Research Article Multilinear Singular Integrals and ...singular integral on Herz space with variable exponent. Similar results for the boundedness of commutators gen-erated by these

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 © 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 Calderón commutators and pseudosdifferential oper-ators having multiparameter function input. Subsequently,many authors including Christ and Journé [2], Kenig andStein [3], and Grafakos and Torres [4] have substantiallyadded to the exiting theory.

Let 𝐾 be a locally integrable function defined away fromthe diagonal 𝑥 = 𝑦

1= ⋅ ⋅ ⋅ = 𝑦

𝑚in (R𝑛)𝑚+1, which for 𝐶 > 0

satisfies the estimates

𝐾 (𝑥, 𝑦1, . . . , 𝑦𝑚) ≤

𝐶

(𝑥 − 𝑦1

+ ⋅ ⋅ ⋅ +𝑥 − 𝑦𝑚

)𝑚𝑛, (1)

and for 𝜖 > 0,

𝐾 (𝑥, 𝑦

1, . . . , 𝑦

𝑚) − 𝐾 (𝑥

, 𝑦1, . . . , 𝑦

𝑚)

≤

𝐶𝑥 − 𝑥

𝜖

(∑𝑚

𝑗=1

𝑥 − 𝑦𝑗

)𝑚𝑛+𝜖

,

(2)

whenever |𝑥 − 𝑥| ≤ (1/2)max{|𝑥 − 𝑦1|, . . . , |𝑥 − 𝑦

𝑚|}, and

𝐾 (𝑥, 𝑦

1, . . . , 𝑦

𝑖, . . . , 𝑦

𝑚) − 𝐾 (𝑥

, 𝑦1, . . . , 𝑦

𝑖, . . . , 𝑦

𝑚)

≤

𝐶𝑦𝑖− 𝑦

𝑖

𝜖

(∑𝑚

𝑗=1

𝑥 − 𝑦𝑗

)𝑚𝑛+𝜖

,

(3)

whenever |𝑦𝑖−𝑦

𝑖| ≤ (1/2)max{|𝑥−𝑦

1|, . . . , |𝑥−𝑦

𝑚|} for all 1 ≤

𝑖 ≤ 𝑚. Then 𝐾 is called 𝑚-linear Calderón-Zygmund kernel.In this paper, we consider an 𝑚-linear singular integraloperator 𝑇 associated with the kernel 𝐾, which is initiallydefined on product of the Schwartz space S(R𝑛) and takesits values in the space of tempered distribution S(R𝑛) suchthat

𝑇 (𝑓1, . . . , 𝑓

𝑚) (𝑥)

= ∫(R𝑛)𝑚

𝐾(𝑥, 𝑦1, . . . , 𝑦

𝑚) 𝑓1(𝑦1) ⋅ ⋅ ⋅ 𝑓

𝑚(𝑦𝑚) 𝑑𝑦1⋅ ⋅ ⋅ 𝑑𝑦𝑚,

(4)

for 𝑥 ∉ ⋂𝑚𝑗=1

supp𝑓𝑗, where 𝑓

1, . . . , 𝑓

𝑚∈ 𝐿∞

𝐶(R𝑛), the space

of compactly supported bounded functions. If 𝑇 is boundedfrom 𝐿𝑝1 × ⋅ ⋅ ⋅ × 𝐿𝑝𝑚 to 𝐿𝑝 with 1 < 𝑝

1⋅ ⋅ ⋅ 𝑝𝑚

< ∞ and1/𝑝1+ ⋅ ⋅ ⋅ + 1/𝑝

𝑚= 1/𝑝, then we say that 𝑇 is an 𝑚-linear

Calderón-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

2 ISRNMathematical Analysis

𝑇 is a bounded operator on product of Lebesgue spaces andendpoint weak estimates hold. For the boundedness of 𝑇 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 𝐿𝑝(𝑥) with the exponent 𝑝depending on the point 𝑥 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 𝐿𝑝(𝑥) spaces. Despitethe progress made, the problems of boundedness of mul-tilinear singular integral operators and their commutatorson 𝐿𝑝(𝑥) 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 𝑝, by keepingthe remaining two exponents 𝛼 and 𝑞 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 �̇�𝛼,𝑞

𝑝(⋅)introduced in [12].

Throughout this paper, 𝐶 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 Ω be a measurable subset of R𝑛 with Lebesgue measure|Ω| > 0. Given a measurable function 𝑝(⋅) : Ω → [1,∞],then for some𝜆 > 0wedefine the variable exponent Lebesguespace as

𝐿𝑝(⋅)

(Ω) := {𝑓 is measurable : ∫Ω

(

𝑓 (𝑥)

𝜆)

𝑝(𝑥)

𝑑𝑥 < ∞} ,

(5)

and the space 𝐿𝑝(⋅)loc (Ω) as

𝐿𝑝(⋅)

loc (Ω) := {𝑓 : 𝑓 ∈ 𝐿𝑝(⋅)

(𝐾) ∀ compact subset 𝐾 ⊂ Ω} .(6)

The Lebesgue space 𝐿𝑝(⋅)(Ω) becomes a Banach functionspace when equipped with the norm

𝑓𝐿𝑝(⋅)(Ω)

:= inf {𝜆 > 0 : ∫Ω

(

𝑓 (𝑥)

𝜆)

𝑝(𝑥)

𝑑𝑥 ≤ 1} . (7)

Given a locally integrable function 𝑓 on Ω, the Hardy-Littlewood maximal operator𝑀 is defined by

𝑀𝑓(𝑥) := sup𝑟>0

𝑟−𝑛∫𝐵(𝑥,𝑟)∩Ω

𝑓 (𝑦) 𝑑𝑦, (8)

where 𝐵(𝑥, 𝑟) := {𝑦 ∈ R𝑛 : |𝑥 − 𝑦| < 𝑟}. We denote

𝑝−:= ess inf {𝑝 (𝑥) : 𝑥 ∈ Ω} ,

𝑝+:= ess sup {𝑝 (𝑥) : 𝑥 ∈ Ω} .

(9)

We also defineP (Ω) := {𝑝 (⋅) : 𝑝

−> 1, 𝑝

+< ∞} ,

B (Ω) := {𝑝 (⋅) : 𝑝 (⋅) ∈ P (Ω) ,

𝑀 is bounded on 𝐿𝑝(⋅) (Ω)} .

(10)

Cruz-Uribe et al. [15] and Nekvinda [16] independentlyproved the following sufficient conditions for the bounded-ness of𝑀 on 𝐿𝑝(⋅)(Ω).

Proposition 1. Let Ω be an open set. If 𝑝(⋅) ∈ P(Ω) satisfies

𝑝 (𝑥) − 𝑝 (𝑦) ≤

𝐶

− log (𝑥 − 𝑦), if 𝑥 − 𝑦

≤1

2,

𝑝 (𝑥) − 𝑝 (𝑦) ≤

𝐶

log (𝑒 + |𝑥|), if 𝑦

≥ |𝑥| ,

(11)

then one has 𝑝(⋅) ∈ B(Ω).

Define P0(Ω) to be the set of measurable functions 𝑝 :Ω → (0,∞) such that

𝑝−:= ess inf {𝑝 (𝑥) : 𝑥 ∈ Ω} > 0,

𝑝+:= ess sup {𝑝 (𝑥) : 𝑥 ∈ Ω} < ∞.

