenhoupt in the study of weighted 119871
119901 boundedness ofHardy-Littlewood maximal functions in [35] A weight119908 is anonnegative locally integrable function onR119899 119861 = 119861(119909
0 119903119861)
denotes the ball with the center 1199090and radius 119903
119861 For 1 lt 119901 lt
infin a weight function 119908 is said to belong to 119860119901if there is a
constant 119862 gt 0 such that for every ball 119861 sube R119899
(
1
|119861|
int
119861
119908 (119909) 119889119909)(
1
|119861|
int
119861
119908(119909)minus1(119901minus1)
119889119909)
119901minus1
le 119862 (15)
where |119861| denotes the Lebesgue measure of 119861 For the case119901 = 1 119908 isin 119860
1 if there is a constant 119862 gt 0 such that for every
ball 119861 sube R119899
1
|119861|
int
119861
119908 (119909) 119889119909 le 119862 sdot ess inf119909isin119861
119908 (119909) (16)
A weight function 119908 isin 119860infinif it satisfies the 119860
119901condition for
some 1 lt 119901 lt infin We also need another weight class 119860119901119902
introduced by Muckenhoupt andWheeden in [36] A weightfunction 119908 belongs to 119860
119901119902for 1 lt 119901 lt 119902 lt infin if there is a
constant 119862 gt 0 such that for every ball 119861 sube R119899
(
1
|119861|
int
119861
119908(119909)119902
119889119909)
1119902
(
1
|119861|
int
119861
119908(119909)minus1199011015840
119889119909)
11199011015840
le 119862 (17)
When 119901 = 1 119908 is in the class 1198601119902
with 1 lt 119902 lt infin if there isa constant 119862 gt 0 such that for every ball 119861 sube R119899
(
1
|119861|
int
119861
119908(119909)119902
119889119909)
1119902
(ess sup119909isin119861
1
119908 (119909)
) le 119862 (18)
Now let us recall the definitions of multiple weightsFor 119898 exponents 119901
1 119901
119898 we will write for the vector
= (1199011 119901
119898) Let 119901
1 119901
119898isin [1infin) and let 119901 isin
(0infin) with 1119901 = sum119898
119896=11119901119896 Given = (119908
1 119908
119898) set
]= prod119898
119894=1119908119901119901119894
119894 We say that satisfies the 119860
condition if it
satisfies
sup119861
(
1
|119861|
int
119861
](119909) 119889119909)
1119901 119898
prod
119894=1
(
1
|119861|
int
119861
119908119894(119909)1minus1199011015840
119894119889119909)
11199011015840
119894
lt infin
(19)
When 119901119894= 1 ((1|119861|) int
119861
119908119894(119909)1minus1199011015840
119894119889119909)11199011015840
119894 is understood as(inf119909isin119861
119908119894(119909))minus1
Let 1199011 119901
119898isin [1infin) let 1119901 = sum
119898
119896=11119901119896 and let 119902 gt
0 Given = (1199081 119908
119898) set ]
= prod119898
119894=1119908119894 We say that
satisfies the 119860119902
condition if it satisfies
sup119861
(
1
|119861|
int
119861
](119909)119902
119889119909)
1119902
times
119898
prod
119894=1
(
1
|119861|
int
119861
119908119894(119909)minus1199011015840
119894119889119909)
11199011015840
119894
lt infin
(20)
When 119901119894= 1 ((1|119861|) int
119861
119908119894(119909)minus1199011015840
119894119889119909)11199011015840
119894 is understood as(inf119909isin119861
119908119894(119909))minus1
Given a ball 119861 and 120582 gt 0 120582119861 denotes the ball with thesame center as 119861whose radius is 120582 times that of 119861 For a givenweight function119908 and a measurable set 119864 we also denote theLebesgue measure of 119864 by |119864| and the weighted measure of 119864by 119908(119864) where 119908(119864) = int
119864
119908(119909)119889119909
Lemma 5 (see [37]) Let 119908 isin 119860119901with 1 le 119901 lt infin Then for
any ball 119861 there exists an absolute constant 119862 gt 0 such that
119908 (2119861) le 119862119908 (119861) (21)
Lemma 6 (see [38]) Let 119908 isin 119860infin Then for all balls 119861 sube R119899
the following reverse Jensen inequality holds
int
119861
119908 (119909) 119889119909 le 119862 |119861| sdot exp( 1
|119861|
int
119861
log 119908 (119909) 119889119909) (22)
Lemma 7 (see [37]) Let119908 isin 119860infin Then for all balls 119861 and all
measurable subsets 119864 of 119861 there exists 120575 gt 0 such that
119908 (119864)
119908 (119861)
le 119862(
|119864|
|119861|
)
120575
(23)
Lemma 8 (see [9]) Let 1199011 119901
119898isin [1infin) and let 1119901 =
sum119898
119896=11119901119896 Then = (119908
1 119908
119898) isin 119860if and only if
]isin 119860119898119901
1199081minus1199011015840
119894
119894isin 1198601198981199011015840
119894
119894 = 1 119898
(24)
where ]= prod119898
119894=1119908119901119901119894
119894and the condition 1199081minus119901
1015840
119894
119894isin 1198601198981199011015840
119894
in thecase 119901
119894= 1 is understood as 1199081119898
119894isin 1198601
4 Journal of Function Spaces and Applications
Lemma 9 (see [17 18]) Let 0 lt 120572 lt 119898119899 and 1199011 119901
119898isin
[1infin) let 1119901 = sum119898
119896=11119901119896 and let 1119902 = 1119901 minus 120572119899 If =
(1199081 119908
119898) isin 119860119902
then
(])119902
isin 119860119898119902
119908minus1199011015840
119894
119894isin 1198601198981199011015840
119894
119894 = 1 119898
(25)
where ]= prod119898
119894=1119908119894
Given a weight function 119908 on R119899 for 0 lt 119901 lt infin theweighted Lebesgue space 119871119901(119908) is defined as the set of allfunctions 119891 such that
10038171003817100381710038171198911003817100381710038171003817119871119901(119908)
= (int
R119899
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119908 (119909) 119889119909)
1119901
lt infin (26)
We also denote by119882119871119901
(119908) the weighted weak space consist-ing of all measurable functions 119891 such that
10038171003817100381710038171198911003817100381710038171003817119882119871119901(119908)
= sup120582gt0
120582 sdot 119908(119909 isin R119899
1003816100381610038161003816119891 (119909)
1003816100381610038161003816gt 120582)1119901
lt infin
(27)
In 2009 Komori and Shirai [26] first defined the weightedMorrey spaces 119871119901120581(119908) for 1 le 119901 lt infin In order to deal withthe multilinear case 119898 ge 2 we will define 119871119901120581(119908) for all 0 lt119901 lt infin
Definition 10 Let 0 lt 119901 lt infin let 0 lt 120581 lt 1 and let 119908 bea weight function on R119899 Then the weighted Morrey space isdefined by
119871119901120581
(119908) = 119891 isin 119871119901
loc (119908) 10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
lt infin (28)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
= sup119861
(
1
119908(119861)120581int
119861
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119908 (119909) 119889119909)
1119901
(29)
and the supremum is taken over all balls 119861 in R119899
Definition 11 Let 0 lt 119901 lt infin let 0 lt 120581 lt 1 and let 119908 be aweight function onR119899Then theweightedweakMorrey spaceis defined by
119882119871119901120581
(119908) = 119891 measurable 10038171003817100381710038171198911003817100381710038171003817119882119871119901120581(119908)
lt infin (30)
where
10038171003817100381710038171198911003817100381710038171003817119882119871119901120581(119908)
= sup119861
sup120582gt0
1
119908(119861)120581119901
120582 sdot 119908(119909 isin 119861 1003816100381610038161003816119891 (119909)
1003816100381610038161003816gt 120582)1119901
(31)
Furthermore in order to deal with the fractional ordercase we need to consider the weighted Morrey spaces withtwo weights
Definition 12 Let 0 lt 119901 lt infin and 0 lt 120581 lt 1 Then for twoweights 119906 and V the weighted Morrey space is defined by
119871119901120581
(119906 V) = 119891 isin 119871119901
loc (119906) 10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V)
lt infin (32)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V)
= sup119861
(
1
V(119861)120581int
119861
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119906 (119909) 119889119909)
1119901
(33)
Throughout this paper we will use 119862 to denote a positiveconstant which is independent of the main parameters andnot necessarily the same at each occurrence Moreover wewill denote the conjugate exponent of119901 gt 1 by1199011015840 = 119901(119901minus1)
3 Proofs of Theorems 1 and 2
Before proving the main theorems of this section we need toestablish the following lemma
Lemma 13 Let 119898 ge 2 let 1199011 119901
119898isin [1infin) and let 119901 isin
(0infin) with 1119901 = sum119898
119896=11119901119896 Assume that 119908
1 119908
119898isin 119860infin
and ]= prod119898
119894=1119908119901119901119894
119894 then for any ball119861 there exists a constant
119862 gt 0 such that
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862int
119861
](119909) 119889119909 (34)
Proof Since 1199081 119908
119898isin 119860infin then by using Lemma 6 we
have
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862
119898
prod
119894=1
(|119861| sdot exp( 1
|119861|
int
119861
log119908119894(119909) 119889119909))
119901119901119894
= 119862
119898
prod
119894=1
(|119861|119901119901119894sdot exp( 1
|119861|
int
119861
log119908119894(119909)119901119901119894119889119909))
= 119862 sdot (|119861|)sum119898
119894=1119901119901119894sdot exp(
119898
sum
119894=1
1
|119861|
int
119861
log119908119894(119909)119901119901119894119889119909)
(35)
Note that sum119898119894=1
119901119901119894= 1 and ]
(119909) = prod
119898
119894=1119908119894(119909)119901119901119894 Then by
Jensen inequality we obtain
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862 sdot |119861| sdot exp( 1
|119861|
int
119861
log ](119909) 119889119909)
le 119862int
119861
](119909) 119889119909
(36)
We are done
Journal of Function Spaces and Applications 5
Proof of Theorem 1 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
letting 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898 and
1205942119861
denotes the characteristic function of 2119861 then we write
119898
prod
119894=1
119891119894(119910119894) =
119898
prod
119894=1
(1198910
119894(119910119894) + 119891infin
119894(119910119894))
= sum
1205721120572119898isin0infin
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
=
119898
prod
119894=1
1198910
119894(119910119894) +sum
1015840
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
(37)
where each term of sum1015840 contains at least one 120572119894= 0 Since 119879 is
an119898-linear operator then we have
1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911 119891
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
le
1
](119861)120581119901
(int
119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119901
](119909)119889119909)
1119901
+sum
1015840 1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
= 1198680
+sum
1015840
1198681205721120572119898
(38)
In view of Lemma 8 we have that ]
isin 119860119898119901 Applying
Theorem A and Lemmas 13 and 5 we get
1198680
le 119862 sdot
1
](119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
prod119898
119894=1119908119894(2119861)120581119901119894
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
](2119861)120581119901
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(39)
For the other terms let us first consider the case when 1205721=
sdot sdot sdot = 120572119898= infin By the size condition for any 119909 isin 119861 we obtain
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot+
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(40)
where we have used the notation 119864119898
= 119864 times sdot sdot sdot times 119864Furthermore by using Holderrsquos inequality the multiple 119860
condition and Lemma 13 we deduce that
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119901+sum119898
119894=1(1minus1119901
119894)
](2119895+1119861)1119901
