research article multilinear singular integrals and ...singular integral on herz space with variable...
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Research ArticleMultilinear Singular Integrals and Commutators onHerz Space with Variable Exponent
Amjad Hussain1,2 and Guilian Gao1,3
1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China2Department of Mathematics, HITEC University, Taxila, Pakistan3 School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
Correspondence should be addressed to Amjad Hussain; [email protected]
Received 7 October 2013; Accepted 23 December 2013; Published 12 February 2014
Academic Editors: V. Maiorov and Q. Xu
Copyright Β© 2014 A. Hussain and G. Gao. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
The paper establishes some sufficient conditions for the boundedness of singular integral operators and their commutators fromproducts of variable exponent Herz spaces to variable exponent Herz spaces.
1. Introduction
In recent years, the interest in multilinear analysis for study-ing boundedness properties of multilinear integral operatorshas grown rapidly. The subject was founded by Coifman andMeyer [1] in their seminal work on singular integral operatorslike CalderoΜn commutators and pseudosdifferential oper-ators having multiparameter function input. Subsequently,many authors including Christ and JourneΜ [2], Kenig andStein [3], and Grafakos and Torres [4] have substantiallyadded to the exiting theory.
Let πΎ be a locally integrable function defined away fromthe diagonal π₯ = π¦
1= β β β = π¦
πin (Rπ)π+1, which for πΆ > 0
satisfies the estimates
πΎ (π₯, π¦1, . . . , π¦π) β€
πΆ
(π₯ β π¦1
+ β β β +π₯ β π¦π
)ππ, (1)
and for π > 0,
πΎ (π₯, π¦
1, . . . , π¦
π) β πΎ (π₯
, π¦1, . . . , π¦
π)
β€
πΆπ₯ β π₯
π
(βπ
π=1
π₯ β π¦π
)ππ+π
,
(2)
whenever |π₯ β π₯| β€ (1/2)max{|π₯ β π¦1|, . . . , |π₯ β π¦
π|}, and
πΎ (π₯, π¦
1, . . . , π¦
π, . . . , π¦
π) β πΎ (π₯
, π¦1, . . . , π¦
π, . . . , π¦
π)
β€
πΆπ¦πβ π¦
π
π
(βπ
π=1
π₯ β π¦π
)ππ+π
,
(3)
whenever |π¦πβπ¦
π| β€ (1/2)max{|π₯βπ¦
1|, . . . , |π₯βπ¦
π|} for all 1 β€
π β€ π. Then πΎ is called π-linear CalderoΜn-Zygmund kernel.In this paper, we consider an π-linear singular integraloperator π associated with the kernel πΎ, which is initiallydefined on product of the Schwartz space S(Rπ) and takesits values in the space of tempered distribution S(Rπ) suchthat
π (π1, . . . , π
π) (π₯)
= β«(Rπ)π
πΎ(π₯, π¦1, . . . , π¦
π) π1(π¦1) β β β π
π(π¦π) ππ¦1β β β ππ¦π,
(4)
for π₯ β βππ=1
suppππ, where π
1, . . . , π
πβ πΏβ
πΆ(Rπ), the space
of compactly supported bounded functions. If π is boundedfrom πΏπ1 Γ β β β Γ πΏππ to πΏπ with 1 < π
1β β β ππ
< β and1/π1+ β β β + 1/π
π= 1/π, then we say that π is an π-linear
CalderoΜn-Zygmund operator. It has been proved in [4] that
Hindawi Publishing CorporationISRN Mathematical AnalysisVolume 2014, Article ID 626327, 10 pageshttp://dx.doi.org/10.1155/2014/626327
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2 ISRNMathematical Analysis
π is a bounded operator on product of Lebesgue spaces andendpoint weak estimates hold. For the boundedness of π andits commutators on the product of Herz-type spaces we referthe reader to see [5, 6] and [7], respectively.
In the last few decades, however, a number of researchpapers have appeared in the literature which study theboundedness of integral operators, including the maximalfunction, singular operators, and fractional integral andcommutators on function spaces with nonstandard growthconditions. Such kind of spaces is named as variable exponentfunction spaces which include variable exponent Lebesgue,Sobolev, Lorentz, Orlicz, and Herz-type function spaces.Among them the most fundamental and widely exploredspace is the Lebesgue space πΏπ(π₯) with the exponent πdepending on the point π₯ of the space. We will describe itbriefly in the next section; however, we refer to the book[8] and the survey paper [9] for historical background andrecent developments in the theory of πΏπ(π₯) spaces. Despitethe progress made, the problems of boundedness of mul-tilinear singular integral operators and their commutatorson πΏπ(π₯) spaces remain open. Recently, Huang and Xu [10]proved the boundedness of such integral operators on theproduct of variable exponent Lebesgue space. Motivated bytheir results, in this paper we will study the multilinearsingular integral on Herz space with variable exponent.Similar results for the boundedness of commutators gen-erated by these operators and BMO functions are alsoprovided.
Herz-type spaces are an important class of functionspaces in harmonic analysis. In [11, 12], Izuki independentlyintroduced Herz space with variable exponent π, by keepingthe remaining two exponents πΌ and π as constants. Variabilityof alpha was recently considered by Almeida and Drihem[13] in proving the boundedness results for some classicaloperators on such spaces. More recently, Samko [14] intro-duced the generalized Herz-type spaces where all the threeexponents were allowed to vary. In this paper, we will studythe multilinear singular operators on variable exponent Herzspace οΏ½ΜοΏ½πΌ,π
π(β )introduced in [12].
Throughout this paper, πΆ denotes a positive constantwhich may change from one occurrence to another. The nextsection contains some basic definitions and the main resultsof this study. Finally, the last section includes the proofs ofmain results along with some supporting lemmas.
2. Main Results
Let Ξ© be a measurable subset of Rπ with Lebesgue measure|Ξ©| > 0. Given a measurable function π(β ) : Ξ© β [1,β],then for someπ > 0wedefine the variable exponent Lebesguespace as
πΏπ(β )
(Ξ©) := {π is measurable : β«Ξ©
(
π (π₯)
π)
π(π₯)
ππ₯ < β} ,
(5)
and the space πΏπ(β )loc (Ξ©) as
πΏπ(β )
loc (Ξ©) := {π : π β πΏπ(β )
(πΎ) β compact subset πΎ β Ξ©} .(6)
The Lebesgue space πΏπ(β )(Ξ©) becomes a Banach functionspace when equipped with the norm
ππΏπ(β )(Ξ©)
:= inf {π > 0 : β«Ξ©
(
π (π₯)
π)
π(π₯)
ππ₯ β€ 1} . (7)
Given a locally integrable function π on Ξ©, the Hardy-Littlewood maximal operatorπ is defined by
ππ(π₯) := supπ>0
πβπβ«π΅(π₯,π)β©Ξ©
π (π¦) ππ¦, (8)
where π΅(π₯, π) := {π¦ β Rπ : |π₯ β π¦| < π}. We denote
πβ:= ess inf {π (π₯) : π₯ β Ξ©} ,
π+:= ess sup {π (π₯) : π₯ β Ξ©} .