(12)

Given 𝑝 ∈ P0(Ω), one can define the space 𝐿𝑝(⋅) 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 𝐵𝑘:= {𝑥 : |𝑥| ≤ 2

𝑘}, 𝐴𝑘= 𝐵𝑘\ 𝐵𝑘−1

and 𝜒𝑘= 𝜒𝐴𝑘

bethe characteristic function of the set 𝐴

𝑘for 𝑘 ∈ Z.

Definition 2. For 𝛼 ∈ R, 0 < 𝑞 ≤ ∞ and 𝑝(⋅) ∈ P(R𝑛), thehomogeneous Herz space with variable exponent𝐾𝛼,𝑞

𝑝(⋅)(R𝑛) is

defined by

�̇�𝛼,𝑞

𝑝(⋅)(R𝑛) := {𝐿

𝑝(⋅)

loc (R𝑛\ {0}) :

𝑓�̇�𝛼,𝑞

𝑝(⋅)(R𝑛)

< ∞} , (13)

ISRNMathematical Analysis 3

where

𝑓�̇�𝛼,𝑞

𝑝(⋅)(R𝑛)

:= {

∞

∑

𝑘=−∞

2𝑘𝛼𝑞𝑓𝜒𝑘

𝑞

𝐿𝑝(⋅)(R𝑛)

}

1/𝑞

. (14)

In the sequel, unless stated otherwise, we will work onthe whole space R𝑛 and will not mention it. Taking 𝑏

1and 𝑏2

in 𝐵𝑀𝑂, Huang and Xu in [10] define the three commutatoroperators for suitable functions 𝑓 and 𝑔. One of them is theoperator

[𝑏1, 𝑏2, 𝑇] (𝑓, 𝑔) (𝑥)

= 𝑏1(𝑥) 𝑏2(𝑥) 𝑇 (𝑓, 𝑔) (𝑥) − 𝑏

1(𝑥) 𝑇 (𝑓, 𝑏

2𝑔) (𝑥)

− 𝑏2(𝑥) 𝑇 (𝑏

1𝑓, 𝑔) (𝑥) + 𝑇 (𝑏

1𝑓, 𝑏2𝑔) (𝑥) .

(15)

As corollaries of their main results they give the followingestimates.

Theorem A. Let 𝑇 be 2-linear Calderón-Zygmund operatorand 𝑝(⋅) ∈ P0(R𝑛). If 𝑝

1(⋅), 𝑝2(⋅) ∈ B(R𝑛) such that 1/𝑝(𝑥) =

1/𝑝1(𝑥) + 1/𝑝

2(𝑥), then there exists a constant 𝐶 independent

of the functions 𝑓, 𝑓ℎ∈ 𝐿𝑝1(⋅), 𝑔, 𝑔

ℎ∈ 𝐿𝑝2(⋅) and ℎ ∈ N such

that𝑇(𝑓, 𝑔)

𝐿𝑝(⋅)≤ 𝐶

𝑓𝐿𝑝1(⋅)

𝑔𝐿𝑝2(⋅)

,

(

∞

∑

ℎ=1

𝑇 (𝑓ℎ, 𝑔ℎ)

𝑟

)

1/𝑟𝐿𝑝(⋅)

≤ 𝐶

(

∞

∑

ℎ=1

𝑓ℎ

𝑟1

)

1/𝑟1𝐿𝑝1(⋅)

×

(

∞

∑

ℎ=1

𝑔ℎ

𝑟2

)

1/𝑟2𝐿𝑝2(⋅)

(16)

hold, where 1 < 𝑟𝑙< ∞ for 𝑙 = 1, 2 and 1/𝑟 = 1/𝑟

1+ 1/𝑟2.

Theorem B. Let 𝑇 be 2-linear Calderón-Zygmund operator,𝑏1, 𝑏2∈ 𝐵𝑀𝑂(R𝑛), and𝑝(⋅) ∈ P0(R𝑛). If𝑝

1(⋅), 𝑝2(⋅) ∈ B(R𝑛)

such that 1/𝑝(𝑥) = 1/𝑝1(𝑥) + 1/𝑝

2(𝑥), then there exists a

constant 𝐶 independent of the functions 𝑓, 𝑓ℎ∈ 𝐿𝑝1(⋅), 𝑔, 𝑔

ℎ∈

𝐿𝑝2(⋅) and ℎ ∈ N such that[𝑏1, 𝑏2, 𝑇](𝑓, 𝑔)

L𝑝(⋅) ≤ 𝐶𝑏1

∗

𝑏2∗

𝑓𝐿𝑝1(⋅)

𝑔𝐿𝑝2(⋅)

,

(

∞

∑

ℎ=1

[𝑏1, 𝑏2, 𝑇] (𝑓ℎ, 𝑔ℎ)

𝑟

)

1/𝑟𝐿𝑝(⋅)

≤

(

∞

∑

ℎ=1

𝑓ℎ

𝑟1

)

1/𝑟1𝐿𝑝1(⋅)

(

∞

∑

ℎ=1

𝑔ℎ

𝑟2

)

1/𝑟2𝐿𝑝2(⋅)

× 𝐶𝑏1

∗

𝑏2∗

(17)

hold, where 1 < 𝑟𝑙< ∞ for 𝑙 = 1, 2 and 1/𝑟 = 1/𝑟

1+ 1/𝑟2.

Motivated by these results, here we give the following twotheorems.

Theorem 3. Let 𝑇 be 2-linear Calderón-Zygmund operatorand 𝑝(⋅) ∈ P(R𝑛). Furthermore, let 𝑝

1(⋅), 𝑝2(⋅) ∈ B(R𝑛),

0 < 𝑞𝑙< ∞, −𝑛𝛿

𝑝𝑙

< 𝛼𝑙< 𝑛𝛿𝑝

𝑙

, 𝑙 = 1, 2, where 𝛿𝑝𝑙

, 𝛿𝑝

𝑙

> 0

are constants defined in the next section such that 1/𝑝(𝑥) =1/𝑝1(𝑥) + 1/𝑝

2(𝑥), 1/𝑞 = 1/𝑞

1+ 1/𝑞2, and 𝛼 = 𝛼

1+ 𝛼2. If 𝑇 is

bounded from 𝐿𝑝1(⋅) × 𝐿𝑝2(⋅) to 𝐿𝑝(⋅), then𝑇(𝑓, 𝑔)

�̇�𝛼,𝑞

𝑝(⋅)

≤ 𝐶𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

𝑔�̇�𝛼2,𝑞2

𝑝2(⋅)

,

(

∞

∑

ℎ=1


𝑟

)

1/𝑟�̇�𝛼,𝑞

𝑝(⋅)

≤ 𝐶

(

∞

∑

ℎ=1

𝑓ℎ

𝑟1

)

1/𝑟1�̇�𝛼1,𝑞1

𝑝1(⋅)

×

(

∞

∑

ℎ=1

𝑔ℎ

𝑟2

)

1/𝑟2�̇�𝛼2,𝑞2

𝑝2(⋅)

(18)

hold for all 𝑓, 𝑓ℎ∈ �̇�𝛼1,𝑞1

𝑝1(⋅), 𝑔, 𝑔ℎ∈ �̇�𝛼2,𝑞2

𝑝2(⋅), where 1 < 𝑟

𝑙< ∞ for

𝑙 = 1,2 and 1/𝑟 = 1/𝑟1+ 1/𝑟2.