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)119908119894(2119895+1
119861)
120581119901119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
prod119898
119894=1119908119894(2119895+1
119861)
120581119901119894
](2119895+1119861)1119901
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(41)
Since ]isin 119860119898119901
sub 119860infin then it follows directly fromLemma 7
that
](119861)
](2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575
(42)
Hence
119868infininfin
le ](119861)(1minus120581)119901 1003816
100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
|2119895+1119861|
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(43)
where the last inequality holds since 0 lt 120581 lt 1 and 120575 gt 0We now consider the case where exactly ℓ of the 120572
119894are infin
for some 1 le ℓ lt 119898 We only give the arguments for oneof these cases The rest are similar and can easily be obtainedfrom the arguments below by permuting the indices Usingthe size condition again we deduce that for any 119909 isin 119861
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899
times 1198891199101sdot sdot sdot 119889119910119898
6 Journal of Function Spaces and Applications
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(44)
and we arrive at the expression considered in the previouscase So for any 119909 isin 119861 we also have
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(45)
Therefore by the inequality (42) and the above pointwiseinequality we have
1198681205721120572119898
le ](119861)(1minus120581)119901
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(46)
Combining the above estimates and then taking the supre-mum over all balls 119861 sube R119899 we complete the proof ofTheorem 1
Proof of Theorem 2 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
decomposing 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898
then for any given 120582 gt 0 we can write
](119909 isin 119861
1003816100381610038161003816119879 (1198911 119891
119898)1003816100381610038161003816gt 120582)1119901
le ](119909 isin 119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119901
+sum
1015840
](119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119901
= 1198680
lowast+sum
1015840
1198681205721120572119898
lowast
(47)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 8
again we know that ]isin 119860119898119901
with 1 le 119898119901 lt infin ApplyingTheorem B and Lemmas 13 and 5 we have
1198680
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119894(2119861)120581119901119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(48)
In the proof of Theorem 1 we have already showed thefollowing pointwise estimate (see (40) and (44)) Consider
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(49)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1
Using Holderrsquos inequality the multiple 119860condition and
Lemma 13 we obtain
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894
times ( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(50)
Observe that ]isin 119860119898119901
with 1 le 119898119901 lt infin Thus it followsfrom the inequality (42) that for any 119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
Journal of Function Spaces and Applications 7
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(51)
If 119909 isin 119861 |119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198681205721120572119898
lowastle
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(52)
holds trivially Now if instead we suppose that 119909 isin 119861
|119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (51) we have
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(53)
which is equivalent to
](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (54)
Therefore
1198681205721120572119898
lowastle ](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (55)
Summing up all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we completethe proof of Theorem 2
By usingHolderrsquos inequality it is easy to check that if each119908119894is in 119860
119901119894
then
119898
prod
119894=1
119860119901119894
sub 119860 (56)
and this inclusion is strict (see [9]) Thus as direct conse-quences of Theorems 1 and 2 we immediately obtain thefollowing
Corollary 14 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If119901
1 119901
119898isin (1infin) and119901 isin (0infin)with
1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898) isin prod
119898
119894=1119860119901119894
then forany 0 lt 120581 lt 1 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (57)
where ]= prod119898
119894=1119908119901119901119894
119894
Corollary 15 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)min119901
1 119901
119898 =
1 and 119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896 and =
(1199081 119908
119898) isin prod
119898
119894=1119860119901119894
then for any 0 lt 120581 lt 1 there existsa constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (58)
where ]= prod119898
119894=1119908119901119901119894
119894
4 Proofs of Theorems 3 and 4
Following along the same lines as those of Lemma 13 we canalso show the following result which plays an important rolein our proofs of Theorems 3 and 4
Lemma 16 Let 119898 ge 2 1199021 119902
119898isin [1infin) and 119902 isin (0infin)
with 1119902 = sum119898
119896=11119902119896 Assume that 1199081199021
1 119908
119902119898
119898isin 119860infin
and]= prod119898
119894=1119908119894 then for any ball 119861 there exists a constant119862 gt 0
such that119898
prod
119894=1
(int
119861
119908119902119894
119894(119909) 119889119909)
119902119902119894
le 119862int
119861
](119909)119902
119889119909 (59)
Proof of Theorem 3 Arguing as in the proof ofTheorem 1 fixa ball119861 = 119861(119909
0 119903119861) sube R119899 and decompose119891
119894= 1198910
119894+119891infin
119894 where
1198910
119894= 1198911198941205942119861 119894 = 1 119898 Since 119868
120572is an119898-linear operator then
we have1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911 119891
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
le
1
]119902
(119861)120581119901
(int
119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
+sum
1015840 1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
= 1198690
+sum
1015840
1198691205721120572119898
(60)
where each term of sum1015840 contains at least one 120572119894= 0 In view of
Lemma 9 we can see that (])119902
isin 119860119898119902 UsingTheoremC and
Lemmas 16 and 5 we get
1198690
le 119862 sdot
1
]119902
(119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
]119902
(119861)120581119901
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
(prod119898
119894=1119908119902119894
119894(2119861)119902119902119894)
120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
]119902
(2119861)120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(61)
8 Journal of Function Spaces and Applications
For the other terms let us first deal with the case when 1205721=
sdot sdot sdot = 120572119898= infin By the definition of 119868
120572 for any 119909 isin 119861 we
obtain
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
= int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
1198891199101sdot sdot sdot119889119910119898
=
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899minus120572119898
119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(62)
Moreover by using Holderrsquos inequality the multiple 119860119902
condition and Lemma 16 we deduce that
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times (int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119902+sum119898
119894=1(1minus1119901
119894)
]119902
(2119895+1119861)1119902
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)119908119902119894
119894(2119895+1
119861)
120581119902119901119902119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
[
[
[
[
(prod119898
119894=1119908119902119894
119894(2119895+1
119861)
119902119902119894
)
120581119901
]119902
(2119895+1119861)1119902
]
]
]
]
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(63)
Since (])119902
isin 119860119898119902
sub 119860infin then it follows immediately from
Lemma 7 that
]119902
(119861)
]119902
(2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840
(64)
Hence
119869infininfin
le ]119902
(119861)1119902minus120581119901 1003816
100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(65)
where in the last inequality we have used the fact that 0 lt 120581 lt119901119902 and 1205751015840 gt 0 We now consider the case where exactly ℓ ofthe 120572119894areinfin for some 1 le ℓ lt 119898 We only give the arguments
for one of these cases The rest are similar and can easily beobtained from the arguments belowby permuting the indicesUsing the definition of 119868
120572again we can see that for any 119909 isin 119861
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
= int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(66)
and we arrive at the expression considered in the previouscase Thus for any 119909 isin 119861 we also have
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(67)
Journal of Function Spaces and Applications 9
Therefore by the inequality (64) and the above pointwiseinequality we obtain
1198691205721120572119898
le ]119902
(119861)1119902minus120581119901
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(68)
Summarizing the estimates derived above and then takingthe supremum over all balls 119861 sube R119899 we finish the proof ofTheorem 3
Proof of Theorem 4 As before fix a ball 119861 = 119861(1199090 119903119861) sube R119899
and split119891119894into119891
119894= 1198910
119894+119891infin
119894 where1198910
119894= 1198911198941205942119861 119894 = 1 119898
Then for each fixed 120582 gt 0 we can write
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911 119891
119898)1003816100381610038161003816gt 120582)1119902
le ]119902
(119909 isin 119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119902
+sum
1015840
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119902
= 1198690
lowast+sum
1015840
1198691205721120572119898
lowast
(69)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 9
again we know that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin UsingTheorem D and Lemmas 16 and 5 we have
1198690
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(70)
In the proof of Theorem 3 we have already proved thefollowing pointwise estimate (see (62) and (66)) Consider
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(71)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1 By
using Holderrsquos inequality the multiple 119860119902
condition andLemma 16 we obtain1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(72)
Note that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin Hence it followsfrom the inequality (64) that for any 119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(73)
If 119909 isin 119861 |119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198691205721120572119898
lowastle
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(74)
holds trivially Now if instead we assume that 119909 isin 119861
|119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (73) we get
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(75)
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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 and Applications
Lemma 9 (see [17 18]) Let 0 lt 120572 lt 119898119899 and 1199011 119901
119898isin
[1infin) let 1119901 = sum119898
119896=11119901119896 and let 1119902 = 1119901 minus 120572119899 If =
(1199081 119908
119898) isin 119860119902
then
(])119902
isin 119860119898119902
119908minus1199011015840
119894
119894isin 1198601198981199011015840