(9)
We also defineP (Ξ©) := {π (β ) : π
β> 1, π
+< β} ,
B (Ξ©) := {π (β ) : π (β ) β P (Ξ©) ,
π is bounded on πΏπ(β ) (Ξ©)} .
(10)
Cruz-Uribe et al. [15] and Nekvinda [16] independentlyproved the following sufficient conditions for the bounded-ness ofπ on πΏπ(β )(Ξ©).
Proposition 1. Let Ξ© be an open set. If π(β ) β P(Ξ©) satisfies
π (π₯) β π (π¦) β€
πΆ
β log (π₯ β π¦), if π₯ β π¦
β€1
2,
π (π₯) β π (π¦) β€
πΆ
log (π + |π₯|), if π¦
β₯ |π₯| ,
(11)
then one has π(β ) β B(Ξ©).
Define P0(Ξ©) to be the set of measurable functions π :Ξ© β (0,β) such that
πβ:= ess inf {π (π₯) : π₯ β Ξ©} > 0,
π+:= ess sup {π (π₯) : π₯ β Ξ©} < β.
(12)
Given π β P0(Ξ©), one can define the space πΏπ(β ) as above.Since we will not use it in the this paper, we omit the detailshere and refer the reader to see [10, 17].
Let π΅π:= {π₯ : |π₯| β€ 2
π}, π΄π= π΅π\ π΅πβ1
and ππ= ππ΄π
bethe characteristic function of the set π΄
πfor π β Z.
Definition 2. For πΌ β R, 0 < π β€ β and π(β ) β P(Rπ), thehomogeneous Herz space with variable exponentπΎπΌ,π
π(β )(Rπ) is
defined by
οΏ½ΜοΏ½πΌ,π
π(β )(Rπ) := {πΏ
π(β )
loc (Rπ\ {0}) :
ποΏ½ΜοΏ½πΌ,π
π(β )(Rπ)
< β} , (13)
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ISRNMathematical Analysis 3
where
ποΏ½ΜοΏ½πΌ,π
π(β )(Rπ)
:= {
β
β
π=ββ
2ππΌππππ
π
πΏπ(β )(Rπ)
}
1/π
. (14)
In the sequel, unless stated otherwise, we will work onthe whole space Rπ and will not mention it. Taking π
1and π2
in π΅ππ, Huang and Xu in [10] define the three commutatoroperators for suitable functions π and π. One of them is theoperator
[π1, π2, π] (π, π) (π₯)
= π1(π₯) π2(π₯) π (π, π) (π₯) β π
1(π₯) π (π, π
2π) (π₯)
β π2(π₯) π (π
1π, π) (π₯) + π (π
1π, π2π) (π₯) .
(15)
As corollaries of their main results they give the followingestimates.
Theorem A. Let π be 2-linear CalderoΜn-Zygmund operatorand π(β ) β P0(Rπ). If π
1(β ), π2(β ) β B(Rπ) such that 1/π(π₯) =
1/π1(π₯) + 1/π
2(π₯), then there exists a constant πΆ independent
of the functions π, πββ πΏπ1(β ), π, π
ββ πΏπ2(β ) and β β N such
thatπ(π, π)
πΏπ(β )β€ πΆ
ππΏπ1(β )
ππΏπ2(β )
,
(
β
β
β=1
π (πβ, πβ)
π
)
1/ππΏπ(β )
β€ πΆ
(
β
β
β=1
πβ
π1
)
1/π1πΏπ1(β )
Γ
(
β
β
β=1
πβ
π2
)
1/π2πΏπ2(β )
(16)
hold, where 1 < ππ< β for π = 1, 2 and 1/π = 1/π
1+ 1/π2.
Theorem B. Let π be 2-linear CalderoΜn-Zygmund operator,π1, π2β π΅ππ(Rπ), andπ(β ) β P0(Rπ). Ifπ
1(β ), π2(β ) β B(Rπ)
such that 1/π(π₯) = 1/π1(π₯) + 1/π
2(π₯), then there exists a
constant πΆ independent of the functions π, πββ πΏπ1(β ), π, π
ββ
πΏπ2(β ) and β β N such that[π1, π2, π](π, π)
Lπ(β ) β€ πΆπ1
β
π2β
ππΏπ1(β )
ππΏπ2(β )
,
(
β
β
β=1
[π1, π2, π] (πβ, πβ)
π
)
1/ππΏπ(β )
β€
(
β
β
β=1
πβ
π1
)
1/π1πΏπ1(β )
(
β
β
β=1
πβ
π2
)
1/π2πΏπ2(β )
Γ πΆπ1
β
π2β
(17)
hold, where 1 < ππ< β for π = 1, 2 and 1/π = 1/π
1+ 1/π2.
Motivated by these results, here we give the following twotheorems.
Theorem 3. Let π be 2-linear CalderoΜn-Zygmund operatorand π(β ) β P(Rπ). Furthermore, let π
1(β ), π2(β ) β B(Rπ),
0 < ππ< β, βππΏ
ππ
< πΌπ< ππΏπ
π
, π = 1, 2, where πΏππ
, πΏπ
π
> 0
are constants defined in the next section such that 1/π(π₯) =1/π1(π₯) + 1/π
2(π₯), 1/π = 1/π
1+ 1/π2, and πΌ = πΌ
1+ πΌ2. If π is
bounded from πΏπ1(β ) Γ πΏπ2(β ) to πΏπ(β ), thenπ(π, π)
οΏ½ΜοΏ½πΌ,π
π(β )
β€ πΆποΏ½ΜοΏ½πΌ1,π1
π1(β )
ποΏ½ΜοΏ½πΌ2,π2
π2(β )
,
(
β
β
β=1
π (πβ, πβ)
π
)
1/ποΏ½ΜοΏ½πΌ,π
π(β )
β€ πΆ
(
β
β
β=1
πβ
π1
)
1/π1οΏ½ΜοΏ½πΌ1,π1
π1(β )
Γ
(
β
β
β=1
πβ
π2
)
1/π2οΏ½ΜοΏ½πΌ2,π2
π2(β )
(18)
hold for all π, πββ οΏ½ΜοΏ½πΌ1,π1
π1(β ), π, πββ οΏ½ΜοΏ½πΌ2,π2
π2(β ), where 1 < π
π< β for
π = 1,2 and 1/π = 1/π1+ 1/π2.