Theorem 4. Let 𝑇 be 2-linear Calderón-Zygmund operator,𝑏1, 𝑏2

∈ 𝐵𝑀𝑂(R𝑛), and 𝑝(⋅) ∈ P(R𝑛). Furthermore, let𝑝1(⋅), 𝑝2(⋅) ∈ B(R𝑛), 0 < 𝑞

𝑙< ∞, −𝑛𝛿

𝑝𝑙


𝑙

, 𝑙 = 1, 2,where 𝛿

𝑝𝑙

, 𝛿𝑝

𝑙

> 0 are constants defined in the next section,such that 1/𝑝(𝑥) = 1/𝑝

1(𝑥) + 1/𝑝

2(𝑥), 1/𝑞 = 1/𝑞

1+ 1/𝑞

2,

𝛼 = 𝛼1+𝛼2. If [𝑏1, 𝑏2, 𝑇] is bounded from 𝐿𝑝1(⋅) ×𝐿𝑝2(⋅) to 𝐿𝑝(⋅),

then[𝑏1, 𝑏2, 𝑇](𝑓, 𝑔)

�̇�𝛼,𝑞

𝑝(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗

𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

𝑔�̇�𝛼2,𝑞2

𝑝2(⋅)

,

(

∞

∑

ℎ=1

[𝑏1, 𝑏2, 𝑇] (𝑓ℎ, 𝑔ℎ)

𝑟

)

1/𝑟�̇�𝛼,𝑞

𝑝(⋅)

≤

(

∞

∑

ℎ=1

𝑓ℎ

𝑟1

)

1/𝑟1�̇�𝛼1,𝑞1

𝑝1(⋅)

(

∞

∑

ℎ=1

𝑔ℎ

𝑟2

)

1/𝑟2�̇�𝛼2,𝑞2

𝑝2(⋅)

× 𝐶𝑏1

∗

𝑏2∗

(19)

hold for all 𝑓, 𝑓ℎ∈ �̇�𝛼1,𝑞1

𝑝1(⋅), 𝑔, 𝑔ℎ∈ �̇�𝛼2,𝑞2

𝑝2(⋅), where 1 < 𝑟

𝑙< ∞ for

𝑙 = 1, 2 and 1/𝑟 = 1/𝑟1+ 1/𝑟2.

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 𝑝(𝑥) to denote the conjugate index of 𝑝(𝑥).Here we give some lemmas which will be helpful in provingTheorems 3 and 4.

Lemma 5 (see [10, 18], generalized Hölder’s inequality). Let𝑝, 𝑝1, 𝑝2∈ P(R𝑛).

(a) If 𝑓 ∈ L𝑝(⋅), 𝑔 ∈ 𝐿𝑝

(⋅), then one has𝑓𝑔

𝐿1≤ 𝐶𝑝

𝑓𝐿𝑝(⋅)

𝑔𝐿𝑝(⋅), (20)

where 𝐶𝑝= 1 + 1/𝑝

−− 1/𝑝

+.


(b) If 𝑓 ∈ 𝐿𝑝1(⋅), 𝑔 ∈ 𝐿𝑝2(⋅) and 1/𝑝(𝑥) = 1/𝑝1(𝑥) +

1/𝑝2(𝑥), then there exists a constant 𝐶

𝑝,𝑝1

such that

𝑓𝑔𝐿𝑝(⋅)

≤ 𝐶𝑝,𝑝1

𝑓𝐿𝑝1(⋅)

𝑔𝐿𝑝2(⋅) (21)

holds, where 𝐶𝑝,𝑝1

= (1 + 1/(𝑝1)−− 1/(𝑝

1)+)1/𝑝− .

Lemma 6 (see [12]). If 𝑝 ∈ B(R𝑛), then there exist a constant𝐶 > 0 such that for all balls 𝐵 in R𝑛,

𝜒𝐵𝐿𝑝(⋅)

𝜒𝐵𝐿𝑝(⋅)≤ 𝐶 |𝐵| . (22)

Lemma 7 (see [12]). If 𝑝 ∈ B(R𝑛), then there exists constants𝛿, 𝐶 > 0 such that for all balls 𝐵 in R𝑛 and all measurablesubsets 𝑆 ⊂ 𝐵,


𝜒𝑆𝐿𝑝(⋅)

≤ 𝐶|𝐵|

|𝑆|,

𝜒𝑆𝐿𝑝(⋅)


≤ 𝐶(|𝑆|

|𝐵|)

𝛿

. (23)

Lemma 8 (see [19, Remark 1]). If 𝑝𝑙∈ B(R𝑛), 𝑙 = 1, 2,

then by Proposition 1 one has 𝑝𝑙∈ B(R𝑛). Therefore, applying

Lemma 7, one can take positive constants 𝛿𝑝𝑙

, 𝛿𝑝

𝑙

> 0 such that

𝜒𝑆𝐿𝑝𝑙(⋅)

𝜒𝐵𝐿𝑝l(⋅)

≤ 𝐶(|𝑆|

|𝐵|)

𝛿𝑝𝑙

,

𝜒𝑆𝐿𝑝

𝑙(⋅)

𝜒𝐵𝐿𝑝

𝑙(⋅)

≤ 𝐶(|𝑆|

|𝐵|)

𝛿𝑝

𝑙

, (24)

for all balls 𝐵 in R𝑛 and all measurable subsets 𝑆 ⊂ 𝐵.

Recently, Izuki [19] established a relationship betweenLebesgue space with variable exponent and 𝐵𝑀𝑂 spacewhich can be stated in the form of the following lemma.

Lemma 9. One has that for all 𝑏 ∈ 𝐵𝑀𝑂(R𝑛) and all 𝑖, 𝑗 ∈ Zwith 𝑗 > 𝑖,

𝐶−1‖𝑏‖∗ ≤ sup

𝐵:ball

1


(𝑏 − 𝑏𝐵)𝜒𝐵𝐿𝑝(⋅)

≤ 𝐶‖𝑏‖∗

(𝑏 − 𝑏

𝐵𝑖

) 𝜒𝐵𝑗

𝐿𝑝(⋅)≤ 𝐶 (𝑗 − 𝑖) ‖𝑏‖∗

𝜒𝐵𝑗

𝐿𝑝(⋅).

(25)

Thenext lemma is the generalizedMinkowski’s inequalityand is useful in proving vector valued inequalities.

Lemma 10 (see [19]). If 1 < 𝑟 < ∞, then there exists aconstant 𝐶 > 0 such that for all sequences of functions {𝑓

ℎ}∞

ℎ=1

satisfying ‖‖{𝑓ℎ}ℎ‖ℓ𝑟‖𝐿1R𝑛

< ∞,

{

∞

∑

ℎ=1

(∫R𝑛

𝑓ℎ (𝑦) 𝑑𝑦)

𝑟

}

1/𝑟

≤ 𝐶∫R𝑛{

∞

∑

ℎ=1

𝑓ℎ (𝑦)

𝑟

}

1/𝑟

𝑑𝑦.

(26)

Proof of Theorem 3. In order to make computations easy, firstwe have to prove the following inequality:

2𝑘𝛼

𝑇 (𝑓𝜒𝑖, 𝑔𝜒𝑗) 𝜒𝑘

𝐿𝑝(⋅)≤ 𝐶𝐷

1(𝑘, 𝑖) 2

𝑖𝛼1𝑓𝜒𝑖

𝐿𝑝1(⋅)

× 𝐷2(𝑘, 𝑗) 2

𝑗𝛼2

𝑔𝜒𝑗

𝐿𝑝2(⋅),

(27)

where for 𝑘, 𝑖 ∈ Z, 𝑙 = 1, 2,

𝐷𝑙(𝑘, 𝑖) =

{{

{{

{

2(𝑘−𝑖)(𝛼

𝑙−𝑛𝛿𝑝

𝑙

)

, if 𝑖 ≤ 𝑘 − 2,1, if 𝑘 − 1 ≤ 𝑖 ≤ 𝑘 + 1,2(𝑘−𝑖)(𝛼

𝑙+𝑛𝛿𝑝𝑙

), if 𝑖 ≥ 𝑘 + 2.

