Commutators of Multilinear Calderón-Zygmund Operator and BMO Functions in Herz-Morrey Spaces with Variable Exponents

Abstract

We obtain the boundedness of a commutator generated by multilinear Calderón-Zygmund operator and BMO functions in Herz-Morrey spaces with variable exponents.

1. Introduction

In recent decades, variable exponent function spaces have attracted much attention. Since Kováčik and Rákosník [1] clarified fundamental properties of the variable Lebesgue and Sobolev spaces, there are many spaces studied, such as Besov and Trieble-Lizorkin spaces with variable exponents, Hardy spaces with variable exponents, Bessel potential spaces with a variable exponent, and Herz-Morrey spaces with variable exponents; see [2–13]. Recently, multilinear operators and their commutators in variable exponent function spaces are also intensively studied by a significant number of authors, such as multilinear commutators of multilinear singular integral with Lipschitz functions and BMO functions, respectively, in [14–19], multilinear commutators of BMO functions and multilinear singular integral operators with nonsmooth kernels in [20, 21], a vector-estimate of higher order commutators on Herz-Morrey spaces with variable exponent in [22], maximal multilinear commutators and maximal iterated commutators generated by multilinear operators and Lipschitz functions in [23], and weighted estimates for vector-valued commutators of multilinear operators in [24].

Motivated by the above results, in this paper, we will consider the boundedness of a commutator generated by a multilinear Calderón-Zygmund operator and BMO functions on the variable Herz-Morrey spaces.

To state the main result of this paper, we need to recall more notations.

Let 𝑇 be a multilinear singular integral operator which is initially defined on the 𝑚-fold product of the Schwartz space 𝒮(ℝ^𝑛). Its values are taken in the space of tempered distributions 𝒮^′(ℝ^𝑛) such that, for 𝑥∉∩^𝑚_𝑗=1supp𝑓_𝑗, 𝑓^→=(𝑓₁,…,𝑓_𝑚), 𝑇𝑓^→(𝑥)=_{(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚^𝑚_𝑖=1𝑓_𝑖𝑦_𝑖𝑑𝑦₁⋯𝑑𝑦_𝑚, (1) where 𝑓₁,…,𝑓_𝑚∈𝐿^∞_𝐶(ℝ^𝑛) (the space of compactly supported bounded functions). Here, the kernel 𝐾 is a function in (ℝ^𝑛)^𝑚+1 away from the diagonal 𝑦₀=𝑦₁=⋯=𝑦_𝑚 and satisfies the standard estimates 𝐾𝑥,𝑦₁,…,𝑦_𝑚≤𝐴𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^−𝑚𝑛,𝐾𝑥,𝑦₁,…,𝑦_𝑚−𝐾𝑥^′,𝑦₁,…,𝑦_𝑚≤𝐴𝑥−𝑥^′𝜖𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^{𝑚𝑛+𝜖} (2) provided that |𝑥−𝑥^′|≤(1/2)max{|𝑥−𝑦₁|+⋯+|𝑥−𝑦_𝑚|}, and, for each 1≤𝑖≤𝑚, 𝐾𝑥,𝑦₁,…,𝑦_𝑖,…,𝑦_𝑚−𝐾𝑥,𝑦₁,…,𝑦^′_𝑖,…,𝑦_𝑚≤𝐴𝑦_𝑖−𝑦^′_𝑖^𝜖∑^𝑚_𝑗=1𝑥−𝑦_𝑗^{𝑚𝑛+𝜖} (3) provided that |𝑦_𝑖−𝑦^′_𝑖|≤(1/2)max{|𝑥−𝑦₁|+⋯+|𝑥−𝑦_𝑚|}, where 𝐴 and 𝜖 are positive constants.

Such kernels are called the 𝑚-linear Calderón-Zygmund kernels and the collection of such functions is denoted by 𝑚-CZK(𝐴,𝜖) in [25].

Let 𝑇 be as in (1) with an 𝑚-CZK(𝐴,𝜖) kernel. If, for some 1<𝑞₁, 𝑞₂,…,𝑞_𝑚<∞, 𝑇 is bounded from 𝐿^𝑞1×𝐿^𝑞2×⋯×𝐿^𝑞𝑚 to 𝐿^𝑞 with 1/𝑞₁+1/𝑞₂+⋯+1/𝑞_𝑚=1/𝑞, then we say 𝑇 is an 𝑚-linear Calderón-Zygmund operator. Grafakos and Torres in [25] showed that if 𝑇 is an 𝑚-linear Calderón-Zygmund operator, then 𝑇 is bounded from 𝐿^𝑞1×𝐿^𝑞2×⋯×𝐿^𝑞𝑚 to 𝐿^𝑞 for any 1<𝑞₁, 𝑞₂,…,𝑞_𝑚<∞ such that 1/𝑞₁+1/𝑞₂+⋯+1/𝑞_𝑚=1/𝑞. Then, Grafakos and Torres in [26] obtained weighted norm inequalities for multilinear Calderón-Zygmund operators.