Theorem 4. Let π be 2-linear CalderoΜn-Zygmund operator,π1, π2
β π΅ππ(Rπ), and π(β ) β P(Rπ). Furthermore, letπ1(β ), π2(β ) β B(Rπ), 0 < π
π< β, βππΏ
ππ
< πΌπ< ππΏπ
π
, π = 1, 2,where πΏ
ππ
, πΏπ
π
> 0 are constants defined in the next section,such that 1/π(π₯) = 1/π
1(π₯) + 1/π
2(π₯), 1/π = 1/π
1+ 1/π
2,
πΌ = πΌ1+πΌ2. If [π1, π2, π] is bounded from πΏπ1(β ) ΓπΏπ2(β ) to πΏπ(β ),
then[π1, π2, π](π, π)
οΏ½ΜοΏ½πΌ,π
π(β )
β€ πΆπ1
β
π2β
ποΏ½ΜοΏ½πΌ1,π1
π1(β )
ποΏ½ΜοΏ½πΌ2,π2
π2(β )
,
(
β
β
β=1
[π1, π2, π] (πβ, πβ)
π
)
1/ποΏ½ΜοΏ½πΌ,π
π(β )
β€
(
β
β
β=1
πβ
π1
)
1/π1οΏ½ΜοΏ½πΌ1,π1
π1(β )
(
β
β
β=1
πβ
π2
)
1/π2οΏ½ΜοΏ½πΌ2,π2
π2(β )
Γ πΆπ1
β
π2β
(19)
hold for all π, πββ οΏ½ΜοΏ½πΌ1,π1
π1(β ), π, πββ οΏ½ΜοΏ½πΌ2,π2
π2(β ), where 1 < π
π< β for
π = 1, 2 and 1/π = 1/π1+ 1/π2.
3. Proofs of the Main Results
In this section, we will prove main results stated in the lastsection.The ideas of these proofsmainly come from [5, 7].Weuse the notation π(π₯) to denote the conjugate index of π(π₯).Here we give some lemmas which will be helpful in provingTheorems 3 and 4.
Lemma 5 (see [10, 18], generalized HoΜlderβs inequality). Letπ, π1, π2β P(Rπ).
(a) If π β Lπ(β ), π β πΏπ
(β ), then one hasππ
πΏ1β€ πΆπ
ππΏπ(β )
ππΏπ(β ), (20)
where πΆπ= 1 + 1/π
ββ 1/π
+.
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4 ISRNMathematical Analysis
(b) If π β πΏπ1(β ), π β πΏπ2(β ) and 1/π(π₯) = 1/π1(π₯) +
1/π2(π₯), then there exists a constant πΆ
π,π1
such that
πππΏπ(β )
β€ πΆπ,π1
ππΏπ1(β )
ππΏπ2(β ) (21)
holds, where πΆπ,π1
= (1 + 1/(π1)ββ 1/(π
1)+)1/πβ .
Lemma 6 (see [12]). If π β B(Rπ), then there exist a constantπΆ > 0 such that for all balls π΅ in Rπ,
ππ΅πΏπ(β )
ππ΅πΏπ(β )β€ πΆ |π΅| . (22)
Lemma 7 (see [12]). If π β B(Rπ), then there exists constantsπΏ, πΆ > 0 such that for all balls π΅ in Rπ and all measurablesubsets π β π΅,
ππ΅πΏπ(β )
πππΏπ(β )
β€ πΆ|π΅|
|π|,
πππΏπ(β )
ππ΅πΏπ(β )
β€ πΆ(|π|
|π΅|)
πΏ
. (23)
Lemma 8 (see [19, Remark 1]). If ππβ B(Rπ), π = 1, 2,
then by Proposition 1 one has ππβ B(Rπ). Therefore, applying
Lemma 7, one can take positive constants πΏππ
, πΏπ
π
> 0 such that
πππΏππ(β )
ππ΅πΏπl(β )
β€ πΆ(|π|
|π΅|)
πΏππ
,
πππΏπ
π(β )
ππ΅πΏπ
π(β )
β€ πΆ(|π|
|π΅|)
πΏπ
π
, (24)
for all balls π΅ in Rπ and all measurable subsets π β π΅.
Recently, Izuki [19] established a relationship betweenLebesgue space with variable exponent and π΅ππ spacewhich can be stated in the form of the following lemma.
Lemma 9. One has that for all π β π΅ππ(Rπ) and all π, π β Zwith π > π,
πΆβ1βπββ β€ sup
π΅:ball
1
ππ΅πΏπ(β )
(π β ππ΅)ππ΅πΏπ(β )
β€ πΆβπββ
(π β π
π΅π
) ππ΅π
πΏπ(β )β€ πΆ (π β π) βπββ
ππ΅π
πΏπ(β ).
(25)
Thenext lemma is the generalizedMinkowskiβs inequalityand is useful in proving vector valued inequalities.
Lemma 10 (see [19]). If 1 < π < β, then there exists aconstant πΆ > 0 such that for all sequences of functions {π
β}β
β=1
satisfying ββ{πβ}βββπβπΏ1Rπ
< β,
{
β
β
β=1
(β«Rπ
πβ (π¦) ππ¦)
π
}
1/π
β€ πΆβ«Rπ{
β
β
β=1
πβ (π¦)
π
}
1/π
ππ¦.
(26)
Proof of Theorem 3. In order to make computations easy, firstwe have to prove the following inequality:
2ππΌ
π (πππ, πππ) ππ
πΏπ(β )β€ πΆπ·
1(π, π) 2
ππΌ1πππ
πΏπ1(β )
Γ π·2(π, π) 2
ππΌ2
πππ
πΏπ2(β ),
(27)
where for π, π β Z, π = 1, 2,
π·π(π, π) =
{{
{{
{
2(πβπ)(πΌ
πβππΏπ
π
)
, if π β€ π β 2,1, if π β 1 β€ π β€ π + 1,2(πβπ)(πΌ
π+ππΏππ
), if π β₯ π + 2.
(28)
If πβ1 β€ π β€ π+1, πβ1 β€ π β€ π+1, then we have 2π βΌ 2π βΌ 2π;hence by the πΏπ(β ) boundedness of π, we obtain
2ππΌ
π (πππ, πππ) ππ
πΏπ(β )β€ πΆ2ππΌ1πππ
πΏπ1(β )2ππΌ2
Γπππ
πΏπ2(β ).