119894
119894 = 1 119898
(25)
where ]= prod119898
119894=1119908119894
Given a weight function 119908 on R119899 for 0 lt 119901 lt infin theweighted Lebesgue space 119871119901(119908) is defined as the set of allfunctions 119891 such that
10038171003817100381710038171198911003817100381710038171003817119871119901(119908)
= (int
R119899
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119908 (119909) 119889119909)
1119901
lt infin (26)
We also denote by119882119871119901
(119908) the weighted weak space consist-ing of all measurable functions 119891 such that
10038171003817100381710038171198911003817100381710038171003817119882119871119901(119908)
= sup120582gt0
120582 sdot 119908(119909 isin R119899
1003816100381610038161003816119891 (119909)
1003816100381610038161003816gt 120582)1119901
lt infin
(27)
In 2009 Komori and Shirai [26] first defined the weightedMorrey spaces 119871119901120581(119908) for 1 le 119901 lt infin In order to deal withthe multilinear case 119898 ge 2 we will define 119871119901120581(119908) for all 0 lt119901 lt infin
Definition 10 Let 0 lt 119901 lt infin let 0 lt 120581 lt 1 and let 119908 bea weight function on R119899 Then the weighted Morrey space isdefined by
119871119901120581
(119908) = 119891 isin 119871119901
loc (119908) 10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
lt infin (28)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
= sup119861
(
1
119908(119861)120581int
119861
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119908 (119909) 119889119909)
1119901
(29)
and the supremum is taken over all balls 119861 in R119899
Definition 11 Let 0 lt 119901 lt infin let 0 lt 120581 lt 1 and let 119908 be aweight function onR119899Then theweightedweakMorrey spaceis defined by
119882119871119901120581
(119908) = 119891 measurable 10038171003817100381710038171198911003817100381710038171003817119882119871119901120581(119908)
lt infin (30)
where
10038171003817100381710038171198911003817100381710038171003817119882119871119901120581(119908)
= sup119861
sup120582gt0
1
119908(119861)120581119901
120582 sdot 119908(119909 isin 119861 1003816100381610038161003816119891 (119909)
1003816100381610038161003816gt 120582)1119901
(31)
Furthermore in order to deal with the fractional ordercase we need to consider the weighted Morrey spaces withtwo weights
Definition 12 Let 0 lt 119901 lt infin and 0 lt 120581 lt 1 Then for twoweights 119906 and V the weighted Morrey space is defined by
119871119901120581
(119906 V) = 119891 isin 119871119901
loc (119906) 10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V)
lt infin (32)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V)
= sup119861
(
1
V(119861)120581int
119861
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119906 (119909) 119889119909)
1119901
(33)
Throughout this paper we will use 119862 to denote a positiveconstant which is independent of the main parameters andnot necessarily the same at each occurrence Moreover wewill denote the conjugate exponent of119901 gt 1 by1199011015840 = 119901(119901minus1)
3 Proofs of Theorems 1 and 2
Before proving the main theorems of this section we need toestablish the following lemma
Lemma 13 Let 119898 ge 2 let 1199011 119901
119898isin [1infin) and let 119901 isin
(0infin) with 1119901 = sum119898
119896=11119901119896 Assume that 119908
1 119908
119898isin 119860infin
and ]= prod119898
119894=1119908119901119901119894
119894 then for any ball119861 there exists a constant
119862 gt 0 such that
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862int
119861
](119909) 119889119909 (34)
Proof Since 1199081 119908
119898isin 119860infin then by using Lemma 6 we
have
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862
119898
prod
119894=1
(|119861| sdot exp( 1
|119861|
int
119861
log119908119894(119909) 119889119909))
119901119901119894
= 119862
119898
prod
119894=1
(|119861|119901119901119894sdot exp( 1
|119861|
int
119861
log119908119894(119909)119901119901119894119889119909))
= 119862 sdot (|119861|)sum119898
119894=1119901119901119894sdot exp(
119898
sum
119894=1
1
|119861|
int
119861
log119908119894(119909)119901119901119894119889119909)
(35)
Note that sum119898119894=1
119901119901119894= 1 and ]
(119909) = prod
119898
119894=1119908119894(119909)119901119901119894 Then by
Jensen inequality we obtain
119898
prod
119894=1
(int
119861
119908119894(119909) 119889119909)
119901119901119894
le 119862 sdot |119861| sdot exp( 1
|119861|
int
119861
log ](119909) 119889119909)
le 119862int
119861
](119909) 119889119909
(36)
We are done
Journal of Function Spaces and Applications 5
Proof of Theorem 1 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
letting 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898 and
1205942119861
denotes the characteristic function of 2119861 then we write
119898
prod
119894=1
119891119894(119910119894) =
119898
prod
119894=1
(1198910
119894(119910119894) + 119891infin
119894(119910119894))
= sum
1205721120572119898isin0infin
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
=
119898
prod
119894=1
1198910
119894(119910119894) +sum
1015840
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
(37)
where each term of sum1015840 contains at least one 120572119894= 0 Since 119879 is
an119898-linear operator then we have
1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911 119891
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
le
1
](119861)120581119901
(int
119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119901
](119909)119889119909)
1119901
+sum
1015840 1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
= 1198680
+sum
1015840
1198681205721120572119898
(38)
In view of Lemma 8 we have that ]
isin 119860119898119901 Applying
Theorem A and Lemmas 13 and 5 we get
1198680
le 119862 sdot
1
](119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
prod119898
119894=1119908119894(2119861)120581119901119894
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
](2119861)120581119901
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(39)
For the other terms let us first consider the case when 1205721=
sdot sdot sdot = 120572119898= infin By the size condition for any 119909 isin 119861 we obtain
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot+
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(40)
where we have used the notation 119864119898
= 119864 times sdot sdot sdot times 119864Furthermore by using Holderrsquos inequality the multiple 119860
condition and Lemma 13 we deduce that
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119901+sum119898
119894=1(1minus1119901
119894)
](2119895+1119861)1119901
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)119908119894(2119895+1
119861)
120581119901119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
prod119898
119894=1119908119894(2119895+1
119861)
120581119901119894
](2119895+1119861)1119901
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(41)
Since ]isin 119860119898119901
sub 119860infin then it follows directly fromLemma 7
that
](119861)
](2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575
(42)
Hence
119868infininfin
le ](119861)(1minus120581)119901 1003816
100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
|2119895+1119861|
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(43)
where the last inequality holds since 0 lt 120581 lt 1 and 120575 gt 0We now consider the case where exactly ℓ of the 120572
119894are infin
for some 1 le ℓ lt 119898 We only give the arguments for oneof these cases The rest are similar and can easily be obtainedfrom the arguments below by permuting the indices Usingthe size condition again we deduce that for any 119909 isin 119861
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899
times 1198891199101sdot sdot sdot 119889119910119898
6 Journal of Function Spaces and Applications
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(44)
and we arrive at the expression considered in the previouscase So for any 119909 isin 119861 we also have
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(45)
Therefore by the inequality (42) and the above pointwiseinequality we have
1198681205721120572119898
le ](119861)(1minus120581)119901
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(46)
Combining the above estimates and then taking the supre-mum over all balls 119861 sube R119899 we complete the proof ofTheorem 1
Proof of Theorem 2 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
decomposing 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898
then for any given 120582 gt 0 we can write
](119909 isin 119861
1003816100381610038161003816119879 (1198911 119891
119898)1003816100381610038161003816gt 120582)1119901
le ](119909 isin 119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119901
+sum
1015840
](119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119901
= 1198680
lowast+sum
1015840
1198681205721120572119898
lowast
(47)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 8
again we know that ]isin 119860119898119901
with 1 le 119898119901 lt infin ApplyingTheorem B and Lemmas 13 and 5 we have
1198680
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119894(2119861)120581119901119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(48)
In the proof of Theorem 1 we have already showed thefollowing pointwise estimate (see (40) and (44)) Consider
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(49)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1
Using Holderrsquos inequality the multiple 119860condition and
Lemma 13 we obtain
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894
times ( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(50)
Observe that ]isin 119860119898119901
with 1 le 119898119901 lt infin Thus it followsfrom the inequality (42) that for any 119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
Journal of Function Spaces and Applications 7
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(51)
If 119909 isin 119861 |119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198681205721120572119898
lowastle
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(52)
holds trivially Now if instead we suppose that 119909 isin 119861
|119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (51) we have
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(53)
which is equivalent to
](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (54)
Therefore
1198681205721120572119898
lowastle ](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (55)
Summing up all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we completethe proof of Theorem 2
By usingHolderrsquos inequality it is easy to check that if each119908119894is in 119860
119901119894
then
119898
prod
119894=1
119860119901119894
sub 119860 (56)
and this inclusion is strict (see [9]) Thus as direct conse-quences of Theorems 1 and 2 we immediately obtain thefollowing
Corollary 14 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If119901
1 119901
119898isin (1infin) and119901 isin (0infin)with
1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898) isin prod
119898
119894=1119860119901119894
then forany 0 lt 120581 lt 1 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (57)
where ]= prod119898
119894=1119908119901119901119894
119894
Corollary 15 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)min119901
1 119901
119898 =
1 and 119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896 and =
(1199081 119908
119898) isin prod
119898
119894=1119860119901119894
then for any 0 lt 120581 lt 1 there existsa constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (58)
where ]= prod119898
119894=1119908119901119901119894
119894
4 Proofs of Theorems 3 and 4
