Multilinear Singular Integrals and Commutators on Herz Space with Variable Exponent

Abstract

The paper establishes some sufficient conditions for the boundedness of singular integral operators and their commutators from products of variable exponent Herz spaces to variable exponent Herz spaces.

1. Introduction

In recent years, the interest in multilinear analysis for studying boundedness properties of multilinear integral operators has grown rapidly. The subject was founded by Coifman and Meyer [1] in their seminal work on singular integral operators like Calderón commutators and pseudosdifferential operators having multiparameter function input. Subsequently, many authors including Christ and Journé [2], Kenig and Stein [3], and Grafakos and Torres [4] have substantially added to the exiting theory.

Let 𝐾 be a locally integrable function defined away from the diagonal 𝑥=𝑦₁=⋯=𝑦_𝑚 in (ℝ^𝑛)^𝑚+1, which for 𝐶>0 satisfies the estimates 𝐾𝑥,𝑦₁,…,𝑦_𝑚≤𝐶𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^𝑚𝑛, (1) and for 𝜖>0, 𝐾𝑥,𝑦₁,…,𝑦_𝑚−𝐾𝑥^′,𝑦₁,…,𝑦_𝑚≤𝐶𝑥−𝑥^′𝜖∑^𝑚_𝑗=1𝑥−𝑦_𝑗^{𝑚𝑛+𝜖}, (2) whenever |𝑥−𝑥^′|≤(1/2)max{|𝑥−𝑦₁|,…,|𝑥−𝑦_𝑚|}, and 𝐾𝑥,𝑦₁,…,𝑦_𝑖,…,𝑦_𝑚−𝐾𝑥^′,𝑦₁,…,𝑦_𝑖,…,𝑦_𝑚≤𝐶𝑦_𝑖−𝑦^′_𝑖^𝜖∑^𝑚_𝑗=1𝑥−𝑦_𝑗^{𝑚𝑛+𝜖}, (3) whenever |𝑦_𝑖−𝑦^′_𝑖|≤(1/2)max{|𝑥−𝑦₁|,…,|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 integral operator 𝑇 associated with the kernel 𝐾, which is initially defined on product of the Schwartz space 𝒮(ℝ^𝑛) and takes its values in the space of tempered distribution 𝒮^′(ℝ^𝑛) such that 𝑇𝑓₁,…,𝑓_𝑚(𝑥)=_{(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚, (4) for 𝑥∉⋂^𝑚_𝑗=1supp𝑓_𝑗, where 𝑓₁,…,𝑓_𝑚∈𝐿^∞_𝐶(ℝ^𝑛), the space of compactly supported bounded functions. If 𝑇 is bounded from 𝐿^𝑝1×⋯×𝐿^𝑝𝑚 to 𝐿^𝑝 with 1<𝑝₁⋯𝑝_𝑚<∞ and 1/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 𝑇 is a bounded operator on product of Lebesgue spaces and endpoint weak estimates hold. For the boundedness of 𝑇 and its commutators on the product of Herz-type spaces we refer the reader to see [5, 6] and [7], respectively.

In the last few decades, however, a number of research papers have appeared in the literature which study the boundedness of integral operators, including the maximal function, singular operators, and fractional integral and commutators on function spaces with nonstandard growth conditions. Such kind of spaces is named as variable exponent function spaces which include variable exponent Lebesgue, Sobolev, Lorentz, Orlicz, and Herz-type function spaces. Among them the most fundamental and widely explored space is the Lebesgue space 𝐿^𝑝(𝑥) with the exponent 𝑝 depending on the point 𝑥 of the space. We will describe it briefly in the next section; however, we refer to the book [8] and the survey paper [9] for historical background and recent developments in the theory of 𝐿^𝑝(𝑥) spaces. Despite the progress made, the problems of boundedness of multilinear singular integral operators and their commutators on 𝐿^𝑝(𝑥) spaces remain open. Recently, Huang and Xu [10] proved the boundedness of such integral operators on the product of variable exponent Lebesgue space. Motivated by their results, in this paper we will study the multilinear singular integral on Herz space with variable exponent. Similar results for the boundedness of commutators generated by these operators and BMO functions are also provided.

Herz-type spaces are an important class of function spaces in harmonic analysis. In [11, 12], Izuki independently introduced Herz space with variable exponent 𝑝, by keeping the remaining two exponents 𝛼 and 𝑞 as constants. Variability of alpha was recently considered by Almeida and Drihem [13] in proving the boundedness results for some classical operators on such spaces. More recently, Samko [14] introduced the generalized Herz-type spaces where all the three exponents were allowed to vary. In this paper, we will study the multilinear singular operators on variable exponent Herz space 𝐾̇^𝛼,𝑞_𝑝(⋅) introduced in [12].

Throughout this paper, 𝐶 denotes a positive constant which may change from one occurrence to another. The next section contains some basic definitions and the main results of this study. Finally, the last section includes the proofs of main results along with some supporting lemmas.

2. Main Results

Let Ω be a measurable subset of ℝ^𝑛 with Lebesgue measure |Ω|>0. Given a measurable function 𝑝(⋅)∶Ω→[1,∞], then for some 𝜆>0 we define the variable exponent Lebesgue space as 𝐿^𝑝(⋅)(Ω)∶=𝑓ismeasurable∶_Ω𝑓(𝑥)𝜆^𝑝(𝑥)𝑑𝑥<∞, (5) and the space 𝐿^𝑝(⋅)_loc(Ω) as 𝐿^𝑝(⋅)_loc(Ω)∶=𝑓∶𝑓∈𝐿^𝑝(⋅)(𝐾)∀compactsubset𝐾⊂Ω. (6) The Lebesgue space 𝐿^𝑝(⋅)(Ω) becomes a Banach function space when equipped with the norm 𝑓_{𝐿𝑝(⋅)(Ω)}∶=inf𝜆>0∶_Ω𝑓(𝑥)𝜆^𝑝(𝑥)𝑑𝑥≤1. (7)

Given a locally integrable function 𝑓 on Ω, the Hardy-Littlewood maximal operator 𝑀 is defined by 𝑀𝑓(𝑥)∶=sup_𝑟>0𝑟^−𝑛_{𝐵(𝑥,𝑟)∩Ω}𝑓𝑦𝑑𝑦, (8) where 𝐵(𝑥,𝑟)∶={𝑦∈ℝ^𝑛∶|𝑥−𝑦|<𝑟}. We denote 𝑝₋𝑝(𝑥),𝑝∶=essinf∶𝑥∈Ω₊∶=esssup𝑝(𝑥)∶𝑥∈Ω. (9) We also define 𝒫(Ω)∶=𝑝(⋅)∶𝑝₋>1,𝑝₊,ℬ(Ω)𝑝(⋅)(⋅)(Ω),<∞∶=∶𝑝∈𝒫𝑀isboundedon𝐿^𝑝(⋅)(Ω). (10)

Cruz-Uribe et al. [15] and Nekvinda [16] independently proved the following sufficient conditions for the boundedness of 𝑀 on 𝐿^𝑝(⋅)(Ω).