(29)
In the other cases, we see that |π₯βπ¦1|+|π₯βπ¦
2| βΌ 2
max(π,π,π), forπ₯ β π΄
π, π¦1β π΄π, π¦2β π΄π. Thus by the generalized HoΜlderβs
inequality,
π (ππ
π, πππ) (π₯) π
π(π₯)
β€ πΆ2β2max(π,π,π)ππππ
1
πππ
1β ππ(π₯)
β€ πΆ2β2max(π,π,π)ππππ
πΏπ1(β )ππ
πΏπ
1(β )
Γπππ
πΏπ2(β )
ππ
πΏπ
2(β )β ππ(π₯) .
(30)
Applying Lemma 5, we have
π (ππ
π, πππ) ππ
πΏπ(β )
β€ πΆ2β2max(π,π,π)ππππ
πΏπ1(β )ππ
πΏπ
1(β )
Γπππ
πΏπ2(β )
ππ
πΏπ
2(β )
πππΏπ(β )
β€ πΆ2βmax(π,π)ππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
ππ΅π
πΏπ1(β )
Γ 2βmax(π,π)ππππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
ππ΅π
πΏπ2(β ).
(31)
Now, for π β€ π β 2, π β€ π β 2, by Lemmas 6 and 8, it is easy toshow that
π (ππ
π, πππ) ππ
πΏπ(β )
β€ πΆπππ
πΏπ1(β )
ππ΅π
πΏπ
1(β )
ππ΅π
πΏπ
1(β )
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
ππ΅π
πΏπ
2(β )
β€ πΆπππ
πΏπ1(β )2β(πβπ)ππΏ
π
1
πππ
πΏπ2(β )2β(πβπ)ππΏ
π
2 .
(32)
By a similar argument for π β₯ π + 2, π β₯ π + 2, we get
π(πππ, πππ)ππ
πΏπ(β )
β€ πΆπππ
πΏπ1(β )
ππ΅π
Lπ1(β )ππ΅π
πΏπ1(β )
πππ
πΏπ2(β )
ππ΅π
πΏπ2(β )
ππ΅π
πΏπ2(β )
β€ πΆπππ
πΏπ1(β )2(πβπ)ππΏ
π1
πππ
πΏπ2(β )2(πβπ)ππΏ
π2 .
(33)
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ISRNMathematical Analysis 5
In view of (29)β(33), (27) is obvious. Now byMinkowskiβsinequality and (27), we get
2ππΌπ(π, π)ππ
πΏπ(β )β€ 2ππΌ
β
β
π=ββ
β
β
π=ββ
π(πππ, πππ)ππ
πΏπ(β )
β€
β
β
π=ββ
β
β
π=ββ
2ππΌ
π(πππ, πππ)ππ
πΏπ(β )
β€ πΆ
β
β
π=ββ
π·1(π, π) 2
ππΌ1πππ
πΏπ1(β )
Γ
β
β
π=ββ
π·2(π, π) 2
ππΌ2
πππ
πΏπ2(β ).
(34)
Since 1/π = 1/π1+ 1/π2, then by definition
π (π, π)οΏ½ΜοΏ½πΌ,π
π(β )
= {
β
β
π=ββ
2ππΌππ (π, π) ππ
π
πΏπ(β )}
1/π
β€
{
{
{
β
β
π=ββ
(
β
β
π=ββ
π·1(π, π) 2
ππΌ1πππ
πΏπ1(β )
Γ
β
β
π=ββ
π·2(π, π) 2
ππΌ2
πππ
πΏπ2(β ))
π
}
}
}
1/π
β€ πΆ{
β
β
π=ββ
(
β
β
π=ββ
π·1(π, π) 2
ππΌ1πππ
πΏπ1(β ))
π1
}
1/π1
Γ
{
{
{
β
β
π=ββ
(
β
β
π=ββ
π·2(π, π) 2
ππΌ2
πππ
πΏπ2(β ))
π2
}
}
}
1/π2
:= πΌ1Γ πΌ2.
(35)
It remains to show that πΌ1
β€ πΆβπβοΏ½ΜοΏ½πΌ1,π1
π1(β )
and πΌ2
β€
πΆβπβοΏ½ΜοΏ½πΌ2,π2
π2(β )
. By symmetry, we only approximate πΌ1. If 0 < π
1<
1, then by the well-known inequality (β |ππ|)π1
β€ β |ππ|π1 and
the inequality
β
β
π=ββ
π·1(π, π)πΎ+
β
β
π=ββ
π·2(π, π)πΎ
< β, for any πΎ > 0, (36)
we have
πΌ1β€ πΆ{
β
β
π=ββ
2ππΌ1π1πππ
π1
πΏπ1(β )
β
β
π=ββ
π·1(π, π)π1}
1/π1
β€ πΆ{
β
β
π=ββ
2ππΌ1π1πππ
π1
πΏπ1(β )}
1/π1
= πΆποΏ½ΜοΏ½πΌ1,π1
π1(β )
.
(37)
If π1> 1, HoΜlderβs inequality and inequality (36) yield
πΌ1β€ πΆ
{
{
{
β
β
π=ββ
β
β
π=ββ
π·1(π, π)π1/22ππΌ1π1πππ
π1
πΏπ1(β )
Γ {
β
β
π=ββ
π·1(π, π)π
1/2}
π1/π
1
}
}
}
1/π1
β€ πΆ{
β
β
π=ββ
2ππΌ1π1πππ
π1
πΏπ1(β )
β
β
π=ββ
π·1(π, π)π1/2}
1/π1
β€ πΆ{
β
β
π=ββ
2ππΌ1π1πππ
π1
πΏπ1(β )}
1/π1
= πΆποΏ½ΜοΏ½πΌ1,π1
π1(β )
.
(38)
Therefore, for 0 < π1< β
πΌ1β€ πΆ
ποΏ½ΜοΏ½πΌ1,π1
π1(β )
. (39)
By symmetry, for 0 < π2< β we have
πΌ2β€ πΆ
ποΏ½ΜοΏ½πΌ2,π2
π2(β )
. (40)
Finally, we obtainπ(π, π)
οΏ½ΜοΏ½πΌ,π
π(β )
β€ πΆποΏ½ΜοΏ½πΌ1,π1
π1(β )
ποΏ½ΜοΏ½πΌ2,π2
π2(β )
. (41)
By virtue of Lemma 10 and the fact that 1/π = 1/π1+ 1/π2, it
is easy to show that
(
β
β
β=1
π (πβ, πβ)
π
)
1/ποΏ½ΜοΏ½πΌ,π
π(β )
β€ πΆ
(
β
β
β=1
πβ
π1
)
1/π1οΏ½ΜοΏ½πΌ1,π1
π1(β )
Γ
(
β
β
β=1
πβ
π2
)
1/π2οΏ½ΜοΏ½πΌ2,π2
π2(β )
(42)
holds for all πββ οΏ½ΜοΏ½πΌ1,π1
π1(β ), πββ οΏ½ΜοΏ½πΌ2,π2
π2(β ), where 1 < π
π< β for
π = 1, 2.Thus the proof of Theorem 3 is complete.