Following along the same lines as those of Lemma 13 we canalso show the following result which plays an important rolein our proofs of Theorems 3 and 4
Lemma 16 Let 119898 ge 2 1199021 119902
119898isin [1infin) and 119902 isin (0infin)
with 1119902 = sum119898
119896=11119902119896 Assume that 1199081199021
1 119908
119902119898
119898isin 119860infin
and]= prod119898
119894=1119908119894 then for any ball 119861 there exists a constant119862 gt 0
such that119898
prod
119894=1
(int
119861
119908119902119894
119894(119909) 119889119909)
119902119902119894
le 119862int
119861
](119909)119902
119889119909 (59)
Proof of Theorem 3 Arguing as in the proof ofTheorem 1 fixa ball119861 = 119861(119909
0 119903119861) sube R119899 and decompose119891
119894= 1198910
119894+119891infin
119894 where
1198910
119894= 1198911198941205942119861 119894 = 1 119898 Since 119868
120572is an119898-linear operator then
we have1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911 119891
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
le
1
]119902
(119861)120581119901
(int
119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
+sum
1015840 1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
= 1198690
+sum
1015840
1198691205721120572119898
(60)
where each term of sum1015840 contains at least one 120572119894= 0 In view of
Lemma 9 we can see that (])119902
isin 119860119898119902 UsingTheoremC and
Lemmas 16 and 5 we get
1198690
le 119862 sdot
1
]119902
(119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
]119902
(119861)120581119901
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
(prod119898
119894=1119908119902119894
119894(2119861)119902119902119894)
120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
]119902
(2119861)120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(61)
8 Journal of Function Spaces and Applications
For the other terms let us first deal with the case when 1205721=
sdot sdot sdot = 120572119898= infin By the definition of 119868
120572 for any 119909 isin 119861 we
obtain
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
= int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
1198891199101sdot sdot sdot119889119910119898
=
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899minus120572119898
119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(62)
Moreover by using Holderrsquos inequality the multiple 119860119902
condition and Lemma 16 we deduce that
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times (int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119902+sum119898
119894=1(1minus1119901
119894)
]119902
(2119895+1119861)1119902
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)119908119902119894
119894(2119895+1
119861)
120581119902119901119902119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
[
[
[
[
(prod119898
119894=1119908119902119894
119894(2119895+1
119861)
119902119902119894
)
120581119901
]119902
(2119895+1119861)1119902
]
]
]
]
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(63)
Since (])119902
isin 119860119898119902
sub 119860infin then it follows immediately from
Lemma 7 that
]119902
(119861)
]119902
(2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840
(64)
Hence
119869infininfin
le ]119902
(119861)1119902minus120581119901 1003816
100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(65)
where in the last inequality we have used the fact that 0 lt 120581 lt119901119902 and 1205751015840 gt 0 We now consider the case where exactly ℓ ofthe 120572119894areinfin for some 1 le ℓ lt 119898 We only give the arguments
for one of these cases The rest are similar and can easily beobtained from the arguments belowby permuting the indicesUsing the definition of 119868
120572again we can see that for any 119909 isin 119861
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
= int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(66)
and we arrive at the expression considered in the previouscase Thus for any 119909 isin 119861 we also have
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(67)
Journal of Function Spaces and Applications 9
Therefore by the inequality (64) and the above pointwiseinequality we obtain
1198691205721120572119898
le ]119902
(119861)1119902minus120581119901
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(68)
Summarizing the estimates derived above and then takingthe supremum over all balls 119861 sube R119899 we finish the proof ofTheorem 3
Proof of Theorem 4 As before fix a ball 119861 = 119861(1199090 119903119861) sube R119899
and split119891119894into119891
119894= 1198910
119894+119891infin
119894 where1198910
119894= 1198911198941205942119861 119894 = 1 119898
Then for each fixed 120582 gt 0 we can write
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911 119891
119898)1003816100381610038161003816gt 120582)1119902
le ]119902
(119909 isin 119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119902
+sum
1015840
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119902
= 1198690
lowast+sum
1015840
1198691205721120572119898
lowast
(69)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 9
again we know that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin UsingTheorem D and Lemmas 16 and 5 we have
1198690
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(70)
In the proof of Theorem 3 we have already proved thefollowing pointwise estimate (see (62) and (66)) Consider
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(71)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1 By
using Holderrsquos inequality the multiple 119860119902
condition andLemma 16 we obtain1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(72)
Note that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin Hence it followsfrom the inequality (64) that for any 119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(73)
If 119909 isin 119861 |119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198691205721120572119898
lowastle
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(74)
holds trivially Now if instead we assume that 119909 isin 119861
|119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (73) we get
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(75)
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces and Applications 5
Proof of Theorem 1 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
letting 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898 and
1205942119861
denotes the characteristic function of 2119861 then we write
119898
prod
119894=1
119891119894(119910119894) =
119898
prod
119894=1
(1198910
119894(119910119894) + 119891infin
119894(119910119894))
= sum
1205721120572119898isin0infin
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
=
119898
prod
119894=1
1198910
119894(119910119894) +sum
1015840
1198911205721
1(1199101) sdot sdot sdot 119891
120572119898
119898(119910119898)
(37)
where each term of sum1015840 contains at least one 120572119894= 0 Since 119879 is
an119898-linear operator then we have
1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911 119891
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
le
1
](119861)120581119901
(int
119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119901
](119909)119889119909)
1119901
+sum
1015840 1
](119861)120581119901
(int
119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119901
](119909) 119889119909)
1119901
= 1198680
+sum
1015840
1198681205721120572119898
(38)
In view of Lemma 8 we have that ]
isin 119860119898119901 Applying
Theorem A and Lemmas 13 and 5 we get
1198680
le 119862 sdot
1
](119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
prod119898
119894=1119908119894(2119861)120581119901119894
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
](2119861)120581119901
](119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(39)
For the other terms let us first consider the case when 1205721=
sdot sdot sdot = 120572119898= infin By the size condition for any 119909 isin 119861 we obtain
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot+
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)1198981198991198891199101sdot sdot sdot119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(40)
where we have used the notation 119864119898
= 119864 times sdot sdot sdot times 119864Furthermore by using Holderrsquos inequality the multiple 119860
condition and Lemma 13 we deduce that
1003816100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119901+sum119898
119894=1(1minus1119901
119894)
](2119895+1119861)1119901
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)119908119894(2119895+1
119861)
120581119901119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
prod119898
119894=1119908119894(2119895+1
119861)
120581119901119894
](2119895+1119861)1119901
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(41)
Since ]isin 119860119898119901
sub 119860infin then it follows directly fromLemma 7
that
](119861)
](2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575
(42)
Hence
119868infininfin
le ](119861)(1minus120581)119901 1003816
100381610038161003816119879 (119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
|2119895+1119861|
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(43)
where the last inequality holds since 0 lt 120581 lt 1 and 120575 gt 0We now consider the case where exactly ℓ of the 120572
119894are infin
for some 1 le ℓ lt 119898 We only give the arguments for oneof these cases The rest are similar and can easily be obtainedfrom the arguments below by permuting the indices Usingthe size condition again we deduce that for any 119909 isin 119861
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899
times 1198891199101sdot sdot sdot 119889119910119898
6 Journal of Function Spaces and Applications
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(44)
and we arrive at the expression considered in the previouscase So for any 119909 isin 119861 we also have
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(45)
Therefore by the inequality (42) and the above pointwiseinequality we have
1198681205721120572119898
le ](119861)(1minus120581)119901
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(46)
Combining the above estimates and then taking the supre-mum over all balls 119861 sube R119899 we complete the proof ofTheorem 1
Proof of Theorem 2 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
decomposing 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898
then for any given 120582 gt 0 we can write
](119909 isin 119861
1003816100381610038161003816119879 (1198911 119891
119898)1003816100381610038161003816gt 120582)1119901
le ](119909 isin 119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119901
+sum
1015840
](119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119901
= 1198680
lowast+sum
1015840
1198681205721120572119898
lowast
(47)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 8
again we know that ]isin 119860119898119901
with 1 le 119898119901 lt infin ApplyingTheorem B and Lemmas 13 and 5 we have
1198680
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119894(2119861)120581119901119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(48)
In the proof of Theorem 1 we have already showed thefollowing pointwise estimate (see (40) and (44)) Consider
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(49)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1
Using Holderrsquos inequality the multiple 119860condition and
Lemma 13 we obtain
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894
times ( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(50)
Observe that ]isin 119860119898119901
with 1 le 119898119901 lt infin Thus it followsfrom the inequality (42) that for any 119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