(28)

If 𝑘−1 ≤ 𝑖 ≤ 𝑘+1, 𝑘−1 ≤ 𝑗 ≤ 𝑘+1, then we have 2𝑘 ∼ 2𝑖 ∼ 2𝑗;hence by the 𝐿𝑝(⋅) boundedness of 𝑇, we obtain

2𝑘𝛼

𝑇 (𝑓𝜒𝑖, 𝑔𝜒𝑗) 𝜒𝑘

𝐿𝑝(⋅)≤ 𝐶2𝑖𝛼1𝑓𝜒𝑖

𝐿𝑝1(⋅)2𝑗𝛼2

×𝑔𝜒𝑗

𝐿𝑝2(⋅).

(29)

In the other cases, we see that |𝑥−𝑦1|+|𝑥−𝑦

2| ∼ 2

max(𝑘,𝑖,𝑗), for𝑥 ∈ 𝐴

𝑘, 𝑦1∈ 𝐴𝑖, 𝑦2∈ 𝐴𝑗. Thus by the generalized Hölder’s

inequality,

𝑇 (𝑓𝜒

𝑖, 𝑔𝜒𝑗) (𝑥) 𝜒

𝑘(𝑥)

≤ 𝐶2−2max(𝑘,𝑖,𝑗)𝑛𝑓𝜒𝑖

1

𝑔𝜒𝑗

1⋅ 𝜒𝑘(𝑥)


𝐿𝑝1(⋅)𝜒𝑖

𝐿𝑝

1(⋅)

×𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝑗

𝐿𝑝

2(⋅)⋅ 𝜒𝑘(𝑥) .

(30)

Applying Lemma 5, we have

𝑇 (𝑓𝜒

𝑖, 𝑔𝜒𝑗) 𝜒𝑘

𝐿𝑝(⋅)



𝐿𝑝

1(⋅)

×𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝑗

𝐿𝑝

2(⋅)

𝜒𝑘𝐿𝑝(⋅)

≤ 𝐶2−max(𝑘,𝑖)𝑛𝑓𝜒𝑖

𝐿𝑝1(⋅)𝜒𝐵𝑖

𝐿𝑝

1(⋅)

𝜒𝐵𝑘

𝐿𝑝1(⋅)

× 2−max(𝑘,𝑗)𝑛𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

𝜒𝐵𝑘

𝐿𝑝2(⋅).

(31)

Now, for 𝑖 ≤ 𝑘 − 2, 𝑗 ≤ 𝑘 − 2, by Lemmas 6 and 8, it is easy toshow that

𝑇 (𝑓𝜒


𝐿𝑝(⋅)

≤ 𝐶𝑓𝜒𝑖

𝐿𝑝1(⋅)

𝜒𝐵𝑖

𝐿𝑝

1(⋅)

𝜒𝐵𝑘

𝐿𝑝

1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

𝜒𝐵𝑘

𝐿𝑝

2(⋅)


𝐿𝑝1(⋅)2−(𝑘−𝑖)𝑛𝛿

𝑝

1

𝑔𝜒𝑗

𝐿𝑝2(⋅)2−(𝑘−𝑗)𝑛𝛿

𝑝

2 .

(32)

By a similar argument for 𝑖 ≥ 𝑘 + 2, 𝑗 ≥ 𝑘 + 2, we get

𝑇(𝑓𝜒𝑖, 𝑔𝜒𝑗)𝜒𝑘

𝐿𝑝(⋅)


𝐿𝑝1(⋅)

𝜒𝐵𝑘

L𝑝1(⋅)𝜒𝐵𝑖

𝐿𝑝1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑘

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝2(⋅)


𝐿𝑝1(⋅)2(𝑘−𝑖)𝑛𝛿

𝑝1

𝑔𝜒𝑗

𝐿𝑝2(⋅)2(𝑘−𝑗)𝑛𝛿

𝑝2 .

(33)


In view of (29)–(33), (27) is obvious. Now byMinkowski’sinequality and (27), we get

2𝑘𝛼𝑇(𝑓, 𝑔)𝜒𝑘

𝐿𝑝(⋅)≤ 2𝑘𝛼

∞

∑

𝑖=−∞

∞

∑

𝑗=−∞


𝐿𝑝(⋅)

≤

∞

∑

𝑖=−∞

∞

∑

𝑗=−∞

2𝑘𝛼


𝐿𝑝(⋅)

≤ 𝐶

∞

∑

𝑖=−∞

𝐷1(𝑘, 𝑖) 2


𝐿𝑝1(⋅)

×

∞

∑

𝑗=−∞

𝐷2(𝑘, 𝑗) 2

𝑗𝛼2

𝑔𝜒𝑗

𝐿𝑝2(⋅).

(34)

Since 1/𝑞 = 1/𝑞1+ 1/𝑞2, then by definition

𝑇 (𝑓, 𝑔)�̇�𝛼,𝑞

𝑝(⋅)

= {

∞

∑

𝑘=−∞

2𝑘𝛼𝑞𝑇 (𝑓, 𝑔) 𝜒𝑘

𝑞

𝐿𝑝(⋅)}

1/𝑞

≤

{

{

{

∞

∑

𝑘=−∞

(

∞

∑

𝑖=−∞

𝐷1(𝑘, 𝑖) 2


𝐿𝑝1(⋅)

×

∞

∑

𝑗=−∞

𝐷2(𝑘, 𝑗) 2

𝑗𝛼2

𝑔𝜒𝑗

𝐿𝑝2(⋅))

𝑞

}

}

}

1/𝑞

≤ 𝐶{

∞

∑

𝑘=−∞

(

∞

∑

𝑖=−∞

𝐷1(𝑘, 𝑖) 2


𝐿𝑝1(⋅))

𝑞1

}

1/𝑞1

×

{

{

{

∞

∑

𝑘=−∞

(

∞

∑

𝑗=−∞

𝐷2(𝑘, 𝑗) 2

𝑗𝛼2

𝑔𝜒𝑗

𝐿𝑝2(⋅))

𝑞2

}

}

}

1/𝑞2

:= 𝐼1× 𝐼2.

(35)

It remains to show that 𝐼1

≤ 𝐶‖𝑓‖�̇�𝛼1,𝑞1

𝑝1(⋅)

and 𝐼2

≤

𝐶‖𝑔‖�̇�𝛼2,𝑞2

𝑝2(⋅)

. By symmetry, we only approximate 𝐼1. If 0 < 𝑞

1<

1, then by the well-known inequality (∑ |𝑎𝑖|)𝑞1

≤ ∑ |𝑎𝑖|𝑞1 and

the inequality

∞

∑

𝑖=−∞

𝐷1(𝑘, 𝑖)𝛾+

∞

∑

𝑗=−∞

𝐷2(𝑘, 𝑗)𝛾

< ∞, for any 𝛾 > 0, (36)

we have

𝐼1≤ 𝐶{

∞

∑

𝑖=−∞

2𝑖𝛼1𝑞1𝑓𝜒𝑖

𝑞1

𝐿𝑝1(⋅)

∞

∑

𝑘=−∞

𝐷1(𝑘, 𝑖)𝑞1}

1/𝑞1

≤ 𝐶{

∞

∑

𝑖=−∞


𝑞1

𝐿𝑝1(⋅)}

1/𝑞1

= 𝐶𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

.