Proposition 1. Let Ω be an open set. If 𝑝(⋅)∈𝒫(Ω) satisfies 𝑝(𝑥)𝑦≤𝐶−𝑝≤1−log𝑥−𝑦,𝑖𝑓𝑥−𝑦2,𝑝(𝑥)𝑦≤𝐶−𝑝(|𝑥𝑦≥|𝑥|,log𝑒+|),𝑖𝑓 (11) then one has 𝑝(⋅)∈ℬ(Ω).

Define 𝒫⁰(Ω) to be the set of measurable functions 𝑝∶Ω→(0,∞) such that 𝑝₋𝑝(𝑥)𝑝∶=essinf∶𝑥∈Ω>0,₊∶=esssup𝑝(𝑥)∶𝑥∈Ω<∞. (12) Given 𝑝∈𝒫⁰(Ω), one can define the space 𝐿^𝑝(⋅) as above. Since we will not use it in the this paper, we omit the details here and refer the reader to see [10, 17].

Let 𝐵_𝑘∶={𝑥∶|𝑥|≤2^𝑘}, 𝐴_𝑘=𝐵_𝑘⧵𝐵_𝑘−1 and 𝜒_𝑘=𝜒_𝐴𝑘 be the characteristic function of the set 𝐴_𝑘 for 𝑘∈ℤ.

Definition 2. For 𝛼∈ℝ, 0<𝑞≤∞ and 𝑝(⋅)∈𝒫(ℝ^𝑛), the homogeneous Herz space with variable exponent 𝐾^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) is defined by 𝐾̇^𝛼,𝑞_𝑝(⋅)ℝ^𝑛∶=𝐿^𝑝(⋅)_locℝ^𝑛⧵{0}∶𝑓_̇𝐾^𝛼,𝑞_{𝑝(⋅)(ℝ𝑛)}<∞, (13) where 𝑓_̇𝐾^𝛼,𝑞_{𝑝(⋅)(ℝ𝑛)}∶=^∞_𝑘=−∞2^𝑘𝛼𝑞𝑓𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞. (14)

In the sequel, unless stated otherwise, we will work on the whole space ℝ^𝑛 and will not mention it. Taking 𝑏₁ and 𝑏₂ in 𝐵𝑀𝑂, Huang and Xu in [10] define the three commutator operators for suitable functions 𝑓 and 𝑔. One of them is the operator 𝑏₁,𝑏₂,𝑇𝑓,𝑔(𝑥)=𝑏₁(𝑥)𝑏₂(𝑥)𝑇𝑓,𝑔(𝑥)−𝑏₁(𝑥)𝑇𝑓,𝑏₂𝑔(𝑥)−𝑏₂(𝑥)𝑇𝑏₁𝑓,𝑔(𝑥)+𝑇𝑏₁𝑓,𝑏₂𝑔(𝑥). (15) As corollaries of their main results they give the following estimates.

Theorem A. Let 𝑇 be 2-linear Calderón-Zygmund operator and 𝑝(⋅)∈𝒫⁰(ℝ^𝑛). If 𝑝₁(⋅),𝑝₂(⋅)∈ℬ(ℝ^𝑛) such that 1/𝑝(𝑥)=1/𝑝₁(𝑥)+1/𝑝₂(𝑥), then there exists a constant 𝐶 independent of the functions 𝑓,𝑓_ℎ∈𝐿^𝑝1(⋅), 𝑔,𝑔_ℎ∈𝐿^𝑝2(⋅) and ℎ∈ℕ such that 𝑇(𝑓,𝑔)_{𝐿𝑝(⋅)}≤𝐶𝑓_{𝐿𝑝1(⋅)}𝑔_{𝐿𝑝2(⋅)},_∞ℎ=1𝑇𝑓_ℎ,𝑔_ℎ^𝑟1/𝑟_{𝐿𝑝(⋅)}≤𝐶^∞_ℎ=1𝑓_ℎ^𝑟11/𝑟1_{𝐿𝑝1(⋅)}×^∞_ℎ=1𝑔_ℎ^𝑟21/𝑟2_{𝐿𝑝2(⋅)} (16) hold, where 1<𝑟_𝑙<∞ for 𝑙=1,2 and 1/𝑟=1/𝑟₁+1/𝑟₂.

Theorem B. Let 𝑇 be 2-linear Calderón-Zygmund operator, 𝑏₁,𝑏₂∈𝐵𝑀𝑂(ℝ^𝑛), and 𝑝(⋅)∈𝒫⁰(ℝ^𝑛). If 𝑝₁(⋅),𝑝₂(⋅)∈ℬ(ℝ^𝑛) such that 1/𝑝(𝑥)=1/𝑝₁(𝑥)+1/𝑝₂(𝑥), then there exists a constant 𝐶 independent of the functions 𝑓,𝑓_ℎ∈𝐿^𝑝1(⋅), 𝑔,𝑔_ℎ∈𝐿^𝑝2(⋅) and ℎ∈ℕ such that [𝑏₁,𝑏₂,𝑇](𝑓,𝑔)_L𝑝(⋅)≤𝐶𝑏_1∗𝑏_2∗𝑓_{𝐿𝑝1(⋅)}𝑔_{𝐿𝑝2(⋅)},_∞ℎ=1𝑏₁,𝑏₂,𝑇𝑓_ℎ,𝑔_ℎ^𝑟1/𝑟_{𝐿𝑝(⋅)}≤^∞_ℎ=1𝑓_ℎ^𝑟11/𝑟1_{𝐿𝑝1(⋅)}^∞_ℎ=1𝑔_ℎ^𝑟21/𝑟2_{𝐿𝑝2(⋅)}×𝐶𝑏_1∗𝑏_2∗ (17) hold, where 1<𝑟_𝑙<∞ for 𝑙=1,2 and 1/𝑟=1/𝑟₁+1/𝑟₂.

Motivated by these results, here we give the following two theorems.