Journal of Function Spaces and Applications 7
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(51)
If 119909 isin 119861 |119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198681205721120572119898
lowastle
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(52)
holds trivially Now if instead we suppose that 119909 isin 119861
|119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (51) we have
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(53)
which is equivalent to
](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (54)
Therefore
1198681205721120572119898
lowastle ](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (55)
Summing up all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we completethe proof of Theorem 2
By usingHolderrsquos inequality it is easy to check that if each119908119894is in 119860
119901119894
then
119898
prod
119894=1
119860119901119894
sub 119860 (56)
and this inclusion is strict (see [9]) Thus as direct conse-quences of Theorems 1 and 2 we immediately obtain thefollowing
Corollary 14 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If119901
1 119901
119898isin (1infin) and119901 isin (0infin)with
1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898) isin prod
119898
119894=1119860119901119894
then forany 0 lt 120581 lt 1 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (57)
where ]= prod119898
119894=1119908119901119901119894
119894
Corollary 15 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)min119901
1 119901
119898 =
1 and 119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896 and =
(1199081 119908
119898) isin prod
119898
119894=1119860119901119894
then for any 0 lt 120581 lt 1 there existsa constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (58)
where ]= prod119898
119894=1119908119901119901119894
119894
4 Proofs of Theorems 3 and 4
Following along the same lines as those of Lemma 13 we canalso show the following result which plays an important rolein our proofs of Theorems 3 and 4
Lemma 16 Let 119898 ge 2 1199021 119902
119898isin [1infin) and 119902 isin (0infin)
with 1119902 = sum119898
119896=11119902119896 Assume that 1199081199021
1 119908
119902119898
119898isin 119860infin
and]= prod119898
119894=1119908119894 then for any ball 119861 there exists a constant119862 gt 0
such that119898
prod
119894=1
(int
119861
119908119902119894
119894(119909) 119889119909)
119902119902119894
le 119862int
119861
](119909)119902
119889119909 (59)
Proof of Theorem 3 Arguing as in the proof ofTheorem 1 fixa ball119861 = 119861(119909
0 119903119861) sube R119899 and decompose119891
119894= 1198910
119894+119891infin
119894 where
1198910
119894= 1198911198941205942119861 119894 = 1 119898 Since 119868
120572is an119898-linear operator then
we have1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911 119891
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
le
1
]119902
(119861)120581119901
(int
119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
+sum
1015840 1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
= 1198690
+sum
1015840
1198691205721120572119898
(60)
where each term of sum1015840 contains at least one 120572119894= 0 In view of
Lemma 9 we can see that (])119902
isin 119860119898119902 UsingTheoremC and
Lemmas 16 and 5 we get
1198690
le 119862 sdot
1
]119902
(119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
]119902
(119861)120581119901
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
(prod119898
119894=1119908119902119894
119894(2119861)119902119902119894)
120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
]119902
(2119861)120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(61)
8 Journal of Function Spaces and Applications
For the other terms let us first deal with the case when 1205721=
sdot sdot sdot = 120572119898= infin By the definition of 119868
120572 for any 119909 isin 119861 we
obtain
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
= int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
1198891199101sdot sdot sdot119889119910119898
=
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899minus120572119898
119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(62)
Moreover by using Holderrsquos inequality the multiple 119860119902
condition and Lemma 16 we deduce that
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times (int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119902+sum119898
119894=1(1minus1119901
119894)
]119902
(2119895+1119861)1119902
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)119908119902119894
119894(2119895+1
119861)
120581119902119901119902119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
[
[
[
[
(prod119898
119894=1119908119902119894
119894(2119895+1
119861)
119902119902119894
)
120581119901
]119902
(2119895+1119861)1119902
]
]
]
]
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(63)
Since (])119902
isin 119860119898119902
sub 119860infin then it follows immediately from
Lemma 7 that
]119902
(119861)
]119902
(2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840
(64)
Hence
119869infininfin
le ]119902
(119861)1119902minus120581119901 1003816
100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(65)
where in the last inequality we have used the fact that 0 lt 120581 lt119901119902 and 1205751015840 gt 0 We now consider the case where exactly ℓ ofthe 120572119894areinfin for some 1 le ℓ lt 119898 We only give the arguments
for one of these cases The rest are similar and can easily beobtained from the arguments belowby permuting the indicesUsing the definition of 119868
120572again we can see that for any 119909 isin 119861
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
= int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(66)
and we arrive at the expression considered in the previouscase Thus for any 119909 isin 119861 we also have
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(67)
Journal of Function Spaces and Applications 9
Therefore by the inequality (64) and the above pointwiseinequality we obtain
1198691205721120572119898
le ]119902
(119861)1119902minus120581119901
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(68)
Summarizing the estimates derived above and then takingthe supremum over all balls 119861 sube R119899 we finish the proof ofTheorem 3
Proof of Theorem 4 As before fix a ball 119861 = 119861(1199090 119903119861) sube R119899
and split119891119894into119891
119894= 1198910
119894+119891infin
119894 where1198910
119894= 1198911198941205942119861 119894 = 1 119898
Then for each fixed 120582 gt 0 we can write
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911 119891
119898)1003816100381610038161003816gt 120582)1119902
le ]119902
(119909 isin 119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119902
+sum
1015840
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119902
= 1198690
lowast+sum
1015840
1198691205721120572119898
lowast
(69)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 9
again we know that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin UsingTheorem D and Lemmas 16 and 5 we have
1198690
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(70)
In the proof of Theorem 3 we have already proved thefollowing pointwise estimate (see (62) and (66)) Consider
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(71)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1 By
using Holderrsquos inequality the multiple 119860119902
condition andLemma 16 we obtain1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(72)
Note that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin Hence it followsfrom the inequality (64) that for any 119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(73)
If 119909 isin 119861 |119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198691205721120572119898
lowastle
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(74)
holds trivially Now if instead we assume that 119909 isin 119861
|119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (73) we get
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(75)
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
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
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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 and Applications
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(44)
and we arrive at the expression considered in the previouscase So for any 119909 isin 119861 we also have
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(45)
Therefore by the inequality (42) and the above pointwiseinequality we have
1198681205721120572119898
le ](119861)(1minus120581)119901
10038161003816100381610038161003816119879 (119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(46)
Combining the above estimates and then taking the supre-mum over all balls 119861 sube R119899 we complete the proof ofTheorem 1
Proof of Theorem 2 For any ball 119861 = 119861(1199090 119903119861) sube R119899 and
decomposing 119891119894= 1198910
119894+ 119891infin
119894 where 1198910
119894= 1198911198941205942119861 119894 = 1 119898
then for any given 120582 gt 0 we can write
](119909 isin 119861
1003816100381610038161003816119879 (1198911 119891
119898)1003816100381610038161003816gt 120582)1119901
le ](119909 isin 119861
10038161003816100381610038161003816119879 (1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119901
+sum
1015840
](119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119901
= 1198680
lowast+sum
1015840
1198681205721120572119898
lowast
(47)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 8
again we know that ]isin 119860119898119901
with 1 le 119898119901 lt infin ApplyingTheorem B and Lemmas 13 and 5 we have
1198680
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909) 119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119894(2119861)120581119901119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(48)
In the proof of Theorem 1 we have already showed thefollowing pointwise estimate (see (40) and (44)) Consider
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(49)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1
Using Holderrsquos inequality the multiple 119860condition and
Lemma 13 we obtain
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894
times ( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894) 119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)1minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
infin
sum
119895=1
](2119895+1
119861)
(120581minus1)119901
(50)
Observe that ]isin 119860119898119901
with 1 le 119898119901 lt infin Thus it followsfrom the inequality (42) that for any 119909 isin 119861
1003816100381610038161003816119879 (1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
](119861)(1minus120581)119901
](2119895+1119861)(1minus120581)119901
Journal of Function Spaces and Applications 7
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(51)
If 119909 isin 119861 |119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198681205721120572119898
lowastle
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(52)
holds trivially Now if instead we suppose that 119909 isin 119861
|119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (51) we have
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(53)
which is equivalent to
](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (54)
Therefore
1198681205721120572119898
lowastle ](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (55)
Summing up all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we completethe proof of Theorem 2
By usingHolderrsquos inequality it is easy to check that if each119908119894is in 119860
119901119894
then
119898
prod
119894=1
119860119901119894
sub 119860 (56)