Proof of Theorem 4. Similar to the proof ofTheorem 3, for thecase πβ1 β€ π β€ π+1, πβ1 β€ π β€ π+1, we useπΏπ(β ) boundednessof [π1, π2, π] to obtain
2ππΌ
[π1, π2, π](ππ
π, πππ)ππ
πΏπ(β )
β€ πΆπ1
β
π2β2ππΌ1πππ
πΏπ1(β )2ππΌ2
πππ
πΏπ2(β ).
(43)
For other possibilities we have |π₯βπ¦1|+|π₯βπ¦
2| βΌ 2
max(π,π,π)
for π₯ β π΄π, π¦1β π΄π, π¦2β π΄π.Thus, we consider the following
two cases.Case I (π β€ π β 2, π β€ π β 2). We denote (π
π)π΅π
by πππ, where
(ππ)π΅π
=1
π΅π
β«π΅π
ππ(π₯) ππ₯, π = 1, 2 (44)
-
6 ISRNMathematical Analysis
and consider the following decomposition:
[π1, π2, π] (ππ
π, πππ) (π₯)
= (π1(π₯) β π
1π) (π2(π₯) β π
2π) π (ππ
π, πππ) (π₯)
β (π1(π₯) β π
1π) π (ππ
π, (π2(β ) β π
2π) πππ) (π₯)
β (π2(π₯) β π
2π) π ((π
1(β ) β π
1π) πππ, πππ) (π₯)
+ π ((π1(β ) β π
1π) πππ, (π2(β ) β π
2π) πππ) (π₯)
= πΏ1(π₯) + πΏ
2(π₯) + πΏ
3(π₯) + πΏ
4(π₯) .
(45)
Thus,
[π1, π2, π] (ππ
π, πππ) ππ
πΏπ(β )β€
4
β
π=1
(πΏπ) πππΏπ(β )
=
4
β
π=1
π½π.
(46)
Now, we will estimate each π½π(π = 1, . . . , 4), separately.
Applying Lemma 5, we haveπΏ1 (π₯)
β€ πΆ2β2ππ π1 (π₯) β π1π
π2(π₯) β π
2π
Γπππ
πΏ1πππ
πΏ1
β€ πΆ2β2πππππ
πΏπ1(β )ππ
πΏπ
1(β )
πππ
πΏπ2(β )
ππ
πΏπ
2(β )
Γπ1 (π₯) β π1π
π2(π₯) β π
2π
.
(47)
Therefore, by virtue of generalized HoΜlderβs inequality andLemmas 6, 8, and 9, we get
π½1=(πΏ1) ππ
πΏπ(β )
β€ πΆ2β2πππππ
πΏπ1(β )ππ
πΏπ
1(β )
πππ
πΏπ2(β )
ππ
πΏπ
2(β )
Γ(π1(π₯) β π
1π) (π2(π₯) β π
2π) ππ
πΏπ(β )
β€ πΆ2β2πππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
Γ(π1(π₯) β π
1π) ππ΅π
πΏπ1(β )
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
Γ(π2(π₯) β π
2π) ππ΅π
πΏπ2(β )
β€ πΆπ1
β
π2β2β2ππ
(π β π)πππ
πΏπ1(β )
Γππ΅π
πΏπ
1(β )
ππ΅π
πΏπ1(β )
Γ (π β π)πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
ππ΅π
πΏπ2(β )
β€ πΆπ1
β
π2β (
π β π)πππ
πΏπ1(β )
Γ
ππ΅π
πΏπ
1(β )
ππ΅π
πΏπ
1(β )
(π β π)πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
ππ΅π
πΏπ
2(β )
β€ πΆπ1
β
π2β
ππππΏπ1(β ) (
π β π) 2β(πβπ)ππΏ
π
1
Γπππ
πΏπ2(β )(π β π) 2
β(πβπ)ππΏπ
2 .
(48)
Similarly, using Lemma 9, we approximate πΏ2as
πΏ2 β€ πΆ2
β2ππ π1 (π₯) β π1π
ππππΏ1
(π2(β ) β π
2π) πππ
πΏ1
β€ πΆ2β2ππ π1 (π₯) β π1π
ππππΏπ1(β )
πππΏπ
1(β )
Γπππ
πΏπ2(β )
(π2(β ) β π
2π)ππ
πΏπ
2(β )
β€ πΆπ2
β2β2ππ π1 (π₯) β π1π
ππππΏπ1(β )
Γππ΅π
πΏπ
1(β )
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
.
(49)
Therefore, in view of Lemmas 6, 9, and 8, we have
π½2=(πΏ2)ππ
πΏπ(β )
β€ πΆπ2
β2β2πππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
Γ(π1(π₯) β π1π)ππ
πΏπ(β )πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
β€ πΆπ1
β
π2β2β2πππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
Γ (π β π)ππ΅π
πΏπ(β )
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
β€ πΆπ1
β
π2β (
π β π)πππ
πΏπ1(β )
Γ
ππ΅π
πΏπ
1(β )
ππ΅π
πΏπ
1(β )
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
ππ΅π
πΏπ
2(β )
β€ πΆπ1
β
π2β
ππiπΏπ1(β ) (
π β π) 2β(πβπ)ππΏ
π
1
Γπππ
πΏπ2(β )2β(πβπ)ππΏ
π
2 .
(50)
By symmetry, the estimate for π½3is similar to that for π½
2;
therefore,
π½3β€ πΆ
π1β
π2β
ππππΏπ1(β )
2β(πβπ)ππΏ
π
1
Γπππ
πΏπ2(β )(π β π) 2
β(πβπ)ππΏπ
2 .
(51)
Finally, it remains to estimate π½4. For that, we use Lemma 9 to
write
πΏ4
=π ((π1(β ) β π
1π) πππ, (π2(β ) β π
2π) πππ) (π₯)
β€ πΆ2β2ππ(π1 (β ) β π1π) πππ
πΏ1(π2(β ) β π
2π) πππ
πΏ1
β€ πΆ2β2πππππ
πΏπ1(β )(π1(β ) β π1π)ππ
πΏπ
1(β )
πππ
πΏπ2(β )
Γ(π2(β ) β π
2π)ππ
πΏπ
2(β )
β€ πΆ2β2πππππ
πΏπ1(β )(π1(β ) β π
1π)ππ΅π
πΏπ
1(β )
πππ
πΏπ2(β )
Γ(π2(β ) β π
2π)ππ΅π
πΏπ
2(β )
-
ISRNMathematical Analysis 7
β€ πΆπ1
β
π2β2β2πππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
πππ
πΏπ2(β )
Γππ΅π
πΏπ
2(β )
.