Theorem 3. Let 𝑇 be 2-linear Calderón-Zygmund operator and 𝑝(⋅)∈𝒫(ℝ^𝑛). Furthermore, let 𝑝₁(⋅),𝑝₂(⋅)∈ℬ(ℝ^𝑛), 0<𝑞_𝑙<∞, −𝑛𝛿_𝑝𝑙<𝛼_𝑙<𝑛𝛿_𝑝′𝑙, 𝑙=1,2, where 𝛿_𝑝𝑙,𝛿_𝑝′𝑙>0 are constants defined in the next section such that 1/𝑝(𝑥)=1/𝑝₁(𝑥)+1/𝑝₂(𝑥), 1/𝑞=1/𝑞₁+1/𝑞₂, and 𝛼=𝛼₁+𝛼₂. If 𝑇 is bounded from 𝐿^𝑝1(⋅)×𝐿^𝑝2(⋅) to 𝐿^𝑝(⋅), then 𝑇(𝑓,𝑔)_̇𝐾^𝛼,𝑞_𝑝(⋅)≤𝐶𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅)𝑔_̇𝐾^𝛼_22𝑝2^,𝑞_(⋅),_∞ℎ=1𝑇𝑓_ℎ,𝑔_ℎ^𝑟1/𝑟_{̇𝐾𝛼,𝑞𝑝(⋅)}≤𝐶^∞_ℎ=1𝑓_ℎ^𝑟11/𝑟1_{̇𝐾𝛼11𝑝1,𝑞(⋅)}×^∞_ℎ=1𝑔_ℎ^𝑟21/𝑟2_{̇𝐾𝛼22𝑝2,𝑞(⋅)} (18) hold for all 𝑓,𝑓_ℎ∈𝐾̇^𝛼1,𝑞1_𝑝1(⋅), 𝑔,𝑔_ℎ∈𝐾̇^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(ℝ^𝑛), and 𝑝(⋅)∈𝒫(ℝ^𝑛). Furthermore, let 𝑝₁(⋅),𝑝₂(⋅)∈ℬ(ℝ^𝑛), 0<𝑞_𝑙<∞, −𝑛𝛿_𝑝𝑙<𝛼_𝑙<𝑛𝛿_𝑝′𝑙, 𝑙=1,2, where 𝛿_𝑝𝑙,𝛿_𝑝′𝑙>0 are constants defined in the next section, such that 1/𝑝(𝑥)=1/𝑝₁(𝑥)+1/𝑝₂(𝑥), 1/𝑞=1/𝑞₁+1/𝑞₂, 𝛼=𝛼₁+𝛼₂. If [𝑏₁,𝑏₂,𝑇] is bounded from 𝐿^𝑝1(⋅)×𝐿^𝑝2(⋅) to 𝐿^𝑝(⋅), then [𝑏₁,𝑏₂,𝑇](𝑓,𝑔)_̇𝐾^𝛼,𝑞_𝑝(⋅)≤𝐶𝑏_1∗𝑏_2∗𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅)𝑔_̇𝐾^𝛼_22𝑝2^,𝑞_(⋅),_∞ℎ=1𝑏₁,𝑏₂,𝑇𝑓_ℎ,𝑔_ℎ^𝑟1/𝑟_{̇𝐾𝛼,𝑞𝑝(⋅)}≤^∞_ℎ=1𝑓_ℎ^𝑟11/𝑟1_{̇𝐾𝛼11𝑝1,𝑞(⋅)}^∞_ℎ=1𝑔_ℎ^𝑟21/𝑟2_{̇𝐾𝛼22𝑝2,𝑞(⋅)}×𝐶𝑏_1∗𝑏_2∗ (19) hold for all 𝑓,𝑓_ℎ∈𝐾̇^𝛼1,𝑞1_𝑝1(⋅), 𝑔,𝑔_ℎ∈𝐾̇^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 last section. The ideas of these proofs mainly come from [5, 7]. We use the notation 𝑝^′(𝑥) to denote the conjugate index of 𝑝(𝑥). Here we give some lemmas which will be helpful in proving Theorems 3 and 4.

Lemma 5 (see [10, 18], generalized Hölder's inequality). Let 𝑝,𝑝₁,𝑝₂∈𝒫(ℝ^𝑛).(a)If 𝑓∈𝐿^𝑝(⋅), 𝑔∈𝐿^𝑝′(⋅), then one has 𝑓𝑔_𝐿1≤𝐶_𝑝𝑓_{𝐿𝑝(⋅)}𝑔_{𝐿𝑝′(⋅)}, (20) where 𝐶_𝑝=1+1/𝑝₋−1/𝑝₊.(b)If 𝑓∈𝐿^𝑝1(⋅), 𝑔∈𝐿^𝑝2(⋅) and 1/𝑝(𝑥)=1/𝑝₁(𝑥)+1/𝑝₂(𝑥), then there exists a constant 𝐶_𝑝,𝑝1 such that 𝑓𝑔_{𝐿𝑝(⋅)}≤𝐶_𝑝,𝑝1𝑓_{𝐿𝑝1(⋅)}𝑔_{𝐿𝑝2(⋅)} (21) holds, where 𝐶_𝑝,𝑝1=(1+1/(𝑝₁)₋−1/(𝑝₁)₊)^1/𝑝−.

Lemma 6 (see [12]). If 𝑝∈ℬ(ℝ^𝑛), then there exist a constant 𝐶>0 such that for all balls 𝐵 in ℝ^𝑛, 𝜒_{𝐵𝐿𝑝(⋅)}𝜒_{𝐵𝐿𝑝′(⋅)}≤𝐶|𝐵|. (22)

Lemma 7 (see [12]). If 𝑝∈ℬ(ℝ^𝑛), then there exists constants 𝛿,𝐶>0 such that for all balls 𝐵 in ℝ^𝑛 and all measurable subsets 𝑆⊂𝐵, 𝜒_{𝐵𝐿𝑝(⋅)}𝜒_{𝑆𝐿𝑝(⋅)}|𝐵|≤𝐶|𝑆|,𝜒_{𝑆𝐿𝑝(⋅)}𝜒_{𝐵𝐿𝑝(⋅)}|𝑆|≤𝐶|𝐵|^𝛿. (23)

Lemma 8 (see [19, Remark 1]). If 𝑝_𝑙∈ℬ(ℝ^𝑛), 𝑙=1,2, then by Proposition 1 one has 𝑝^′_𝑙∈ℬ(ℝ^𝑛). Therefore, applying Lemma 7, one can take positive constants 𝛿_𝑝𝑙,𝛿_𝑝′𝑙>0 such that 𝜒_{𝑆𝐿𝑝𝑙(⋅)}𝜒_{𝐵𝐿𝑝𝑙(⋅)}|𝑆|≤𝐶|𝐵|^𝛿𝑝𝑙,𝜒_{𝑆𝐿𝑝′𝑙(⋅)}𝜒_{𝐵𝐿𝑝′𝑙(⋅)}|𝑆|≤𝐶|𝐵|^{𝛿𝑝′𝑙}, (24) for all balls 𝐵 in ℝ^𝑛 and all measurable subsets 𝑆⊂𝐵.