and this inclusion is strict (see [9]) Thus as direct conse-quences of Theorems 1 and 2 we immediately obtain thefollowing
Corollary 14 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If119901
1 119901
119898isin (1infin) and119901 isin (0infin)with
1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898) isin prod
119898
119894=1119860119901119894
then forany 0 lt 120581 lt 1 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (57)
where ]= prod119898
119894=1119908119901119901119894
119894
Corollary 15 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)min119901
1 119901
119898 =
1 and 119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896 and =
(1199081 119908
119898) isin prod
119898
119894=1119860119901119894
then for any 0 lt 120581 lt 1 there existsa constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (58)
where ]= prod119898
119894=1119908119901119901119894
119894
4 Proofs of Theorems 3 and 4
Following along the same lines as those of Lemma 13 we canalso show the following result which plays an important rolein our proofs of Theorems 3 and 4
Lemma 16 Let 119898 ge 2 1199021 119902
119898isin [1infin) and 119902 isin (0infin)
with 1119902 = sum119898
119896=11119902119896 Assume that 1199081199021
1 119908
119902119898
119898isin 119860infin
and]= prod119898
119894=1119908119894 then for any ball 119861 there exists a constant119862 gt 0
such that119898
prod
119894=1
(int
119861
119908119902119894
119894(119909) 119889119909)
119902119902119894
le 119862int
119861
](119909)119902
119889119909 (59)
Proof of Theorem 3 Arguing as in the proof ofTheorem 1 fixa ball119861 = 119861(119909
0 119903119861) sube R119899 and decompose119891
119894= 1198910
119894+119891infin
119894 where
1198910
119894= 1198911198941205942119861 119894 = 1 119898 Since 119868
120572is an119898-linear operator then
we have1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911 119891
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
le
1
]119902
(119861)120581119901
(int
119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
+sum
1015840 1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
= 1198690
+sum
1015840
1198691205721120572119898
(60)
where each term of sum1015840 contains at least one 120572119894= 0 In view of
Lemma 9 we can see that (])119902
isin 119860119898119902 UsingTheoremC and
Lemmas 16 and 5 we get
1198690
le 119862 sdot
1
]119902
(119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
]119902
(119861)120581119901
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
(prod119898
119894=1119908119902119894
119894(2119861)119902119902119894)
120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
]119902
(2119861)120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(61)
8 Journal of Function Spaces and Applications
For the other terms let us first deal with the case when 1205721=
sdot sdot sdot = 120572119898= infin By the definition of 119868
120572 for any 119909 isin 119861 we
obtain
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
= int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
1198891199101sdot sdot sdot119889119910119898
=
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899minus120572119898
119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(62)
Moreover by using Holderrsquos inequality the multiple 119860119902
condition and Lemma 16 we deduce that
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times (int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119902+sum119898
119894=1(1minus1119901
119894)
]119902
(2119895+1119861)1119902
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)119908119902119894
119894(2119895+1
119861)
120581119902119901119902119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
[
[
[
[
(prod119898
119894=1119908119902119894
119894(2119895+1
119861)
119902119902119894
)
120581119901
]119902
(2119895+1119861)1119902
]
]
]
]
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(63)
Since (])119902
isin 119860119898119902
sub 119860infin then it follows immediately from
Lemma 7 that
]119902
(119861)
]119902
(2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840
(64)
Hence
119869infininfin
le ]119902
(119861)1119902minus120581119901 1003816
100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(65)
where in the last inequality we have used the fact that 0 lt 120581 lt119901119902 and 1205751015840 gt 0 We now consider the case where exactly ℓ ofthe 120572119894areinfin for some 1 le ℓ lt 119898 We only give the arguments
for one of these cases The rest are similar and can easily beobtained from the arguments belowby permuting the indicesUsing the definition of 119868
120572again we can see that for any 119909 isin 119861
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
= int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(66)
and we arrive at the expression considered in the previouscase Thus for any 119909 isin 119861 we also have
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(67)
Journal of Function Spaces and Applications 9
Therefore by the inequality (64) and the above pointwiseinequality we obtain
1198691205721120572119898
le ]119902
(119861)1119902minus120581119901
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(68)
Summarizing the estimates derived above and then takingthe supremum over all balls 119861 sube R119899 we finish the proof ofTheorem 3
Proof of Theorem 4 As before fix a ball 119861 = 119861(1199090 119903119861) sube R119899
and split119891119894into119891
119894= 1198910
119894+119891infin
119894 where1198910
119894= 1198911198941205942119861 119894 = 1 119898
Then for each fixed 120582 gt 0 we can write
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911 119891
119898)1003816100381610038161003816gt 120582)1119902
le ]119902
(119909 isin 119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119902
+sum
1015840
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119902
= 1198690
lowast+sum
1015840
1198691205721120572119898
lowast
(69)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 9
again we know that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin UsingTheorem D and Lemmas 16 and 5 we have
1198690
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(70)
In the proof of Theorem 3 we have already proved thefollowing pointwise estimate (see (62) and (66)) Consider
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(71)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1 By
using Holderrsquos inequality the multiple 119860119902
condition andLemma 16 we obtain1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(72)
Note that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin Hence it followsfrom the inequality (64) that for any 119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(73)
If 119909 isin 119861 |119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198691205721120572119898
lowastle
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(74)
holds trivially Now if instead we assume that 119909 isin 119861
|119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (73) we get
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(75)
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
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Stochastic AnalysisInternational Journal of
Journal of Function Spaces and Applications 7
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
120575(1minus120581)119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(51)
If 119909 isin 119861 |119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198681205721120572119898
lowastle
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)
(52)
holds trivially Now if instead we suppose that 119909 isin 119861
|119879(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (51) we have
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894)sdot
1
](119861)(1minus120581)119901
(53)
which is equivalent to
](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (54)
Therefore
1198681205721120572119898
lowastle ](119861)1119901
le
119862 sdot ](119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (55)
Summing up all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we completethe proof of Theorem 2
By usingHolderrsquos inequality it is easy to check that if each119908119894is in 119860
119901119894
then
119898
prod
119894=1
119860119901119894
sub 119860 (56)
and this inclusion is strict (see [9]) Thus as direct conse-quences of Theorems 1 and 2 we immediately obtain thefollowing
Corollary 14 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If119901
1 119901
119898isin (1infin) and119901 isin (0infin)with
1119901 = sum119898
119896=11119901119896and = (119908
1 119908
119898) isin prod
119898
119894=1119860119901119894
then forany 0 lt 120581 lt 1 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (57)
where ]= prod119898
119894=1119908119901119901119894
119894
Corollary 15 Let 119898 ge 2 and let 119879 be an 119898-linear Calderon-Zygmund operator If 119901
1 119901
119898isin [1infin)min119901
1 119901
119898 =
1 and 119901 isin (0infin) with 1119901 = sum119898
119896=11119901119896 and =
(1199081 119908
119898) isin prod
119898
119894=1119860119901119894
then for any 0 lt 120581 lt 1 there existsa constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119879 (
119891)
10038171003817100381710038171003817119882119871119901120581(])
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581(119908
119894) (58)
where ]= prod119898
119894=1119908119901119901119894
119894
4 Proofs of Theorems 3 and 4
Following along the same lines as those of Lemma 13 we canalso show the following result which plays an important rolein our proofs of Theorems 3 and 4
Lemma 16 Let 119898 ge 2 1199021 119902
119898isin [1infin) and 119902 isin (0infin)
with 1119902 = sum119898
119896=11119902119896 Assume that 1199081199021
1 119908
119902119898
119898isin 119860infin
and]= prod119898
119894=1119908119894 then for any ball 119861 there exists a constant119862 gt 0
such that119898
prod
119894=1
(int
119861
119908119902119894
119894(119909) 119889119909)
119902119902119894
le 119862int
119861
](119909)119902
119889119909 (59)
Proof of Theorem 3 Arguing as in the proof ofTheorem 1 fixa ball119861 = 119861(119909
0 119903119861) sube R119899 and decompose119891
119894= 1198910
119894+119891infin
119894 where
1198910
119894= 1198911198941205942119861 119894 = 1 119898 Since 119868
120572is an119898-linear operator then
we have1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911 119891
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
le
1
]119902
(119861)120581119901
(int
119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898) (119909)
10038161003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
+sum
1015840 1
]119902
(119861)120581119901
(int
119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
119902
](119909)119902
119889119909)
1119902
= 1198690
+sum
1015840
1198691205721120572119898
(60)
where each term of sum1015840 contains at least one 120572119894= 0 In view of
Lemma 9 we can see that (])119902
isin 119860119898119902 UsingTheoremC and
Lemmas 16 and 5 we get
1198690
le 119862 sdot
1
]119902
(119861)120581119901
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
]119902
(119861)120581119901
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
(prod119898
119894=1119908119902119894
119894(2119861)119902119902119894)
120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
]119902
(2119861)120581119901
]119902
(119861)120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(61)
8 Journal of Function Spaces and Applications
For the other terms let us first deal with the case when 1205721=
sdot sdot sdot = 120572119898= infin By the definition of 119868