(37)

If 𝑞1> 1, Hölder’s inequality and inequality (36) yield

𝐼1≤ 𝐶

{

{

{

∞

∑

𝑘=−∞

∞

∑

𝑖=−∞

𝐷1(𝑘, 𝑖)𝑞1/22𝑖𝛼1𝑞1𝑓𝜒𝑖

𝑞1

𝐿𝑝1(⋅)

× {

∞

∑

𝑖=−∞

𝐷1(𝑘, 𝑖)𝑞

1/2}

𝑞1/𝑞

1

}

}

}

1/𝑞1

≤ 𝐶{

∞

∑

𝑖=−∞


𝑞1

𝐿𝑝1(⋅)

∞

∑

𝑘=−∞

𝐷1(𝑘, 𝑖)𝑞1/2}

1/𝑞1

≤ 𝐶{

∞

∑

𝑖=−∞


𝑞1

𝐿𝑝1(⋅)}

1/𝑞1

= 𝐶𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

.

(38)

Therefore, for 0 < 𝑞1< ∞

𝐼1≤ 𝐶

𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

. (39)

By symmetry, for 0 < 𝑞2< ∞ we have

𝐼2≤ 𝐶

𝑔�̇�𝛼2,𝑞2

𝑝2(⋅)

. (40)

Finally, we obtain𝑇(𝑓, 𝑔)

�̇�𝛼,𝑞

𝑝(⋅)

≤ 𝐶𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

𝑔�̇�𝛼2,𝑞2

𝑝2(⋅)

. (41)

By virtue of Lemma 10 and the fact that 1/𝑟 = 1/𝑟1+ 1/𝑟2, it

is easy to show that

(

∞

∑

ℎ=1


𝑟

)

1/𝑟�̇�𝛼,𝑞

𝑝(⋅)

≤ 𝐶

(

∞

∑

ℎ=1

𝑓ℎ

𝑟1

)

1/𝑟1�̇�𝛼1,𝑞1

𝑝1(⋅)

×

(

∞

∑

ℎ=1

𝑔ℎ

𝑟2

)

1/𝑟2�̇�𝛼2,𝑞2

𝑝2(⋅)

(42)

holds for all 𝑓ℎ∈ �̇�𝛼1,𝑞1

𝑝1(⋅), 𝑔ℎ∈ �̇�𝛼2,𝑞2

𝑝2(⋅), where 1 < 𝑟

𝑙< ∞ for

𝑙 = 1, 2.Thus the proof of Theorem 3 is complete.

Proof of Theorem 4. Similar to the proof ofTheorem 3, for thecase 𝑘−1 ≤ 𝑖 ≤ 𝑘+1, 𝑘−1 ≤ 𝑗 ≤ 𝑘+1, we use𝐿𝑝(⋅) boundednessof [𝑏1, 𝑏2, 𝑇] to obtain

2𝑘𝛼

[𝑏1, 𝑏2, 𝑇](𝑓𝜒

𝑖, 𝑔𝜒𝑗)𝜒𝑘

𝐿𝑝(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗2𝑖𝛼1𝑓𝜒𝑖

𝐿𝑝1(⋅)2𝑗𝛼2

𝑔𝜒𝑗

𝐿𝑝2(⋅).

(43)

For other possibilities we have |𝑥−𝑦1|+|𝑥−𝑦

2| ∼ 2

max(𝑘,𝑖,𝑗)

for 𝑥 ∈ 𝐴𝑘, 𝑦1∈ 𝐴𝑖, 𝑦2∈ 𝐴𝑗.Thus, we consider the following

two cases.Case I (𝑖 ≤ 𝑘 − 2, 𝑗 ≤ 𝑘 − 2). We denote (𝑏

𝑙)𝐵𝑖

by 𝑏𝑙𝑖, where

(𝑏𝑙)𝐵𝑖

=1

𝐵𝑖

∫𝐵𝑖

𝑏𝑙(𝑥) 𝑑𝑥, 𝑙 = 1, 2 (44)


and consider the following decomposition:

[𝑏1, 𝑏2, 𝑇] (𝑓𝜒

𝑖, 𝑔𝜒𝑗) (𝑥)

= (𝑏1(𝑥) − 𝑏

1𝑖) (𝑏2(𝑥) − 𝑏

2𝑗) 𝑇 (𝑓𝜒


− (𝑏1(𝑥) − 𝑏

1𝑖) 𝑇 (𝑓𝜒

𝑖, (𝑏2(⋅) − 𝑏

2𝑗) 𝑔𝜒𝑗) (𝑥)

− (𝑏2(𝑥) − 𝑏

2𝑗) 𝑇 ((𝑏

1(⋅) − 𝑏

1𝑖) 𝑓𝜒𝑖, 𝑔𝜒𝑗) (𝑥)

+ 𝑇 ((𝑏1(⋅) − 𝑏

1𝑖) 𝑓𝜒𝑖, (𝑏2(⋅) − 𝑏


= 𝐿1(𝑥) + 𝐿

2(𝑥) + 𝐿

3(𝑥) + 𝐿

4(𝑥) .

(45)

Thus,

[𝑏1, 𝑏2, 𝑇] (𝑓𝜒


𝐿𝑝(⋅)≤

4

∑

𝑚=1

(𝐿𝑚) 𝜒𝑘𝐿𝑝(⋅)

=

4

∑

𝑚=1

𝐽𝑚.

(46)

Now, we will estimate each 𝐽𝑚(𝑚 = 1, . . . , 4), separately.

Applying Lemma 5, we have𝐿1 (𝑥)

≤ 𝐶2−2𝑘𝑛 𝑏1 (𝑥) − 𝑏1𝑖

𝑏2(𝑥) − 𝑏

2𝑗

×𝑓𝜒𝑖

𝐿1𝑔𝜒𝑗

𝐿1

≤ 𝐶2−2𝑘𝑛𝑓𝜒𝑖


𝐿𝑝

1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝑗

𝐿𝑝

2(⋅)

×𝑏1 (𝑥) − 𝑏1𝑖

𝑏2(𝑥) − 𝑏

2𝑗

.

(47)

Therefore, by virtue of generalized Hölder’s inequality andLemmas 6, 8, and 9, we get

𝐽1=(𝐿1) 𝜒𝑘

𝐿𝑝(⋅)



𝐿𝑝

1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝑗

𝐿𝑝

2(⋅)

×(𝑏1(𝑥) − 𝑏

1𝑖) (𝑏2(𝑥) − 𝑏

2𝑗) 𝜒𝑘

𝐿𝑝(⋅)



𝐿𝑝

1(⋅)

×(𝑏1(𝑥) − 𝑏

1𝑖) 𝜒𝐵𝑘

𝐿𝑝1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

×(𝑏2(𝑥) − 𝑏

2𝑗) 𝜒𝐵𝑘

𝐿𝑝2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗2−2𝑘𝑛

(𝑘 − 𝑖)𝑓𝜒𝑖

𝐿𝑝1(⋅)

×𝜒𝐵𝑖

𝐿𝑝

1(⋅)

𝜒𝐵𝑘

𝐿𝑝1(⋅)

× (𝑘 − 𝑗)𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

𝜒𝐵𝑘

𝐿𝑝2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗ (

𝑘 − 𝑖)𝑓𝜒𝑖

𝐿𝑝1(⋅)

×

𝜒𝐵𝑖

𝐿𝑝

1(⋅)

𝜒𝐵𝑘

𝐿𝑝

1(⋅)

(𝑘 − 𝑗)𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

𝜒𝐵𝑘

𝐿𝑝

2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗

𝑓𝜒𝑖𝐿𝑝1(⋅) (

𝑘 − 𝑖) 2−(𝑘−𝑖)𝑛𝛿

𝑝

1

×𝑔𝜒𝑗

𝐿𝑝2(⋅)(𝑘 − 𝑗) 2

−(𝑘−𝑗)𝑛𝛿𝑝

2 .