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

Lemma 9. One has that for all 𝑏∈𝐵𝑀𝑂(ℝ^𝑛) and all 𝑖,𝑗∈ℤ with 𝑗>𝑖, 𝐶⁻¹‖𝑏‖_∗≤sup_𝐵∶ball1𝜒_{𝐵𝐿𝑝(⋅)}(𝑏−𝑏_𝐵)𝜒_{𝐵𝐿𝑝(⋅)}≤𝐶‖𝑏‖_∗𝑏−𝑏_𝐵𝑖𝜒_{𝐵𝑗𝐿𝑝(⋅)}≤𝐶𝑗−𝑖‖𝑏‖_∗𝜒_{𝐵𝑗𝐿𝑝(⋅)}. (25)

The next lemma is the generalized Minkowski's inequality and is useful in proving vector valued inequalities.

Lemma 10 (see [19]). If 1<𝑟<∞, then there exists a constant 𝐶>0 such that for all sequences of functions {𝑓_ℎ}^∞_ℎ=1 satisfying ‖‖{𝑓_ℎ}_ℎ‖_ℓ𝑟‖_𝐿1ℝ𝑛<∞, ^∞_ℎ=1ℝ𝑛𝑓_ℎ𝑦𝑑𝑦^𝑟1/𝑟≤𝐶_ℝ𝑛^∞_ℎ=1𝑓_ℎ𝑦^𝑟1/𝑟𝑑𝑦. (26)