120572 for any 119909 isin 119861 we
obtain
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
= int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
1198891199101sdot sdot sdot119889119910119898
=
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899minus120572119898
119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(62)
Moreover by using Holderrsquos inequality the multiple 119860119902
condition and Lemma 16 we deduce that
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times (int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119902+sum119898
119894=1(1minus1119901
119894)
]119902
(2119895+1119861)1119902
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)119908119902119894
119894(2119895+1
119861)
120581119902119901119902119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
[
[
[
[
(prod119898
119894=1119908119902119894
119894(2119895+1
119861)
119902119902119894
)
120581119901
]119902
(2119895+1119861)1119902
]
]
]
]
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(63)
Since (])119902
isin 119860119898119902
sub 119860infin then it follows immediately from
Lemma 7 that
]119902
(119861)
]119902
(2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840
(64)
Hence
119869infininfin
le ]119902
(119861)1119902minus120581119901 1003816
100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(65)
where in the last inequality we have used the fact that 0 lt 120581 lt119901119902 and 1205751015840 gt 0 We now consider the case where exactly ℓ ofthe 120572119894areinfin for some 1 le ℓ lt 119898 We only give the arguments
for one of these cases The rest are similar and can easily beobtained from the arguments belowby permuting the indicesUsing the definition of 119868
120572again we can see that for any 119909 isin 119861
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
= int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(66)
and we arrive at the expression considered in the previouscase Thus for any 119909 isin 119861 we also have
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(67)
Journal of Function Spaces and Applications 9
Therefore by the inequality (64) and the above pointwiseinequality we obtain
1198691205721120572119898
le ]119902
(119861)1119902minus120581119901
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(68)
Summarizing the estimates derived above and then takingthe supremum over all balls 119861 sube R119899 we finish the proof ofTheorem 3
Proof of Theorem 4 As before fix a ball 119861 = 119861(1199090 119903119861) sube R119899
and split119891119894into119891
119894= 1198910
119894+119891infin
119894 where1198910
119894= 1198911198941205942119861 119894 = 1 119898
Then for each fixed 120582 gt 0 we can write
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911 119891
119898)1003816100381610038161003816gt 120582)1119902
le ]119902
(119909 isin 119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119902
+sum
1015840
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119902
= 1198690
lowast+sum
1015840
1198691205721120572119898
lowast
(69)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 9
again we know that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin UsingTheorem D and Lemmas 16 and 5 we have
1198690
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(70)
In the proof of Theorem 3 we have already proved thefollowing pointwise estimate (see (62) and (66)) Consider
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(71)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1 By
using Holderrsquos inequality the multiple 119860119902
condition andLemma 16 we obtain1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(72)
Note that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin Hence it followsfrom the inequality (64) that for any 119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(73)
If 119909 isin 119861 |119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198691205721120572119898
lowastle
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(74)
holds trivially Now if instead we assume that 119909 isin 119861
|119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (73) we get
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(75)
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
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
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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 and Applications
For the other terms let us first deal with the case when 1205721=
sdot sdot sdot = 120572119898= infin By the definition of 119868
120572 for any 119909 isin 119861 we
obtain
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
= int
(R119899)119898(2119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
1198891199101sdot sdot sdot119889119910119898
=
infin
sum
119895=1
int
(2119895+1119861)119898(2119895119861)119898
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
infin
sum
119895=1
119898
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
1003816100381610038161003816119909 minus 119910119894
1003816100381610038161003816
119899minus120572119898
119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(62)
Moreover by using Holderrsquos inequality the multiple 119860119902
condition and Lemma 16 we deduce that
1003816100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times (int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
sdot
100381610038161003816100381610038162119895+1
119861
10038161003816100381610038161003816
1119902+sum119898
119894=1(1minus1119901
119894)
]119902
(2119895+1119861)1119902
times
119898
prod
119894=1
(1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)119908119902119894
119894(2119895+1
119861)
120581119902119901119902119894
)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
[
[
[
[
(prod119898
119894=1119908119902119894
119894(2119895+1
119861)
119902119902119894
)
120581119901
]119902
(2119895+1119861)1119902
]
]
]
]
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(63)
Since (])119902
isin 119860119898119902
sub 119860infin then it follows immediately from
Lemma 7 that
]119902
(119861)
]119902
(2119895+1119861)
le 119862(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840
(64)
Hence
119869infininfin
le ]119902
(119861)1119902minus120581119901 1003816
100381610038161003816119868120572(119891infin
1 119891
infin
119898) (119909)
1003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(65)
where in the last inequality we have used the fact that 0 lt 120581 lt119901119902 and 1205751015840 gt 0 We now consider the case where exactly ℓ ofthe 120572119894areinfin for some 1 le ℓ lt 119898 We only give the arguments
for one of these cases The rest are similar and can easily beobtained from the arguments belowby permuting the indicesUsing the definition of 119868
120572again we can see that for any 119909 isin 119861
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
= int
(R119899)ℓ(2119861)ℓ
int
(2119861)119898minusℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
119898(119910119898)1003816100381610038161003816
(1003816100381610038161003816119909 minus 1199101
1003816100381610038161003816+ sdot sdot sdot +
1003816100381610038161003816119909 minus 119910119898
1003816100381610038161003816)119898119899minus120572
times 1198891199101sdot sdot sdot 119889119910119898
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
times int
(2119895+1119861)ℓ(2119895119861)ℓ
10038161003816100381610038161198911(1199101) sdot sdot sdot 119891
ℓ(119910ℓ)10038161003816100381610038161198891199101sdot sdot sdot 119889119910ℓ
le 119862
119898
prod
119894=ℓ+1
int
2119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
infin
sum
119895=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
119898minus120572119899
ℓ
prod
119894=1
int
2119895+11198612119895119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(66)
and we arrive at the expression considered in the previouscase Thus for any 119909 isin 119861 we also have
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(67)
Journal of Function Spaces and Applications 9
Therefore by the inequality (64) and the above pointwiseinequality we obtain
1198691205721120572119898
le ]119902
(119861)1119902minus120581119901
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(68)
Summarizing the estimates derived above and then takingthe supremum over all balls 119861 sube R119899 we finish the proof ofTheorem 3
Proof of Theorem 4 As before fix a ball 119861 = 119861(1199090 119903119861) sube R119899
and split119891119894into119891
119894= 1198910
119894+119891infin
119894 where1198910
119894= 1198911198941205942119861 119894 = 1 119898
Then for each fixed 120582 gt 0 we can write
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911 119891
119898)1003816100381610038161003816gt 120582)1119902
le ]119902
(119909 isin 119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119902
+sum
1015840
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119902
= 1198690
lowast+sum
1015840
1198691205721120572119898
lowast
(69)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 9
again we know that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin UsingTheorem D and Lemmas 16 and 5 we have
1198690
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(70)
In the proof of Theorem 3 we have already proved thefollowing pointwise estimate (see (62) and (66)) Consider
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(71)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1 By
using Holderrsquos inequality the multiple 119860119902
condition andLemma 16 we obtain1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(72)
Note that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin Hence it followsfrom the inequality (64) that for any 119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(73)
If 119909 isin 119861 |119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198691205721120572119898
lowastle
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(74)
holds trivially Now if instead we assume that 119909 isin 119861
|119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (73) we get
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(75)
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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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
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces and Applications 9
Therefore by the inequality (64) and the above pointwiseinequality we obtain
1198691205721120572119898
le ]119902
(119861)1119902minus120581119901
10038161003816100381610038161003816119868120572(119891infin
1 119891
infin
ℓ 1198910
ℓ+1 119891
0
119898) (119909)
10038161003816100381610038161003816
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(68)
Summarizing the estimates derived above and then takingthe supremum over all balls 119861 sube R119899 we finish the proof ofTheorem 3
Proof of Theorem 4 As before fix a ball 119861 = 119861(1199090 119903119861) sube R119899
and split119891119894into119891
119894= 1198910
119894+119891infin
119894 where1198910
119894= 1198911198941205942119861 119894 = 1 119898
Then for each fixed 120582 gt 0 we can write
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911 119891
119898)1003816100381610038161003816gt 120582)1119902
le ]119902
(119909 isin 119861
10038161003816100381610038161003816119868120572(1198910
1 119891
0
119898)
10038161003816100381610038161003816gt
120582
2119898)
1119902
+sum
1015840
]119902
(119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898)1003816100381610038161003816gt
120582
2119898)
1119902
= 1198690
lowast+sum
1015840
1198691205721120572119898
lowast
(69)
where each termofsum1015840contains at least one120572119894= 0 By Lemma 9