(48)

Similarly, using Lemma 9, we approximate 𝐿2as

𝐿2 ≤ 𝐶2

−2𝑘𝑛 𝑏1 (𝑥) − 𝑏1𝑖

𝑓𝜒𝑖𝐿1

(𝑏2(⋅) − 𝑏

2𝑗) 𝑔𝜒𝑗

𝐿1

≤ 𝐶2−2𝑘𝑛 𝑏1 (𝑥) − 𝑏1𝑖

𝑓𝜒𝑖𝐿𝑝1(⋅)

𝜒𝑖𝐿𝑝

1(⋅)

×𝑔𝜒𝑗

𝐿𝑝2(⋅)

(𝑏2(⋅) − 𝑏

2𝑗)𝜒𝑗

𝐿𝑝

2(⋅)

≤ 𝐶𝑏2

∗2−2𝑘𝑛 𝑏1 (𝑥) − 𝑏1𝑖


×𝜒𝐵𝑖

𝐿𝑝

1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

.

(49)

Therefore, in view of Lemmas 6, 9, and 8, we have

𝐽2=(𝐿2)𝜒𝑘

𝐿𝑝(⋅)

≤ 𝐶𝑏2

∗2−2𝑘𝑛𝑓𝜒𝑖


𝐿𝑝

1(⋅)

×(𝑏1(𝑥) − 𝑏1𝑖)𝜒𝑘

𝐿𝑝(⋅)𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗2−2𝑘𝑛𝑓𝜒𝑖


𝐿𝑝

1(⋅)

× (𝑘 − 𝑖)𝜒𝐵𝑘

𝐿𝑝(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗ (


𝐿𝑝1(⋅)

×

𝜒𝐵𝑖

𝐿𝑝

1(⋅)

𝜒𝐵𝑘

𝐿𝑝

1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

𝜒𝐵𝑘

𝐿𝑝

2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗

𝑓𝜒i𝐿𝑝1(⋅) (

𝑘 − 𝑖) 2−(𝑘−𝑖)𝑛𝛿

𝑝

1

×𝑔𝜒𝑗


𝑝

2 .

(50)

By symmetry, the estimate for 𝐽3is similar to that for 𝐽

2;

therefore,

𝐽3≤ 𝐶

𝑏1∗

𝑏2∗


2−(𝑘−𝑖)𝑛𝛿

𝑝

1

×𝑔𝜒𝑗

𝐿𝑝2(⋅)(𝑘 − 𝑗) 2

−(𝑘−𝑗)𝑛𝛿𝑝

2 .

(51)

Finally, it remains to estimate 𝐽4. For that, we use Lemma 9 to

write

𝐿4

=𝑇 ((𝑏1(⋅) − 𝑏

1𝑖) 𝑓𝜒𝑖, (𝑏2(⋅) − 𝑏


≤ 𝐶2−2𝑘𝑛(𝑏1 (⋅) − 𝑏1𝑖) 𝑓𝜒𝑖

𝐿1(𝑏2(⋅) − 𝑏

2𝑗) 𝑔𝜒𝑗

𝐿1


𝐿𝑝1(⋅)(𝑏1(⋅) − 𝑏1𝑖)𝜒𝑖

𝐿𝑝

1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

×(𝑏2(⋅) − 𝑏

2𝑗)𝜒𝑗

𝐿𝑝

2(⋅)


𝐿𝑝1(⋅)(𝑏1(⋅) − 𝑏

1𝑖)𝜒𝐵𝑖

𝐿𝑝

1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

×(𝑏2(⋅) − 𝑏

2𝑗)𝜒𝐵𝑗

𝐿𝑝

2(⋅)


≤ 𝐶𝑏1

∗



𝐿𝑝

1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

×𝜒𝐵𝑗

𝐿𝑝

2(⋅)

.

(52)

Thus, by Hölder’s inequality and Lemmas 6 and 8, we obtain

𝐽4=(𝐿4)𝜒𝑘

𝐿𝑝(⋅)

≤ 𝐶𝑏1

∗



𝐿𝑝1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

×𝜒𝐵𝑗

𝐿𝑝2(⋅)


≤ 𝐶𝑏1

∗



𝐿𝑝1(⋅)

𝜒𝐵𝑘

𝐿𝑝1(⋅)

×𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑘

𝐿𝑝2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗ (


𝐿𝑝1(⋅)

𝜒𝐵𝑖

𝐿𝑝

1(⋅)

𝜒𝐵𝑘

𝐿𝑝

1(⋅)

×𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

𝜒𝐵𝑘

𝐿𝑝

2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗


2−(𝑘−𝑖)𝑛𝛿

𝑝

1

𝑔𝜒𝑗


𝑝

2 .

(53)

Combining the estimates for 𝐽1, 𝐽2, 𝐽3, and 𝐽

4, we have

[𝑏1, 𝑏2, 𝑇](𝑓𝜒


𝐿𝑝(⋅)≤ 𝐶

𝑏1∗

𝑏2∗


× (𝑘 − 𝑖) 2−(𝑘−𝑖)𝑛𝛿

𝑝

1

𝑔𝜒𝑗

𝐿𝑝2(⋅)

× (𝑘 − 𝑗) 2−(𝑘−𝑗)𝑛𝛿

𝑝

2 .

(54)

Case II (𝑖 ≥ 𝑘 + 2, 𝑗 ≥ 𝑘 + 2). We denote (𝑏𝑙)𝐵𝑘

by 𝑏𝑙𝑘, where

(𝑏𝑙)𝐵𝑘

= (1/|𝐵𝑘|) ∫𝐵𝑘

𝑏𝑙(𝑥)𝑑𝑥, 𝑙 = 1, 2. In this case, we consider

the following decomposition:

[𝑏1, 𝑏2, 𝑇] (𝑓𝜒


= (𝑏1(𝑥) − 𝑏

1𝑘) (𝑏2(𝑥) − 𝑏

2𝑘) 𝑇 (𝑓𝜒


− (𝑏1(𝑥) − 𝑏

1𝑘) 𝑇 (𝑓𝜒

𝑖, (𝑏2(⋅) − 𝑏

2𝑘) 𝑔𝜒𝑗) (𝑥)

− (𝑏2(𝑥) − 𝑏

2𝑘) 𝑇 ((𝑏

1(⋅) − 𝑏

1𝑘) 𝑓𝜒𝑖, 𝑔𝜒𝑗) (𝑥)

+ 𝑇 ((𝑏1(⋅) − 𝑏

1𝑘) 𝑓𝜒𝑖, (𝑏2(⋅) − 𝑏


= �̃�1(𝑥) + �̃�

2(𝑥) + �̃�

3(𝑥) + �̃�

4(𝑥) .

(55)

Thus,

[𝑏1, 𝑏2, 𝑇] (𝑓𝜒


𝐿𝑝(⋅)≤

4

∑

𝑚=1

(�̃�𝑚) 𝜒𝑘

𝐿𝑝(⋅)

=

4

∑

𝑚=1

𝐽𝑚.