(52)
Thus, by HoΜlderβs inequality and Lemmas 6 and 8, we obtain
π½4=(πΏ4)ππ
πΏπ(β )
β€ πΆπ1
β
π2β2β2πππππ
πΏπ1(β )ππ΅π
πΏπ1(β )
πππ
πΏπ2(β )
Γππ΅π
πΏπ2(β )
πππΏπ(β )
β€ πΆπ1
β
π2β2β2πππππ
πΏπ1(β )ππ΅π
πΏπ1(β )
ππ΅π
πΏπ1(β )
Γπππ
πΏπ2(β )
ππ΅π
πΏπ2(β )
ππ΅π
πΏπ2(β )
β€ πΆπ1
β
π2β (
π β π)πππ
πΏπ1(β )
ππ΅π
πΏπ
1(β )
ππ΅π
πΏπ
1(β )
Γπππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
ππ΅π
πΏπ
2(β )
β€ πΆπ1
β
π2β
ππππΏπ1(β )
2β(πβπ)ππΏ
π
1
πππ
πΏπ2(β )2β(πβπ)ππΏ
π
2 .
(53)
Combining the estimates for π½1, π½2, π½3, and π½
4, we have
[π1, π2, π](ππ
π, πππ)ππ
πΏπ(β )β€ πΆ
π1β
π2β
ππππΏπ1(β )
Γ (π β π) 2β(πβπ)ππΏ
π
1
πππ
πΏπ2(β )
Γ (π β π) 2β(πβπ)ππΏ
π
2 .
(54)
Case II (π β₯ π + 2, π β₯ π + 2). We denote (ππ)π΅π
by πππ, where
(ππ)π΅π
= (1/|π΅π|) β«π΅π
ππ(π₯)ππ₯, π = 1, 2. In this case, we consider
the following decomposition:
[π1, π2, π] (ππ
π, πππ) (π₯)
= (π1(π₯) β π
1π) (π2(π₯) β π
2π) π (ππ
π, πππ) (π₯)
β (π1(π₯) β π
1π) π (ππ
π, (π2(β ) β π
2π) πππ) (π₯)
β (π2(π₯) β π
2π) π ((π
1(β ) β π
1π) πππ, πππ) (π₯)
+ π ((π1(β ) β π
1π) πππ, (π2(β ) β π
2π) πππ) (π₯)
= οΏ½ΜοΏ½1(π₯) + οΏ½ΜοΏ½
2(π₯) + οΏ½ΜοΏ½
3(π₯) + οΏ½ΜοΏ½
4(π₯) .
(55)
Thus,
[π1, π2, π] (ππ
π, πππ) ππ
πΏπ(β )β€
4
β
π=1
(οΏ½ΜοΏ½π) ππ
πΏπ(β )
=
4
β
π=1
π½π.
(56)
Let us first compute π½1. As in the proof of Theorem 3, in this
case we estimate οΏ½ΜοΏ½1as
οΏ½ΜοΏ½1
β€ πΆ
π1 (π₯) β π1π
π2 (π₯) β π2π 2βπππππ
πΏ12βππ
Γπππ
πΏ1
β€ πΆπ1 (π₯) β π1π
π2 (π₯) β π2π 2βπππππ
πΏπ1(β )
Γππ
πΏπ
1(β )2βππ
πππ
πΏπ2(β )
ππ
πΏπ
2(β ).
(57)
By virtue of generalizedHoΜlderβs inequality and Lemmas 6β9,we obtain
π½1=(οΏ½ΜοΏ½1) ππ
πΏπ(β )
β€ πΆ2βπππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )2βππ
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
Γ(π1(π₯) β π1π)(π2(π₯) β b2π)ππ
πΏπ(β )
β€ πΆ2βπππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
(π1(π₯) β π1π)πππΏπ1(β )
Γ 2βππ
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
(π2 (π₯) β π2π) πππΏπ2(β )
β€ πΆπ1
β
π2β2βπππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
ππ΅π
πΏπ1(β )
Γ 2βππ
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
ππ΅π
πΏπ2(β )
β€ πΆπ1
β
π2β
ππππΏπ1(β )
ππ΅π
πΏπ1(β )
ππ΅π
πΏπ1(β )
πππ
πΏπ2(β )
ππ΅π
πΏπ2(β )
ππ΅π
πΏπ2(β )
β€ πΆπ1
β
π2β
ππππΏπ1(β )
2(πβπ)ππΏ
π1
πππ
πΏπ2(β )2(πβπ)ππΏ
π2 .
(58)
Next, we approximate π½2. Using Lemma 9, it is easy to see that
οΏ½ΜοΏ½2
β€ πΆ2β2ππ π1 (π₯) β π1π
ππππΏ1
(π2(β ) β π
2π) πππ
πΏ1
β€ πΆ2βππ π1 (π₯) β π1π
ππππΏπ1(β )
πππΏπ
1(β )
Γ 2βππ
πππ
πΏπ2(β )
(π2(β ) β π
2π)ππ
πΏπ
2(β )
β€ πΆπ2
β2βππ π1 (π₯) β π1π
ππππΏπ1(β )
ππ΅π
πΏπ
1(β )
Γ (π β π) 2βππ
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
.
(59)
Thus, in view of Lemmas 6, 9, and 8, we get
π½2=(οΏ½ΜοΏ½2) ππ
πΏπ(β )
β€ πΆπ2
β2βπππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
(π1(π₯) β π1π)πππΏπ(β )
Γ (π β π) 2βππ
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
-
8 ISRNMathematical Analysis
β€ πΆπ1
β
π2β2βπππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
ππ΅π
πΏπ(β )
Γ (π β π) 2βππ
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
β€ πΆπ1
β
π2β
ππππΏπ1(β )
ππ΅π
πΏπ1(β )
ππ΅π
πΏπ1(β )
(π β π)
Γπππ
πΏπ2(β )
ππ΅π
πΏπ2(β )
ππ΅π
πΏπ2(β )
β€ πΆπ1
β
π2β
ππππΏπ1(β )
2(πβπ)ππΏ
π1
πππ
πΏπ2(β )
Γ (π β π) 2(πβπ)ππΏ
π2 .