Proof of Theorem 3. In order to make computations easy, first we have to prove the following inequality: 2^𝑘𝛼𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝐷₁(𝑘,𝑖)2^𝑖𝛼1𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×𝐷₂𝑘,𝑗2^𝑗𝛼2𝑔𝜒_{𝑗𝐿𝑝2(⋅)}, (27) where for 𝑘,𝑖∈ℤ, 𝑙=1,2, 𝐷_𝑙()=2𝑘,𝑖^{(𝑘−𝑖)(𝛼𝑙−𝑛𝛿𝑝′𝑙)}2,if𝑖≤𝑘−2,1,if𝑘−1≤𝑖≤𝑘+1,_{(𝑘−𝑖)(𝛼𝑙+𝑛𝛿𝑝𝑙)},if𝑖≥𝑘+2. (28) If 𝑘−1≤𝑖≤𝑘+1, 𝑘−1≤𝑗≤𝑘+1, then we have 2^𝑘∼2^𝑖∼2^𝑗; hence by the 𝐿^𝑝(⋅) boundedness of 𝑇, we obtain 2^𝑘𝛼𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶2^𝑖𝛼1𝑓𝜒_{𝑖𝐿𝑝1(⋅)}2^𝑗𝛼2×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}. (29) In the other cases, we see that |𝑥−𝑦₁|+|𝑥−𝑦₂|∼2^{max(𝑘,𝑖,𝑗)}, for 𝑥∈𝐴_𝑘, 𝑦₁∈𝐴_𝑖, 𝑦₂∈𝐴_𝑗. Thus by the generalized Hölder's inequality, 𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗(𝑥)𝜒_𝑘(𝑥)≤𝐶2^{−2max(𝑘,𝑖,𝑗)𝑛}𝑓𝜒_𝑖1𝑔𝜒_𝑗1⋅𝜒_𝑘(𝑥)≤𝐶2^{−2max(𝑘,𝑖,𝑗)𝑛}𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝑖𝐿𝑝′1(⋅)}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝑗𝐿𝑝′2(⋅)}⋅𝜒_𝑘(𝑥). (30) Applying Lemma 5, we have 𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶2^{−2max(𝑘,𝑖,𝑗)𝑛}𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝑖𝐿𝑝′1(⋅)}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝑗𝐿𝑝′2(⋅)}𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶2^{−max(𝑘,𝑖)𝑛}𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝜒_{𝐵𝑘𝐿𝑝1(⋅)}×2^{−max(𝑘,𝑗)𝑛}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}𝜒_{𝐵𝑘𝐿𝑝2(⋅)}. (31) Now, for 𝑖≤𝑘−2, 𝑗≤𝑘−2, by Lemmas 6 and 8, it is easy to show that 𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝜒_{𝐵𝑘𝐿𝑝′1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}𝜒_{𝐵𝑘𝐿𝑝′2(⋅)}≤𝐶𝑓𝜒_{𝑖𝐿𝑝1(⋅)}2^{−(𝑘−𝑖)𝑛𝛿𝑝′1}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}2^{−(𝑘−𝑗)𝑛𝛿𝑝′2}. (32) By a similar argument for 𝑖≥𝑘+2, 𝑗≥𝑘+2, we get 𝑇(𝑓𝜒_𝑖,𝑔𝜒_𝑗)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑘𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑘𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝2(⋅)}≤𝐶𝑓𝜒_{𝑖𝐿𝑝1(⋅)}2^{(𝑘−𝑖)𝑛𝛿𝑝1}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}2^{(𝑘−𝑗)𝑛𝛿𝑝2}. (33)
In view of (29)–(33), (27) is obvious. Now by Minkowski’s inequality and (27), we get 2^𝑘𝛼𝑇(𝑓,𝑔)𝜒_{𝑘𝐿𝑝(⋅)}≤2^{𝑘𝛼∞∞}_{𝑖=−∞𝑗=−∞}𝑇(𝑓𝜒_𝑖,𝑔𝜒_𝑗)𝜒_{𝑘𝐿𝑝(⋅)}≤^∞∞_{𝑖=−∞𝑗=−∞}2^𝑘𝛼𝑇(𝑓𝜒_𝑖,𝑔𝜒_𝑗)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶^∞_𝑖=−∞𝐷₁(𝑘,𝑖)2^𝑖𝛼1𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×^∞_𝑗=−∞𝐷₂𝑘,𝑗2^𝑗𝛼2𝑔𝜒_{𝑗𝐿𝑝2(⋅)}. (34) Since 1/𝑞=1/𝑞₁+1/𝑞₂, then by definition 𝑇𝑓,𝑔_̇𝐾^𝛼,𝑞_𝑝(⋅)=^∞_𝑘=−∞2^𝑘𝛼𝑞𝑇𝑓,𝑔𝜒_𝑘^𝑞_{𝐿𝑝(⋅)}^1/𝑞≤^∞_𝑘=−∞^∞_𝑖=−∞𝐷₁(𝑘,𝑖)2^𝑖𝛼1𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×^∞_𝑗=−∞𝐷₂𝑘,𝑗2^𝑗𝛼2𝑔𝜒_{𝑗𝐿𝑝2(⋅)}^𝑞1/𝑞≤𝐶^∞_𝑘=−∞^∞_𝑖=−∞𝐷₁(𝑘,𝑖)2^𝑖𝛼1𝑓𝜒_{𝑖𝐿𝑝1(⋅)}^𝑞11/𝑞1×^∞_𝑘=−∞^∞_𝑗=−∞𝐷₂𝑘,𝑗2^𝑗𝛼2𝑔𝜒_{𝑗𝐿𝑝2(⋅)}^𝑞21/𝑞2∶=𝐼₁×𝐼₂. (35)
It remains to show that 𝐼₁≤𝐶‖𝑓‖_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅) and 𝐼₂≤𝐶‖𝑔‖_̇𝐾^𝛼_22𝑝2^,𝑞_(⋅). By symmetry, we only approximate 𝐼₁. If 0<𝑞₁<1, then by the well-known inequality (∑|𝑎_𝑖|)^𝑞1≤∑|𝑎_𝑖|^𝑞1 and the inequality ^∞_𝑖=−∞𝐷₁(𝑘,𝑖)^𝛾+^∞_𝑗=−∞𝐷₂𝑘,𝑗^𝛾<∞,forany𝛾>0, (36) we have 𝐼₁≤𝐶^∞_𝑖=−∞2^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}^∞_𝑘=−∞𝐷₁(𝑘,𝑖)^𝑞11/𝑞1≤𝐶^∞_𝑖=−∞2^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}^1/𝑞1=𝐶𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅). (37)
If 𝑞₁>1, Hölder's inequality and inequality (36) yield 𝐼₁≤𝐶^∞∞_{𝑘=−∞𝑖=−∞}𝐷₁(𝑘,𝑖)^𝑞1/22^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}×^∞_𝑖=−∞𝐷₁(𝑘,𝑖)^{𝑞′1/2𝑞1/𝑞′11/𝑞1}≤𝐶^∞_𝑖=−∞2^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}^∞_𝑘=−∞𝐷₁(𝑘,𝑖)^{𝑞1/21/𝑞1}≤𝐶^∞_𝑖=−∞2^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}^1/𝑞1=𝐶𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅). (38) Therefore, for 0<𝑞₁<∞𝐼₁≤𝐶𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅). (39) By symmetry, for 0<𝑞₂<∞ we have 𝐼₂≤𝐶𝑔_̇𝐾^𝛼_22𝑝2^,𝑞_(⋅). (40) Finally, we obtain 𝑇(𝑓,𝑔)_̇𝐾^𝛼,𝑞_𝑝(⋅)≤𝐶𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅)𝑔_̇𝐾^𝛼_22𝑝2^,𝑞_(⋅). (41) By virtue of Lemma 10 and the fact that 1/𝑟=1/𝑟₁+1/𝑟₂, it is easy to show that ^∞_ℎ=1𝑇𝑓_ℎ,𝑔_ℎ^𝑟1/𝑟_{̇𝐾𝛼,𝑞𝑝(⋅)}≤𝐶^∞_ℎ=1𝑓_ℎ^𝑟11/𝑟1_{̇𝐾𝛼11𝑝1,𝑞(⋅)}×^∞_ℎ=1𝑔_ℎ^𝑟21/𝑟2_{̇𝐾𝛼22𝑝2,𝑞(⋅)} (42) holds for all 𝑓_ℎ∈𝐾̇^𝛼1,𝑞1_𝑝1(⋅), 𝑔_ℎ∈𝐾̇^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 of Theorem 3, for the case 𝑘−1≤𝑖≤𝑘+1, 𝑘−1≤𝑗≤𝑘+1, we use 𝐿^𝑝(⋅) boundedness of [𝑏₁,𝑏₂,𝑇] to obtain 2^𝑘𝛼[𝑏₁,𝑏₂,𝑇](𝑓𝜒_𝑖,𝑔𝜒_𝑗)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗2^𝑖𝛼1𝑓𝜒_{𝑖𝐿𝑝1(⋅)}2^𝑗𝛼2𝑔𝜒_{𝑗𝐿𝑝2(⋅)}. (43)