(56)

Let us first compute 𝐽1. As in the proof of Theorem 3, in this

case we estimate �̃�1as

�̃�1

≤ 𝐶

𝑏1 (𝑥) − 𝑏1𝑘

𝑏2 (𝑥) − 𝑏2𝑘 2−𝑖𝑛𝑓𝜒𝑖

𝐿12−𝑗𝑛

×𝑔𝜒𝑗

𝐿1

≤ 𝐶𝑏1 (𝑥) − 𝑏1𝑘

𝑏2 (𝑥) − 𝑏2𝑘 2−𝑖𝑛𝑓𝜒𝑖

𝐿𝑝1(⋅)

×𝜒𝑖

𝐿𝑝

1(⋅)2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝑗

𝐿𝑝

2(⋅).

(57)

By virtue of generalizedHölder’s inequality and Lemmas 6–9,we obtain

𝐽1=(�̃�1) 𝜒𝑘

𝐿𝑝(⋅)

≤ 𝐶2−𝑖𝑛𝑓𝜒𝑖


𝐿𝑝

1(⋅)2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

×(𝑏1(𝑥) − 𝑏1𝑘)(𝑏2(𝑥) − b2𝑘)𝜒𝑘

𝐿𝑝(⋅)



𝐿𝑝

1(⋅)

(𝑏1(𝑥) − 𝑏1𝑘)𝜒𝑘𝐿𝑝1(⋅)

× 2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

(𝑏2 (𝑥) − 𝑏2𝑘) 𝜒𝑘𝐿𝑝2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗2−𝑖𝑛𝑓𝜒𝑖


𝐿𝑝

1(⋅)

𝜒𝐵𝑘

𝐿𝑝1(⋅)

× 2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

𝜒𝐵𝑘

𝐿𝑝2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗


𝜒𝐵𝑘

𝐿𝑝1(⋅)

𝜒𝐵𝑖

𝐿𝑝1(⋅)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑘

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗


2(𝑘−𝑖)𝑛𝛿

𝑝1

𝑔𝜒𝑗


𝑝2 .

(58)

Next, we approximate 𝐽2. Using Lemma 9, it is easy to see that

�̃�2

≤ 𝐶2−2𝑖𝑛 𝑏1 (𝑥) − 𝑏1𝑘

𝑓𝜒𝑖𝐿1

(𝑏2(⋅) − 𝑏

2𝑘) 𝑔𝜒𝑗

𝐿1

≤ 𝐶2−𝑖𝑛 𝑏1 (𝑥) − 𝑏1𝑘


𝜒𝑖𝐿𝑝

1(⋅)

× 2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

(𝑏2(⋅) − 𝑏

2𝑘)𝜒𝑗

𝐿𝑝

2(⋅)

≤ 𝐶𝑏2

∗2−𝑖𝑛 𝑏1 (𝑥) − 𝑏1𝑘


𝜒𝐵𝑖

𝐿𝑝

1(⋅)

× (𝑗 − 𝑘) 2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

.

(59)

Thus, in view of Lemmas 6, 9, and 8, we get

𝐽2=(�̃�2) 𝜒𝑘

𝐿𝑝(⋅)

≤ 𝐶𝑏2

∗2−𝑖𝑛𝑓𝜒𝑖


𝐿𝑝

1(⋅)

(𝑏1(𝑥) − 𝑏1𝑘)𝜒𝑘𝐿𝑝(⋅)

× (𝑗 − 𝑘) 2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)


≤ 𝐶𝑏1

∗

𝑏2∗2−𝑖𝑛𝑓𝜒𝑖


𝐿𝑝

1(⋅)

𝜒𝐵𝑘

𝐿𝑝(⋅)

× (𝑗 − 𝑘) 2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗


𝜒𝐵𝑘

𝐿𝑝1(⋅)

𝜒𝐵𝑖

𝐿𝑝1(⋅)

(𝑗 − 𝑘)

×𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑘

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗


2(𝑘−𝑖)𝑛𝛿

𝑝1

𝑔𝜒𝑗

𝐿𝑝2(⋅)

× (𝑗 − 𝑘) 2(𝑘−𝑗)𝑛𝛿

𝑝2 .

(60)

By symmetry, the estimate for 𝐽3is similar to that for 𝐽

2; thus

𝐽3≤ 𝐶

𝑏1∗

𝑏2∗


𝑖 − 𝑘) 2(𝑘−𝑖)𝑛𝛿

𝑝1

×𝑔𝜒𝑗


𝑝2 .

(61)

Lastly, it remains to compute 𝐽4. For this purpose we use

generalized Höder’s inequality and lemmas 6 and 9 to obtain�̃�4

=𝑇 ((𝑏1(⋅) − 𝑏

1𝑘) 𝑓𝜒𝑖, (𝑏2(⋅) − 𝑏


≤ 𝐶2−𝑖𝑛(𝑏1 (⋅) − 𝑏1𝑘) 𝑓𝜒𝑖

𝐿12−𝑗𝑛

×(𝑏2(⋅) − 𝑏

2𝑘)𝑔𝜒𝑗

𝐿1


𝐿𝑝1(⋅)(𝑏1(⋅) − 𝑏1𝑘)𝜒𝑖

𝐿𝑝

1(⋅)

× 2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

(𝑏2(⋅) − 𝑏

2𝑘)𝜒𝑗

𝐿𝑝

2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗ (

𝑖 − 𝑘) 2−𝑖𝑛𝑓𝜒𝑖


𝐿𝑝

1(⋅)

× (𝑗 − 𝑘) 2−𝑗𝑛

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗ (

𝑖 − 𝑘)


𝜒𝐵𝑖

𝐿𝑝1(⋅)

(𝑗 − 𝑘)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝2(⋅)

.

(62)

Thus, an application of Lemma 8 yields

𝐽4=(�̃�4)𝜒𝑘

𝐿𝑝(⋅)≤ 𝐶

𝑏1∗

𝑏2∗ (

𝑖 − 𝑘)


𝜒𝑖𝐿𝑝1(⋅)

× (𝑗 − 𝑘)

𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝑗

𝐿𝑝2(⋅)


≤ 𝐶𝑏1

∗

𝑏2∗ (

𝑖 − 𝑘)𝑓𝜒𝑖

𝐿𝑝1(⋅)

𝜒𝐵𝑘

𝐿𝑝1(⋅)

𝜒𝐵𝑖

𝐿𝑝1(⋅)

(𝑗 − 𝑘)

×𝑔𝜒𝑗

𝐿𝑝2(⋅)

𝜒𝐵𝑘

𝐿𝑝2(⋅)

𝜒𝐵𝑗

𝐿𝑝2(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗ (

𝑖 − 𝑘)𝑓𝜒𝑖

𝐿𝑝1(⋅)2(𝑘−𝑖)𝑛𝛿

𝑝1

× (𝑗 − 𝑘)𝑔𝜒𝑗


𝑝2 .

(63)

Combining the estimates for 𝐽1, 𝐽2, 𝐽3, and 𝐽

4, we have

[𝑏1, 𝑏2, 𝑇](𝑓𝜒


𝐿𝑝(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗


𝑖 − 𝑘) 2(𝑘−𝑖)𝑛𝛿

𝑝1

×𝑔𝜒𝑗

𝐿𝑝2(⋅)(𝑗 − 𝑘) 2

(𝑘−𝑗)𝑛𝛿𝑝2 .