again we know that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin UsingTheorem D and Lemmas 16 and 5 we have
1198690
lowastle
119862
120582
119898
prod
119894=1
(int
2119861
1003816100381610038161003816119891119894(119909)
1003816100381610038161003816
119901119894
119908119894(119909)119901119894119889119909)
1119901119894
le
119862 sdot prod119898
119894=1119908119902119894
119894(2119861)120581119902119901119902
119894
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(2119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(70)
In the proof of Theorem 3 we have already proved thefollowing pointwise estimate (see (62) and (66)) Consider
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
119898
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
(71)
Without loss of generality we may assume that 1199011= sdot sdot sdot =
119901ℓ
= min1199011 119901
119898 = 1 and 119901
ℓ+1 119901
119898gt 1 By
using Holderrsquos inequality the multiple 119860119902
condition andLemma 16 we obtain1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119889119910119894
le 119862
infin
sum
119895=1
ℓ
prod
119894=1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
times int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816119908119894(119910119894) 119889119910119894( inf119910119894isin2119895+1119861
119908119894(119910119894))
minus1
times
119898
prod
119894=ℓ+1
1
10038161003816100381610038162119895+11198611003816100381610038161003816
1minus120572119898119899
(int
2119895+1119861
1003816100381610038161003816119891119894(119910119894)1003816100381610038161003816
119901119894
119908119894(119910119894)119901119894
119889119910119894)
1119901119894
times (int
2119895+1119861
119908119894(119910119894)minus1199011015840
119894119889119910119894)
11199011015840
119894
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
infin
sum
119895=1
]119902
(2119895+1
119861)
120581119901minus1119902
(72)
Note that (])119902
isin 119860119898119902
with 1 lt 119898119902 lt infin Hence it followsfrom the inequality (64) that for any 119909 isin 119861
1003816100381610038161003816119868120572(1198911205721
1 119891
120572119898
119898) (119909)
1003816100381610038161003816
= 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
]119902
(119861)1119902minus120581119901
]119902
(2119895+1119861)1119902minus120581119901
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
sdot
1
]119902
(119861)1119902minus120581119901
infin
sum
119895=1
(
|119861|
10038161003816100381610038162119895+11198611003816100381610038161003816
)
1205751015840(1119902minus120581119901)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(73)
If 119909 isin 119861 |119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
= Oslash then theinequality
1198691205721120572119898
lowastle
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(74)
holds trivially Now if instead we assume that 119909 isin 119861
|119868120572(1198911205721
1 119891
120572119898
119898)(119909)| gt 1205822
119898
=Oslash then by the pointwiseinequality (73) we get
120582 lt 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)sdot
1
]119902
(119861)1119902minus120581119901
(75)
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
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
10 Journal of Function Spaces and Applications
which in turn gives that
]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (76)
Therefore
1198691205721120572119898
lowastle ]119902
(119861)1119902
le
119862 sdot ]119902
(119861)120581119901
120582
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894)
(77)
Collecting all the above estimates and then taking thesupremum over all balls 119861 sube R119899 and all 120582 gt 0 we concludethe proof of Theorem 4
By using Holderrsquos inequality it is easy to verify that if 1 le119901119894lt 119902119894 1119902 = sum119898
119896=11119902119896 and each119908
119894is in119860
119901119894119902119894
thenwe have
119898
prod
119894=1
119860119901119894119902119894
sub 119860119902 (78)
and this inclusion is strict (see [17]) Also recall that119908 isin 119860119901119902
if and only if 119908119902 isin 1198601+119902119901
1015840 sub 119860infin
(see [36]) Thus asstraightforward consequences ofTheorems 3 and 4 we finallyobtain the following
Corollary 17 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin (1infin)
1119901 = sum119898
119896=11119901119896 1119902119896= 1119901
119896minus120572119898119899 and 1119902 = sum119898
119896=11119902119896=
1119901 minus 120572119899 and = (1199081 119908
119898) isin prod
119898
119894=1119860119901119894119902119894
then for any0 lt 120581 lt 119901119902 there exists a constant 119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (79)
where ]= prod119898
119894=1119908119894
Corollary 18 Let 119898 ge 2 let 0 lt 120572 lt 119898119899 and let 119868120572be an
119898-linear fractional integral operator If 1199011 119901
119898isin [1infin)
min1199011 119901
119898 = 1 1119901 = sum
119898
119896=11119901119896 1119902119896= 1119901
119896minus 120572119898119899
and 1119902 = sum119898
119896=11119902119896= 1119901 minus 120572119899 and = (119908
1 119908
119898) isin
prod119898
119894=1119860119901119894119902119894
then for any 0 lt 120581 lt 119901119902 there exists a constant119862 gt 0 independent of
119891 = (1198911 119891
119898) such that
10038171003817100381710038171003817119868120572(
119891)
10038171003817100381710038171003817119882119871119902120581119902119901((])119902)
le 119862
119898
prod
119894=1
1003817100381710038171003817119891119894
1003817100381710038171003817119871119901119894120581119901119894119902119901119902119894 (119908
119901119894
119894119908119902119894
119894) (80)
where ]= prod119898
119894=1119908119894
References
[1] R R Coifman and Y Meyer ldquoOn commutators of singularintegrals and bilinear singular integralsrdquo Transactions of theAmerican Mathematical Society vol 212 pp 315ndash331 1975
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] L Grafakos and R H Torres ldquoMaximal operator and weightednorm inequalities for multilinear singular integralsrdquo IndianaUniversity Mathematics Journal vol 51 no 5 pp 1261ndash12762002
[4] L Grafakos and R H Torres ldquoOnmultilinear singular integralsof Calderon-Zygmund typerdquo Publicacions Matematiques pp57ndash91 2002
[5] L Grafakos and N Kalton ldquoMultilinear Calderon-Zygmundoperators on Hardy spacesrdquo Collectanea Mathematica vol 52no 2 pp 169ndash179 2001
[6] G E Hu and Y Meng ldquoMultilinear Calderon-Zygmund oper-ator on products of Hardy spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 2 pp 281ndash294 2012
[7] W J Li Q Y Xue and K Yabuta ldquoMaximal operator formultilinear Calderon-Zygmund singular integral operators onweighted Hardy spacesrdquo Journal of Mathematical Analysis andApplications vol 373 no 2 pp 384ndash392 2011
[8] W J Li Q Y Xue and K Yabuta ldquoMultilinear Calderon-Zygmund operators on weighted Hardy spacesrdquo Studia Math-ematica vol 199 no 1 pp 1ndash16 2010
[9] A K Lerner S Ombrosi C Perez R H Torres and R Trujillo-Gonzalez ldquoNew maximal functions and multiple weights forthe multilinear Calderon-Zygmund theoryrdquo Advances in Math-ematics vol 220 no 4 pp 1222ndash1264 2009
[10] L Grafakos ldquoOn multilinear fractional integralsrdquo Studia Math-ematica vol 102 no 1 pp 49ndash56 1992
[11] T Iida E Sato Y Sawano and H Tanaka ldquoWeighted norminequalities for multilinear fractional operators on Morreyspacesrdquo Studia Mathematica vol 205 no 2 pp 139ndash170 2011
[12] T Iida E Sato Y Sawano and H Tanaka ldquoMultilinear frac-tional integrals on Morrey spacesrdquo Acta Mathematica Sinica(English Series) vol 28 no 7 pp 1375ndash1384 2012
[13] T Iida E Sato Y Sawano and H Tanaka ldquoSharp boundsfor multilinear fractional integral operators on Morrey typespacesrdquo Positivity vol 16 no 2 pp 339ndash358 2012
[14] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[15] G Pradolini ldquoWeighted inequalities and pointwise estimatesfor the multilinear fractional integral and maximal operatorsrdquoJournal of Mathematical Analysis and Applications vol 367 no2 pp 640ndash656 2010
[16] L Tang ldquoEndpoint estimates for multilinear fractional inte-gralsrdquo Journal of the Australian Mathematical Society vol 84no 3 pp 419ndash429 2008
[17] K Moen ldquoWeighted inequalities for multilinear fractionalintegral operatorsrdquo Collectanea Mathematica vol 60 no 2 pp213ndash238 2009
[18] X Chen and Q Y Xue ldquoWeighted estimates for a class ofmultilinear fractional type operatorsrdquo Journal of MathematicalAnalysis and Applications vol 362 no 2 pp 355ndash373 2010
[19] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[20] D R Adams ldquoA note on Riesz potentialsrdquo Duke MathematicalJournal vol 42 no 4 pp 765ndash778 1975
[21] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[22] J Peetre ldquoOn the theory of L119901120582
spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969
Journal of Function Spaces and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
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 and Applications 11
[23] D S Fan S Z Lu and D C Yang ldquoRegularity in Morreyspaces of strong solutions to nondivergence elliptic equationswith VMO coefficientsrdquo Georgian Mathematical Journal vol 5no 5 pp 425ndash440 1998
[24] G Di Fazio and M A Ragusa ldquoInterior estimates in Morreyspaces for strong solutions to nondivergence form equationswith discontinuous coefficientsrdquo Journal of Functional Analysisvol 112 no 2 pp 241ndash256 1993
[25] G Di Fazio D K Palagachev and M A Ragusa ldquoGlobalMorrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coefficientsrdquo Journal ofFunctional Analysis vol 166 no 2 pp 179ndash196 1999
[26] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[27] H Wang ldquoSome estimates for commutators of Calderon-Zygmund operators on the weighted Morrey spacesrdquo ScientiaSinica Mathematica vol 42 no 1 pp 31ndash45 2012
[28] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 4 pp 589ndash600 2012
[29] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012
[30] H Wang ldquoBoundedness of fractional integral operators withrough kernels on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 175ndash186 2013
[31] H Wang ldquoSome estimates for commutators of fractional inte-gral operators on weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 56 pp 889ndash906 2013
[32] H Wang ldquoSome estimates for commutators of fractional inte-grals associated to operators with Gaussian kernel bounds onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 72ndash85 2013
[33] HWang ldquoWeak type estimates for intrinsic square functions onweighted Morrey spacesrdquo Analysis in Theory and Applicationsvol 29 pp 104ndash119 2013
[34] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica (Chinese Series) vol 55 no 3 pp 551ndash560 2012
[35] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972
[36] BMuckenhoupt andRWheeden ldquoWeighted norm inequalitiesfor fractional integralsrdquo Transactions of the American Mathe-matical Society vol 192 pp 261ndash274 1974
[37] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Publishing Amsterdam TheNetherlands 1985
[38] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2000
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
![Multilinear commutators of fractional integrals over ...harmonic analysis. Hu et al. [6] studied the Lp-boundedness and certain weak type endpoint estimates for multilinear commutators](https://static.fdocuments.net/doc/165x107/5f707d9ebb35dc2f8c119134/multilinear-commutators-of-fractional-integrals-over-harmonic-analysis-hu-et.jpg)