For other possibilities we have |𝑥−𝑦₁|+|𝑥−𝑦₂|∼2^{max(𝑘,𝑖,𝑗)} for 𝑥∈𝐴_𝑘, 𝑦₁∈𝐴_𝑖, 𝑦₂∈𝐴_𝑗. Thus, we consider the following two cases.
Case I  (𝑖≤𝑘−2, 𝑗≤𝑘−2). We denote (𝑏_𝑙)_𝐵𝑖 by 𝑏_𝑙𝑖, where 𝑏_𝑙𝐵𝑖=1𝐵_𝑖𝐵𝑖𝑏_𝑙(𝑥)𝑑𝑥,𝑙=1,2 (44) and consider the following decomposition: 𝑏₁,𝑏₂,𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗(𝑥)=𝑏₁(𝑥)−𝑏_1𝑖𝑏₂(𝑥)−𝑏_2𝑗𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗(𝑥)−𝑏₁(𝑥)−𝑏_1𝑖𝑇𝑓𝜒_𝑖,𝑏₂(⋅)−𝑏_2𝑗𝑔𝜒_𝑗(𝑥)−𝑏₂(𝑥)−𝑏_2𝑗𝑇𝑏₁(⋅)−𝑏_1𝑖𝑓𝜒_𝑖,𝑔𝜒_𝑗(𝑥)+𝑇𝑏₁(⋅)−𝑏_1𝑖𝑓𝜒_𝑖,𝑏₂(⋅)−𝑏_2𝑗𝑔𝜒_𝑗(𝑥)=𝐿₁(𝑥)+𝐿₂(𝑥)+𝐿₃(𝑥)+𝐿₄(𝑥). (45) Thus, 𝑏₁,𝑏₂,𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗𝜒_{𝑘𝐿𝑝(⋅)}≤⁴_𝑚=1𝐿_𝑚𝜒_{𝑘𝐿𝑝(⋅)}=⁴_𝑚=1𝐽_𝑚. (46) Now, we will estimate each 𝐽_𝑚(𝑚=1,…,4), separately. Applying Lemma 5, we have 𝐿₁(𝑥)≤𝐶2^−2𝑘𝑛𝑏₁(𝑥)−𝑏_1𝑖𝑏₂(𝑥)−𝑏_2𝑗×𝑓𝜒_𝑖𝐿1𝑔𝜒_𝑗𝐿1≤𝐶2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝑖𝐿𝑝′1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝑗𝐿𝑝′2(⋅)}×𝑏₁(𝑥)−𝑏_1𝑖𝑏₂(𝑥)−𝑏_2𝑗. (47) Therefore, by virtue of generalized Hölder's inequality and Lemmas 6, 8, and 9, we get 𝐽₁=𝐿₁𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝑖𝐿𝑝′1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝑗𝐿𝑝′2(⋅)}×𝑏₁(𝑥)−𝑏_1𝑖𝑏₂(𝑥)−𝑏_2𝑗𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}×𝑏₁(𝑥)−𝑏_1𝑖𝜒_{𝐵𝑘𝐿𝑝1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}×𝑏₂(𝑥)−𝑏_2𝑗𝜒_{𝐵𝑘𝐿𝑝2(⋅)}≤𝐶𝑏_1∗𝑏_2∗2^−2𝑘𝑛(𝑘−𝑖)𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝜒_{𝐵𝑘𝐿𝑝1(⋅)}×𝑘−𝑗𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}𝜒_{𝐵𝑘𝐿𝑝2(⋅)}≤𝐶𝑏_1∗𝑏_2∗(𝑘−𝑖)𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝜒_{𝐵𝑘𝐿𝑝′1(⋅)}𝑘−𝑗𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}𝜒_{𝐵𝑘𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}(𝑘−𝑖)2^{−(𝑘−𝑖)𝑛𝛿𝑝′1}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝑘−𝑗2^{−(𝑘−𝑗)𝑛𝛿𝑝′2}. (48) Similarly, using Lemma 9, we approximate 𝐿₂ as 𝐿₂≤𝐶2^−2𝑘𝑛𝑏₁(𝑥)−𝑏_1𝑖𝑓𝜒_𝑖𝐿1𝑏₂(⋅)−𝑏_2𝑗𝑔𝜒_𝑗𝐿1≤𝐶2^−2𝑘𝑛𝑏₁(𝑥)−𝑏_1𝑖𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝑖𝐿𝑝′1(⋅)}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}(𝑏₂(⋅)−𝑏_2𝑗)𝜒_{𝑗𝐿𝑝′2(⋅)}≤𝐶𝑏_2∗2^−2𝑘𝑛𝑏₁(𝑥)−𝑏_1𝑖𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}. (49) Therefore, in view of Lemmas 6, 9, and 8, we have 𝐽₂=(𝐿₂)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_2∗2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}×(𝑏₁(𝑥)−𝑏_1𝑖)𝜒_{𝑘𝐿𝑝(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}×(𝑘−𝑖)𝜒_{𝐵𝑘𝐿𝑝(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗(𝑘−𝑖)𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝜒_{𝐵𝑘𝐿𝑝′1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}𝜒_{𝐵𝑘𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}(𝑘−𝑖)2^{−(𝑘−𝑖)𝑛𝛿𝑝′1}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}2^{−(𝑘−𝑗)𝑛𝛿𝑝′2}. (50) By symmetry, the estimate for 𝐽₃ is similar to that for 𝐽₂; therefore, 𝐽₃≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}2^{−(𝑘−𝑖)𝑛𝛿𝑝′1}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝑘−𝑗2^{−(𝑘−𝑗)𝑛𝛿𝑝′2}. (51) Finally, it remains to estimate 𝐽₄. For that, we use Lemma 9 to write 𝐿₄=𝑇𝑏₁(⋅)−𝑏_1𝑖𝑓𝜒_𝑖,𝑏₂(⋅)−𝑏_2𝑗𝑔𝜒_𝑗(𝑥)≤𝐶2^−2𝑘𝑛𝑏₁(⋅)−𝑏_1𝑖𝑓𝜒_𝑖𝐿1𝑏₂(⋅)−𝑏_2𝑗𝑔𝜒_𝑗𝐿1≤𝐶2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}(𝑏₁(⋅)−𝑏_1𝑖)𝜒_{𝑖𝐿𝑝′1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}×(𝑏₂(⋅)−𝑏_2𝑗)𝜒_{𝑗𝐿𝑝′2(⋅)}≤𝐶2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}(𝑏₁(⋅)−𝑏_1𝑖)𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}×(𝑏₂(⋅)−𝑏_2𝑗)𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}×𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}. (52) Thus, by Hölder's inequality and Lemmas 6 and 8, we obtain 