(60)
By symmetry, the estimate for π½3is similar to that for π½
2; thus
π½3β€ πΆ
π1β
π2β
ππππΏπ1(β ) (
π β π) 2(πβπ)ππΏ
π1
Γπππ
πΏπ2(β )2(πβπ)ππΏ
π2 .
(61)
Lastly, it remains to compute π½4. For this purpose we use
generalized HoΜderβs inequality and lemmas 6 and 9 to obtainοΏ½ΜοΏ½4
=π ((π1(β ) β π
1π) πππ, (π2(β ) β π
2π) πππ) (π₯)
β€ πΆ2βππ(π1 (β ) β π1π) πππ
πΏ12βππ
Γ(π2(β ) β π
2π)πππ
πΏ1
β€ πΆ2βπππππ
πΏπ1(β )(π1(β ) β π1π)ππ
πΏπ
1(β )
Γ 2βππ
πππ
πΏπ2(β )
(π2(β ) β π
2π)ππ
πΏπ
2(β )
β€ πΆπ1
β
π2β (
π β π) 2βπππππ
πΏπ1(β )ππ΅π
πΏπ
1(β )
Γ (π β π) 2βππ
πππ
πΏπ2(β )
ππ΅π
πΏπ
2(β )
β€ πΆπ1
β
π2β (
π β π)
ππππΏπ1(β )
ππ΅π
πΏπ1(β )
(π β π)
πππ
πΏπ2(β )
ππ΅π
πΏπ2(β )
.
(62)
Thus, an application of Lemma 8 yields
π½4=(οΏ½ΜοΏ½4)ππ
πΏπ(β )β€ πΆ
π1β
π2β (
π β π)
ππππΏπ1(β )
πππΏπ1(β )
Γ (π β π)
πππ
πΏπ2(β )
ππ
πΏπ2(β )
πππΏπ(β )
β€ πΆπ1
β
π2β (
π β π)πππ
πΏπ1(β )
ππ΅π
πΏπ1(β )
ππ΅π
πΏπ1(β )
(π β π)
Γπππ
πΏπ2(β )
ππ΅π
πΏπ2(β )
ππ΅π
πΏπ2(β )
β€ πΆπ1
β
π2β (
π β π)πππ
πΏπ1(β )2(πβπ)ππΏ
π1
Γ (π β π)πππ
πΏπ2(β )2(πβπ)ππΏ
π2 .
(63)
Combining the estimates for π½1, π½2, π½3, and π½
4, we have
[π1, π2, π](ππ
π, πππ)ππ
πΏπ(β )
β€ πΆπ1
β
π2β
ππππΏπ1(β ) (
π β π) 2(πβπ)ππΏ
π1
Γπππ
πΏπ2(β )(π β π) 2
(πβπ)ππΏπ2 .
(64)
In view of (43), (54), and (64), we arrive at
2ππΌ
[π1, π2, π](ππ
π, πππ)ππ
πΏπ(β )
β€ πΆπ1
β
π2βπΈ1(π, π) 2
ππΌ1πππ
πΏπ1(β )
Γ πΈ2(π, π) 2
ππΌ2
πππ
πΏπ2(β ),
(65)
where for π, π β Z, π = 1, 2,
πΈπ(π, π) =
{{
{{
{
(π β π) 2(πβπ)(πΌ
πβππΏπ
π
)
, if π β€ π β 2,1, if π β 1 β€ π β€ π + 1,(π β π) 2
(πβπ)(πΌπ+ππΏππ
), if π β₯ π + 2.
(66)
Under the assumption βππΏππ
< πΌπ< ππΏπ
π
, it is easy to see that
β
β
π=ββ
πΈ1(π, π)πΎ+
β
β
π=ββ
πΈ2(π, π)πΎ
< β, for any πΎ > 0. (67)
Now by the Minkowskiβs inequality and (65), we get
2ππΌ[π1, π2, π](π, π)ππ
πΏπ(β )
β€ 2ππΌ
β
β
π=ββ
β
β
π=ββ
[π1, π2, π](ππ
π, πππ)ππ
πΏπ(β )
β€
β
β
π=ββ
β
β
π=ββ
2ππΌ
[π1, π2, π](ππ
π, πππ)ππ
πΏπ(β )
β€ πΆπ1
β
π2β
β
β
π=ββ
πΈ1(π, π) 2
ππΌ1πππ
πΏπ1(β )
Γ
β
β
π=ββ
πΈ2(π, π) 2
ππΌ2
πππ
πΏπ2(β ).
(68)
Finally, by definition and the fact that 1/π = 1/π1+ 1/π2, we
obtain[π1, π2, π](π, π)
οΏ½ΜοΏ½πΌ,π
π(β )
= {
β
β
π=ββ
2ππΌπ[π1, π2, π] (π, π) ππ
π
πΏπ(β )}
1/π
-
ISRNMathematical Analysis 9
β€ πΆπ1
β
π2β
Γ {
β
β
π=ββ
{
β
β
π=ββ
πΈ1(π, π)2
ππΌ1πππ
πΏπ1(β )}
π1
}
1/π1
Γ
{
{
{
β
β
π=ββ
{
{
{
β
β
π=ββ
πΈ2(π, π) 2
ππΌ2
πππ
πΏπ2(β )
}
}
}
π2
}
}
}
1/π2
:=π1
β
π2β(πΌ1Γ πΌ2) .
(69)
It is enough to show that πΌ1β€ πΆβπβ
οΏ½ΜοΏ½πΌ1,π1
π1(β )
and πΌ2β€ πΆβπβ
οΏ½ΜοΏ½πΌ2,π2
π2(β )
.
By symmetry, we only give estimates for πΌ1. For 0 < π
1< β,
by inequality (β |ππ|)π1
β€ β |ππ|π1 and inequality (67), we have
πΌ1β€ πΆ{
β
β
π=ββ
2ππΌ1π1πππ
π1
πΏπ1(β )
β
β
π=ββ
πΈ1(π, π)π1}
1/π1
β€ πΆ{
β
β
π=ββ
2ππΌ1π1πππ
π1
πΏπ1(β )}
1/π1
= πΆποΏ½ΜοΏ½πΌ1,π1
π1(β )
.
(70)
For π1> 1, by HoΜlderβs inequality and inequality (67), we
obtain
πΌ1β€ πΆ
{
{
{
β
β
π=ββ
β
β
π=ββ
πΈ1(π, π)π1/22ππΌ1π1πππ
π1
πΏπ1(β )
Γ {
β
β
π=ββ
πΈ1(π, π)π
1/2}
π1/π
1
}
}
}
1/π1
β€ πΆ{
β
β
π=ββ
2ππΌ1π1πππ
π1
πΏπ1(β )
β
β
k=ββπΈ1(π, π)π1/2}
1/π1
β€ πΆ{
β
β
π=ββ
2ππΌ1π1πππ
π1
πΏπ1(β )}
1/π1
= πΆποΏ½ΜοΏ½πΌ1,π1
π1(β )
.