(64)

In view of (43), (54), and (64), we arrive at

2𝑘𝛼

[𝑏1, 𝑏2, 𝑇](𝑓𝜒


𝐿𝑝(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗𝐸1(𝑘, 𝑖) 2


𝐿𝑝1(⋅)

× 𝐸2(𝑘, 𝑗) 2

𝑗𝛼2

𝑔𝜒𝑗

𝐿𝑝2(⋅),

(65)

where for 𝑘, 𝑖 ∈ Z, 𝑙 = 1, 2,

𝐸𝑙(𝑘, 𝑖) =

{{

{{

{

(𝑘 − 𝑖) 2(𝑘−𝑖)(𝛼

𝑙−𝑛𝛿𝑝

𝑙

)

, if 𝑖 ≤ 𝑘 − 2,1, if 𝑘 − 1 ≤ 𝑖 ≤ 𝑘 + 1,(𝑖 − 𝑘) 2

(𝑘−𝑖)(𝛼𝑙+𝑛𝛿𝑝𝑙

), if 𝑖 ≥ 𝑘 + 2.

(66)

Under the assumption −𝑛𝛿𝑝𝑙


𝑙

, it is easy to see that

∞

∑

𝑖=−∞

𝐸1(𝑘, 𝑖)𝛾+

∞

∑

𝑗=−∞

𝐸2(𝑘, 𝑗)𝛾

< ∞, for any 𝛾 > 0. (67)

Now by the Minkowski’s inequality and (65), we get

2𝑘𝛼[𝑏1, 𝑏2, 𝑇](𝑓, 𝑔)𝜒𝑘

𝐿𝑝(⋅)

≤ 2𝑘𝛼

∞

∑

𝑖=−∞

∞

∑

𝑗=−∞

[𝑏1, 𝑏2, 𝑇](𝑓𝜒


𝐿𝑝(⋅)

≤

∞

∑

𝑖=−∞

∞

∑

𝑗=−∞

2𝑘𝛼

[𝑏1, 𝑏2, 𝑇](𝑓𝜒


𝐿𝑝(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗

∞

∑

𝑖=−∞

𝐸1(𝑘, 𝑖) 2


𝐿𝑝1(⋅)

×

∞

∑

𝑗=−∞

𝐸2(𝑘, 𝑗) 2

𝑗𝛼2

𝑔𝜒𝑗

𝐿𝑝2(⋅).

(68)

Finally, by definition and the fact that 1/𝑞 = 1/𝑞1+ 1/𝑞2, we

obtain[𝑏1, 𝑏2, 𝑇](𝑓, 𝑔)

�̇�𝛼,𝑞

𝑝(⋅)

= {

∞

∑

𝑘=−∞

2𝑘𝛼𝑞[𝑏1, 𝑏2, 𝑇] (𝑓, 𝑔) 𝜒𝑘

𝑞

𝐿𝑝(⋅)}

1/𝑞


≤ 𝐶𝑏1

∗

𝑏2∗

× {

∞

∑

𝑘=−∞

{

∞

∑

𝑖=−∞

𝐸1(𝑘, 𝑖)2


𝐿𝑝1(⋅)}

𝑞1

}

1/𝑞1

×

{

{

{

∞

∑

𝑘=−∞

{

{

{

∞

∑

𝑗=−∞

𝐸2(𝑘, 𝑗) 2

𝑗𝛼2

𝑔𝜒𝑗

𝐿𝑝2(⋅)

}

}

}

𝑞2

}

}

}

1/𝑞2

:=𝑏1

∗

𝑏2∗(𝐼1× 𝐼2) .

(69)

It is enough to show that 𝐼1≤ 𝐶‖𝑓‖

�̇�𝛼1,𝑞1

𝑝1(⋅)

and 𝐼2≤ 𝐶‖𝑔‖

�̇�𝛼2,𝑞2

𝑝2(⋅)

.

By symmetry, we only give estimates for 𝐼1. For 0 < 𝑞

1< ∞,

by inequality (∑ |𝑎𝑖|)𝑞1

≤ ∑ |𝑎𝑖|𝑞1 and inequality (67), we have

𝐼1≤ 𝐶{

∞

∑

𝑖=−∞


𝑞1

𝐿𝑝1(⋅)

∞

∑

𝑘=−∞

𝐸1(𝑘, 𝑖)𝑞1}

1/𝑞1

≤ 𝐶{

∞

∑

𝑖=−∞


𝑞1

𝐿𝑝1(⋅)}

1/𝑞1

= 𝐶𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

.

(70)

For 𝑞1> 1, by Hölder’s inequality and inequality (67), we

obtain

𝐼1≤ 𝐶

{

{

{

∞

∑

𝑘=−∞

∞

∑

𝑖=−∞

𝐸1(𝑘, 𝑖)𝑞1/22𝑖𝛼1𝑞1𝑓𝜒𝑖

𝑞1

𝐿𝑝1(⋅)

× {

∞

∑

𝑖=−∞

𝐸1(𝑘, 𝑖)𝑞

1/2}

𝑞1/𝑞

1

}

}

}

1/𝑞1

≤ 𝐶{

∞

∑

𝑖=−∞


𝑞1

𝐿𝑝1(⋅)

∞

∑

k=−∞𝐸1(𝑘, 𝑖)𝑞1/2}

1/𝑞1

≤ 𝐶{

∞

∑

𝑖=−∞


𝑞1

𝐿𝑝1(⋅)}

1/𝑞1

= 𝐶𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

.

(71)

Hence, for 0 < 𝑞1< ∞

𝐼1≤ 𝐶

𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

. (72)

By symmetry, for 0 < 𝑞2< ∞, we have

𝐼2≤ 𝐶

𝑔�̇�𝛼2,𝑞2

𝑝2(⋅)

. (73)

Therefore,

[𝑏1, 𝑏2, 𝑇] (𝑓, 𝑔)�̇�𝛼,𝑞

𝑝(⋅)

≤ 𝐶𝑏1

∗

𝑏2∗

𝑓�̇�𝛼1,𝑞1

𝑝1(⋅)

𝑔�̇�𝛼2,𝑞2

𝑝2(⋅)

. (74)

By a similar procedure one can prove that

(

∞

∑

ℎ=1

[𝑏1, 𝑏2, 𝑇] (𝑓ℎ, 𝑔ℎ)

𝑟

)

1/𝑟�̇�𝛼,𝑞

𝑝(⋅)

≤

(

∞

∑

ℎ=1

𝑓ℎ

𝑟1

)

1/𝑟1�̇�𝛼1,𝑞1

𝑝1(⋅)

(

∞

∑

ℎ=1

𝑔ℎ

𝑟2

)

1/𝑟2�̇�𝛼2,𝑞2

𝑝2(⋅)

× 𝐶𝑏1

∗

𝑏2∗

(75)

holds for all 𝑓ℎ∈ �̇�𝛼1,𝑞1

𝑝1(⋅), 𝑔ℎ∈ �̇�𝛼2,𝑞2

𝑝2(⋅), where 1 < 𝑟

𝑙< ∞ for

𝑙 = 1, 2 and 1/𝑟 = 1/𝑟1+ 1/𝑟2. Thus, we finish the proof of

Theorem 4.

Remark 11. Note that when 𝛼𝑙≥ 𝑛𝛿𝑝

𝑙

, then the results ofTheorems 3 and 4 are no longer true. It needs to replacethe variable exponent Herz space �̇�𝛼,𝑞

𝑝(⋅)(R𝑛) with 𝐻�̇�𝛼,𝑞

𝑝(⋅)(R𝑛)

and the Herz-type Hardy spaces with the variable exponentrecently introduced in [20]. The boundedness of multilinearsingular integral operators from 𝐻�̇�𝛼,𝑞

𝑝(⋅)(R𝑛) to �̇�𝛼,𝑞

𝑝(⋅)(R𝑛) 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 𝑚-linear case without anyessential difficulty.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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Research Article Multilinear Singular Integrals and ...singular integral on Herz space with variable exponent. Similar results for the boundedness of commutators gen-erated by these