𝐽₄=(𝐿₄)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}×𝜒_{𝐵𝑗𝐿𝑝2(⋅)}𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗2^−2𝑘𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑘𝐿𝑝1(⋅)}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑘𝐿𝑝2(⋅)}≤𝐶𝑏_1∗𝑏_2∗(𝑘−𝑖)𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝜒_{𝐵𝑘𝐿𝑝′1(⋅)}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}𝜒_{𝐵𝑘𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}2^{−(𝑘−𝑖)𝑛𝛿𝑝′1}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}2^{−(𝑘−𝑗)𝑛𝛿𝑝′2}. (53) Combining the estimates for 𝐽₁, 𝐽₂, 𝐽₃, and 𝐽₄, we have [𝑏₁,𝑏₂,𝑇](𝑓𝜒_𝑖,𝑔𝜒_𝑗)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×(𝑘−𝑖)2^{−(𝑘−𝑖)𝑛𝛿𝑝′1}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}×𝑘−𝑗2^{−(𝑘−𝑗)𝑛𝛿𝑝′2}. (54)
Case II(𝑖≥𝑘+2, 𝑗≥𝑘+2). We denote (𝑏_𝑙)_𝐵𝑘 by 𝑏_𝑙𝑘, where (𝑏_𝑙)_𝐵𝑘=(1/|𝐵_𝑘|)∫_𝐵𝑘𝑏_𝑙(𝑥)𝑑𝑥,𝑙=1,2. In this case, we consider the following decomposition: 𝑏₁,𝑏₂,𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗(𝑥)=𝑏₁(𝑥)−𝑏_1𝑘𝑏₂(𝑥)−𝑏_2𝑘𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗(𝑥)−𝑏₁(𝑥)−𝑏_1𝑘𝑇𝑓𝜒_𝑖,𝑏₂(⋅)−𝑏_2𝑘𝑔𝜒_𝑗(𝑥)−𝑏₂(𝑥)−𝑏_2𝑘𝑇𝑏₁(⋅)−𝑏_1𝑘𝑓𝜒_𝑖,𝑔𝜒_𝑗(𝑥)+𝑇𝑏₁(⋅)−𝑏_1𝑘𝑓𝜒_𝑖,𝑏₂(⋅)−𝑏_2𝑘𝑔𝜒_𝑗(𝑥)=𝐿₁(𝑥)+𝐿₂(𝑥)+𝐿₃(𝑥)+𝐿₄(𝑥). (55) Thus, 𝑏₁,𝑏₂,𝑇𝑓𝜒_𝑖,𝑔𝜒_𝑗𝜒_{𝑘𝐿𝑝(⋅)}≤⁴_𝑚=1𝐿_𝑚𝜒_{𝑘𝐿𝑝(⋅)}=⁴_𝑚=1𝐽_𝑚̃. (56) Let us first compute 𝐽₁̃. As in the proof of Theorem 3, in this case we estimate 𝐿₁ as 𝐿₁≤𝐶𝑏₁(𝑥)−𝑏_1𝑘𝑏₂(𝑥)−𝑏_2𝑘2^−𝑖𝑛𝑓𝜒_𝑖𝐿12^−𝑗𝑛×𝑔𝜒_𝑗𝐿1≤𝐶𝑏₁(𝑥)−𝑏_1𝑘𝑏₂(𝑥)−𝑏_2𝑘2^−𝑖𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×𝜒_{𝑖𝐿𝑝′1(⋅)}2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝑗𝐿𝑝′2(⋅)}. (57) By virtue of generalized Hölder's inequality and Lemmas 6–9, we obtain 𝐽₁̃=𝐿₁𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶2^−𝑖𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}×(𝑏₁(𝑥)−𝑏_1𝑘)(𝑏₂(𝑥)−𝑏_2𝑘)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶2^−𝑖𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}(𝑏₁(𝑥)−𝑏_1𝑘)𝜒_{𝑘𝐿𝑝1(⋅)}×2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}𝑏₂(𝑥)−𝑏_2𝑘𝜒_{𝑘𝐿𝑝2(⋅)}≤𝐶𝑏_1∗𝑏_2∗2^−𝑖𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝜒_{𝐵𝑘𝐿𝑝1(⋅)}×2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}𝜒_{𝐵𝑘𝐿𝑝2(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑘𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝1(⋅)}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑘𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝2(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}2^{(𝑘−𝑖)𝑛𝛿𝑝1}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}2^{(𝑘−𝑗)𝑛𝛿𝑝2}. (58) Next, we approximate 𝐽₂̃. Using Lemma 9, it is easy to see that 𝐿₂≤𝐶2^−2𝑖𝑛𝑏₁(𝑥)−𝑏_1𝑘𝑓𝜒_𝑖𝐿1𝑏₂(⋅)−𝑏_2𝑘𝑔𝜒_𝑗𝐿1≤𝐶2^−𝑖𝑛𝑏₁(𝑥)−𝑏_1𝑘𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝑖𝐿𝑝′1(⋅)}×2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}(𝑏₂(⋅)−𝑏_2𝑘)𝜒_{𝑗𝐿𝑝′2(⋅)}≤𝐶𝑏_2∗2^−𝑖𝑛𝑏₁(𝑥)−𝑏_1𝑘𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}×𝑗−𝑘2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}. (59) Thus, in view of Lemmas 6, 9, and 8, we get 𝐽₂̃=𝐿₂𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_2∗2^−𝑖𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}(𝑏₁(𝑥)−𝑏_1𝑘)𝜒_{𝑘𝐿𝑝(⋅)}×𝑗−𝑘2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗2^−𝑖𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}𝜒_{𝐵𝑘𝐿𝑝(⋅)}×𝑗−𝑘2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑘𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝1(⋅)}×𝑗−𝑘𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑘𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝2(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}2^{(𝑘−𝑖)𝑛𝛿𝑝1}𝑔𝜒_{𝑗𝐿𝑝2(⋅)}×𝑗−𝑘2^{(𝑘−𝑗)𝑛𝛿𝑝2}. (60) By symmetry, the estimate for 𝐽₃̃ is similar to that for 𝐽₂̃; thus 𝐽₃̃≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}(𝑖−𝑘)2^{(𝑘−𝑖)𝑛𝛿𝑝1}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}2^{(𝑘−𝑗)𝑛𝛿𝑝2}. (61) Lastly, it remains to compute 𝐽₄̃. For this purpose we use generalized Höder's inequality and lemmas 6 and 9 to obtain 