(71)
Hence, for 0 < π1< β
πΌ1β€ πΆ
ποΏ½ΜοΏ½πΌ1,π1
π1(β )
. (72)
By symmetry, for 0 < π2< β, we have
πΌ2β€ πΆ
ποΏ½ΜοΏ½πΌ2,π2
π2(β )
. (73)
Therefore,
[π1, π2, π] (π, π)οΏ½ΜοΏ½πΌ,π
π(β )
β€ πΆπ1
β
π2β
ποΏ½ΜοΏ½πΌ1,π1
π1(β )
ποΏ½ΜοΏ½πΌ2,π2
π2(β )
. (74)
By a similar procedure one can prove that
(
β
β
β=1
[π1, π2, π] (πβ, πβ)
π
)
1/ποΏ½ΜοΏ½πΌ,π
π(β )
β€
(
β
β
β=1
πβ
π1
)
1/π1οΏ½ΜοΏ½πΌ1,π1
π1(β )
(
β
β
β=1
πβ
π2
)
1/π2οΏ½ΜοΏ½πΌ2,π2
π2(β )
Γ πΆπ1
β
π2β
(75)
holds for all πββ οΏ½ΜοΏ½πΌ1,π1
π1(β ), πββ οΏ½ΜοΏ½πΌ2,π2
π2(β ), where 1 < π
π< β for
π = 1, 2 and 1/π = 1/π1+ 1/π2. Thus, we finish the proof of
Theorem 4.
Remark 11. Note that when πΌπβ₯ ππΏπ
π
, then the results ofTheorems 3 and 4 are no longer true. It needs to replacethe variable exponent Herz space οΏ½ΜοΏ½πΌ,π
π(β )(Rπ) with π»οΏ½ΜοΏ½πΌ,π
π(β )(Rπ)
and the Herz-type Hardy spaces with the variable exponentrecently introduced in [20]. The boundedness of multilinearsingular integral operators from π»οΏ½ΜοΏ½πΌ,π
π(β )(Rπ) to οΏ½ΜοΏ½πΌ,π
π(β )(Rπ) is
still an interesting question that needs to be answered.
Remark 12. Although we considered the 2-linear case, themethod can be extended for any π-linear case without anyessential difficulty.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
References
[1] R. Coifman and Y. Meyer, βOn commutators of singular in-tegrals and bilinear singular integrals,β Transactions of theAmerican Mathematical Society, vol. 212, pp. 315β331, 1975.
[2] M. Christ and J. JourneΜ, βPolynomial growth estimates formultilinear singular integral operators,β Acta Mathematica, vol.159, no. 1, pp. 51β80, 1987.
[3] C. Kenig and E. Stein, βMultilinear estimates and fractionalintegration,β Mathematical Research Letters, vol. 6, no. 1, pp. 1β15, 1999.
[4] L. Grafakos and R. H. Torres, βMultilinear CalderoΜn-Zygmundtheory,β Advances in Mathematics, vol. 165, no. 1, pp. 124β164,2002.
[5] X. Tao and H. Zhang, βOn the boundedness of multilinearoperators on weighted herz-Morrey spaces,β Taiwanese Journalof Mathematics, vol. 15, no. 4, pp. 1527β1543, 2011.
[6] Q.Wu, βWeighted estimates formultilinearCalderon-Zygmundoperators,βAdvances inMathematics, vol. 33, pp. 333β342, 2004.
[7] J. Zhou, Y. Cao, and L. Li, βEstimates of commutators for mul-tilinear operators on Herz-type spaces,β Applied Mathematics,vol. 25, no. 2, pp. 177β184, 2010.
[8] L. Diening, P. Harjulehto, P. HaΜstoΜ, and M. RuzΜicΜka, Lebesgueand Sobolev Spaces with Variable Exponents, vol. 2017 of LectureNotes in Mathematics, Springer, Heidelberg, Germany, 2011.
[9] S. Samko, βOn a progress in the theory of Lebesgue spaces withvariable exponent: maximal and singular operators,β Integral
-
10 ISRNMathematical Analysis
Transforms and Special Functions, vol. 16, no. 5-6, pp. 461β482,2005.
[10] A. Huang and J. Xu, βMultilinear singular integrals and com-mutators in variable exponent Lebesgue spaces,β Applied Math-ematics, vol. 25, no. 1, pp. 69β77, 2010.
[11] M. Izuki, βHerz and amalgam spaces with variable exponent,the Haar wavelets and greediness of the wavelet system,β EastJournal on Approximations, vol. 15, pp. 87β109, 2009.
[12] M. Izuki, βBoundedness of sublinear operators on Herz spaceswith variable exponent and application to wavelet characteriza-tion,β Analysis Mathematica, vol. 36, no. 1, pp. 33β50, 2010.
[13] A. Almeida and D. Drihem, βMaximal, potential and singulartype operators on Herz spaces with variable exponents,β Journalof Mathematical Analysis and Applications, vol. 394, no. 2, pp.781β795, 2012.
[14] S. Samko, βVariable exponent Herz spaces,β MediterraneanJournal of Mathematics, vol. 10, no. 4, pp. 2007β2025, 2013.
[15] D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, βThe max-imal function on variable πΏπ spaces,β Annales Academiae Scien-tiarum Fennicae Mathematica, vol. 28, no. 1, pp. 223β238, 2003.
[16] A. Nekvinda, βHardy-littlewood maxell operator on πΏπ(π₯)(Rπ),βMathematical Inequalities andApplications, vol. 7, no. 2, pp. 255β265, 2004.
[17] D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. PeΜrez, βTheboundedness of classical operators on variable πΏπ spaces,βAnnales Academiae Scientiarum Fennicae Mathematica, vol. 31,no. 1, pp. 239β264, 2006.
[18] O. KovoΜcΜik and J. Rakosnk, βOn spaces πΏπ(π₯) and ππ,π(π₯),βCzechoslovak Mathematical Journal, vol. 41, pp. 592β618, 1991.
[19] M. Izuki, βBoundedness of commutators on Herz spaceswith variable exponent,β Rendiconti del Circolo Matematico diPalermo, vol. 59, no. 2, pp. 199β213, 2010.
[20] H. Wang and Z. Liu, βThe Herz-type Hardy spaces withvariable exponent and their Applications,β Taiwanese Journal ofMathematics, vol. 16, no. 4, pp. 1363β1389, 2012.
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