𝐿₄=𝑇𝑏₁(⋅)−𝑏_1𝑘𝑓𝜒_𝑖,𝑏₂(⋅)−𝑏_2𝑘𝑔𝜒_𝑗(𝑥)≤𝐶2^−𝑖𝑛𝑏₁(⋅)−𝑏_1𝑘𝑓𝜒_𝑖𝐿12^−𝑗𝑛×(𝑏₂(⋅)−𝑏_2𝑘)𝑔𝜒_𝑗𝐿1≤𝐶2^−𝑖𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}(𝑏₁(⋅)−𝑏_1𝑘)𝜒_{𝑖𝐿𝑝′1(⋅)}×2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}(𝑏₂(⋅)−𝑏_2𝑘)𝜒_{𝑗𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗(𝑖−𝑘)2^−𝑖𝑛𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝′1(⋅)}×𝑗−𝑘2^−𝑗𝑛𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝′2(⋅)}≤𝐶𝑏_1∗𝑏_2∗(𝑖−𝑘)𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝1(⋅)}𝑗−𝑘𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝2(⋅)}. (62) Thus, an application of Lemma 8 yields 𝐽₄̃=(𝐿₄)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗(𝑖−𝑘)𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝑖𝐿𝑝1(⋅)}×𝑗−𝑘𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗(𝑖−𝑘)𝑓𝜒_{𝑖𝐿𝑝1(⋅)}𝜒_{𝐵𝑘𝐿𝑝1(⋅)}𝜒_{𝐵𝑖𝐿𝑝1(⋅)}×𝑗−𝑘𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝜒_{𝐵𝑘𝐿𝑝2(⋅)}𝜒_{𝐵𝑗𝐿𝑝2(⋅)}≤𝐶𝑏_1∗𝑏_2∗(𝑖−𝑘)𝑓𝜒_{𝑖𝐿𝑝1(⋅)}2^{(𝑘−𝑖)𝑛𝛿𝑝1}×𝑗−𝑘𝑔𝜒_{𝑗𝐿𝑝2(⋅)}2^{(𝑘−𝑗)𝑛𝛿𝑝2}. (63) Combining the estimates for 𝐽₁̃, 𝐽₂̃, 𝐽₃̃, and 𝐽₄̃, we have [𝑏₁,𝑏₂,𝑇](𝑓𝜒_𝑖,𝑔𝜒_𝑗)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓𝜒_{𝑖𝐿𝑝1(⋅)}(𝑖−𝑘)2^{(𝑘−𝑖)𝑛𝛿𝑝1}×𝑔𝜒_{𝑗𝐿𝑝2(⋅)}𝑗−𝑘2^{(𝑘−𝑗)𝑛𝛿𝑝2}. (64) In view of (43), (54), and (64), we arrive at 2^𝑘𝛼[𝑏₁,𝑏₂,𝑇](𝑓𝜒_𝑖,𝑔𝜒_𝑗)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝐸₁(𝑘,𝑖)2^𝑖𝛼1𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×𝐸₂𝑘,𝑗2^𝑗𝛼2𝑔𝜒_{𝑗𝐿𝑝2(⋅)}, (65) where for 𝑘,𝑖∈ℤ, 𝑙=1,2, 𝐸_𝑙()=()2𝑘,𝑖𝑘−𝑖^{(𝑘−𝑖)(𝛼𝑙−𝑛𝛿𝑝′𝑙)}()2,if𝑖≤𝑘−2,1,if𝑘−1≤𝑖≤𝑘+1,𝑖−𝑘^{(𝑘−𝑖)(𝛼𝑙+𝑛𝛿𝑝𝑙)},if𝑖≥𝑘+2. (66) Under the assumption −𝑛𝛿_𝑝𝑙<𝛼_𝑙<𝑛𝛿_𝑝′𝑙, it is easy to see that ^∞_𝑖=−∞𝐸₁(𝑘,𝑖)^𝛾+^∞_𝑗=−∞𝐸₂𝑘,𝑗^𝛾<∞,forany𝛾>0. (67) Now by the Minkowski's inequality and (65), we get 2^𝑘𝛼[𝑏₁,𝑏₂,𝑇](𝑓,𝑔)𝜒_{𝑘𝐿𝑝(⋅)}≤2^{𝑘𝛼∞∞}_{𝑖=−∞𝑗=−∞}[𝑏₁,𝑏₂,𝑇](𝑓𝜒_𝑖,𝑔𝜒_𝑗)𝜒_{𝑘𝐿𝑝(⋅)}≤^∞∞_{𝑖=−∞𝑗=−∞}2^𝑘𝛼[𝑏₁,𝑏₂,𝑇](𝑓𝜒_𝑖,𝑔𝜒_𝑗)𝜒_{𝑘𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗^∞_𝑖=−∞𝐸₁(𝑘,𝑖)2^𝑖𝛼1𝑓𝜒_{𝑖𝐿𝑝1(⋅)}×^∞_𝑗=−∞𝐸₂𝑘,𝑗2^𝑗𝛼2𝑔𝜒_{𝑗𝐿𝑝2(⋅)}. (68) Finally, by definition and the fact that 1/𝑞=1/𝑞₁+1/𝑞₂, we obtain [𝑏₁,𝑏₂,𝑇](𝑓,𝑔)_̇𝐾^𝛼,𝑞_𝑝(⋅)=^∞_𝑘=−∞2^𝑘𝛼𝑞𝑏₁,𝑏₂,𝑇𝑓,𝑔𝜒_𝑘^𝑞_{𝐿𝑝(⋅)}^1/𝑞≤𝐶𝑏_1∗𝑏_2∗×^∞_𝑘=−∞^∞_𝑖=−∞𝐸₁(𝑘,𝑖)2^𝑖𝛼1𝑓𝜒_{𝑖𝐿𝑝1(⋅)}^𝑞11/𝑞1×^∞_𝑘=−∞^∞_𝑗=−∞𝐸₂𝑘,𝑗2^𝑗𝛼2𝑔𝜒_{𝑗𝐿𝑝2(⋅)}^𝑞21/𝑞2∶=𝑏_1∗𝑏_2∗𝐼₁̃×𝐼₂̃. (69) It is enough to show that 𝐼₁̃≤𝐶‖𝑓‖_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅) and 𝐼₂̃≤𝐶‖𝑔‖_̇𝐾^𝛼_22𝑝2^,𝑞_(⋅). By symmetry, we only give estimates for 𝐼₁̃. For 0<𝑞₁<∞, by inequality (∑|𝑎_𝑖|)^𝑞1≤∑|𝑎_𝑖|^𝑞1 and inequality (67), we have 𝐼₁̃≤𝐶^∞_𝑖=−∞2^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}^∞_𝑘=−∞𝐸₁(𝑘,𝑖)^𝑞11/𝑞1≤𝐶^∞_𝑖=−∞2^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}^1/𝑞1=𝐶𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅). (70) For 𝑞₁>1, by Hölder's inequality and inequality (67), we obtain 𝐼₁̃≤𝐶^∞∞_{𝑘=−∞𝑖=−∞}𝐸₁(𝑘,𝑖)^𝑞1/22^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}×^∞_𝑖=−∞𝐸₁(𝑘,𝑖)^{𝑞′1/2𝑞1/𝑞′11/𝑞1}≤𝐶^∞_𝑖=−∞2^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}^∞_𝑘=−∞𝐸₁(𝑘,𝑖)^{𝑞1/21/𝑞1}≤𝐶^∞_𝑖=−∞2^{𝑖𝛼1𝑞1}𝑓𝜒_𝑖^𝑞1_{𝐿𝑝1(⋅)}^1/𝑞1=𝐶𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅). (71) Hence, for 0<𝑞₁<∞𝐼₁̃≤𝐶𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅). (72) By symmetry, for 0<𝑞₂<∞, we have 𝐼₂̃≤𝐶𝑔_̇𝐾^𝛼_22𝑝2^,𝑞_(⋅). (73) Therefore, 𝑏₁,𝑏₂,𝑇𝑓,𝑔_̇𝐾^𝛼,𝑞_𝑝(⋅)≤𝐶𝑏_1∗𝑏_2∗𝑓_̇𝐾^𝛼_11𝑝1^,𝑞_(⋅)𝑔_̇𝐾^𝛼_22𝑝2^,𝑞_(⋅). (74) By a similar procedure one can prove that ^∞_ℎ=1𝑏₁,𝑏₂,𝑇𝑓_ℎ,𝑔_ℎ^𝑟1/𝑟_{̇𝐾𝛼,𝑞𝑝(⋅)}≤^∞_ℎ=1𝑓_ℎ^𝑟11/𝑟1_{̇𝐾𝛼11𝑝1,𝑞(⋅)}^∞_ℎ=1𝑔_ℎ^𝑟21/𝑟2_{̇𝐾𝛼22𝑝2,𝑞(⋅)}×𝐶𝑏_1∗𝑏_2∗ (75) holds for all 𝑓_ℎ∈𝐾̇^𝛼1,𝑞1_𝑝1(⋅), 𝑔_ℎ∈𝐾̇^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 𝛼_𝑙≥𝑛𝛿_𝑝′𝑙, then the results of Theorems 3 and 4 are no longer true. It needs to replace the variable exponent Herz space 𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) with 𝐻𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) and the Herz-type Hardy spaces with the variable exponent recently introduced in [20]. The boundedness of multilinear singular integral operators from 𝐻𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) to 𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) is still an interesting question that needs to be answered.

Remark 12. Although we considered the 2-linear case, the method can be extended for any 𝑚-linear case without any essential difficulty.

Conflict of Interests

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