If 𝑏∈𝐿¹_loc(ℝ^𝑛) (the set of all complex-valued locally integrable functions on ℝ^𝑛), set ‖𝑏‖_∗∶=sup_𝐵1|𝐵|_𝐵𝑏(𝑥)−𝑏_𝐵𝑑𝑥, (4) where the supremum is taken over all balls in ℝ^𝑛, 𝑏_𝐵 is the mean of 𝑏 on 𝐵, and what follows |𝐸| is the Lebesgue measure of measurable set 𝐸 in ℝ^𝑛. A function 𝑏 is called bounded mean oscillation if ‖𝑏‖_∗<∞ and BMO(ℝ^𝑛) is the set of all locally integrable functions 𝑏 on ℝ^𝑛 such that ‖𝑏‖_∗<∞.

Let ^→𝑏=(𝑏₁,…,𝑏_𝑚) and 𝑏_𝑗∈BMO(ℝ^𝑛) for 1≤𝑗≤𝑚. We will consider the commutator [^→𝑏,𝑇], which is defined for suitable functions 𝑓₁,…,𝑓_𝑚 by ^→𝑏,𝑇𝑓^→(𝑥)∶=_𝜎(−1)^|𝜎′|𝑏_𝜎(𝑥)𝑇𝑏𝑓_𝜎′^→(𝑥); (5) where 𝑓^→=(𝑓₁,…,𝑓_𝑚), 𝜎 denotes a subset of {1,2,…,𝑚},  𝜎^′ is the complement of 𝜎 in {1,2,…,𝑚}, |𝜎^′| denotes the number of elements of 𝜎^′, 𝑏_𝜎(𝑥)=∏_𝑗∈𝜎𝑏_𝑗(𝑥), 𝑏𝑓_𝜎′^→=(𝑔₁,…,𝑔_𝑚), when 𝑗∈𝜎^′, 𝑔_𝑗=𝑏_𝑗𝑓_𝑗, otherwise, 𝑔_𝑗=𝑓_𝑗.

Definition 1. Let 𝑝(⋅)∶ℝ^𝑛→[1,∞) be a measurable function.(i)The Lebesgue space with variable exponent 𝐿^𝑝(⋅)(ℝ^𝑛) is defined by 𝐿^𝑝(⋅)ℝ^𝑛∶=𝑓ismeasurable:_ℝ𝑛𝑓(𝑥)𝜆^𝑝(𝑥).𝑑𝑥<∞forsome𝜆>0 (6)(ii)The space 𝐿^𝑝(⋅)_loc(ℝ^𝑛) is defined by 𝐿^𝑝(⋅)_locℝ^𝑛∶=𝑓∶𝑓𝜒_𝐾∈𝐿^𝑝(⋅)ℝ^𝑛forallcompactsubsets𝐾⊂ℝ^𝑛, (7) where, and in what follows, 𝜒_𝑆 denotes the characteristic function of a measurable set 𝑆⊂ℝ^𝑛.
𝐿^𝑝(⋅)(ℝ^𝑛) is a Banach function space when equipped with the norm 𝑓_{𝐿𝑝(⋅)}∶=inf𝜆>0∶_ℝ𝑛|𝑓(𝑥)|𝜆^𝑝(𝑥)𝑑𝑥≤1. (8)
Letting 𝑝(⋅)∶ℝ^𝑛→(0,∞), we denote 𝑝₋∶=essinf_{𝑥∈ℝ𝑛}𝑝(𝑥),𝑝₊∶=essinf_{𝑥∈ℝ𝑛}𝑝(𝑥). (9) The set 𝒫(ℝ^𝑛) consists of all 𝑝(⋅) satisfying 𝑝₋>1 and 𝑝₊<∞; 𝒫⁰(ℝ^𝑛) consists of all 𝑝(⋅) satisfying 𝑝₋>0 and 𝑝₊<∞.𝐿^𝑝(⋅) can be similarly defined as mentioned above for 𝑝(⋅)∈𝒫⁰(ℝ^𝑛). 𝑝^′(⋅) means the conjugate exponent of 𝑝(⋅) that means 1/𝑝(⋅)+1/𝑝^′(⋅)=1.
Let 𝑓∈𝐿¹_loc(ℝ^𝑛). Then the standard Hardy-Littlewood maximal function of 𝑓 is defined by 𝑀𝑓(𝑥)=sup_𝑥∈𝐵1|𝐵|_𝐵𝑓𝑦𝑑𝑦,∀𝑥∈ℝ^𝑛, (10) where 𝐵 is a ball and |𝐵| is the volume of 𝐵. Let ℬ(ℝ^𝑛) be the set of 𝑝(⋅)∈𝒫(ℝ^𝑛) such that 𝑀 is bounded on 𝐿^𝑝(⋅)(ℝ^𝑛). For more on ℬ(ℝ^𝑛), see [27, 28].

Definition 2. We say that a function 𝛼(⋅)∶ℝ^𝑛→ℝ is locally log-Hölder continuous if there exists a constant 𝐶 such that 𝛼(𝑥)𝑦≤𝐶−𝛼,log𝑒+1/𝑥−𝑦𝑥,𝑦∈ℝ^𝑛,<1𝑥−𝑦2. (11) If |𝛼(𝑥)(0≤𝐶−𝛼)|log(𝑒+1/|𝑥|),∀𝑥∈ℝ^𝑛, (12) then we say that 𝛼(⋅) is log-Hölder continuous at the origin. If there exists 𝛼_∞∈ℝ such that 𝛼(𝑥)−𝛼_∞≤𝐶log(𝑒+|𝑥|),∀𝑥∈ℝ^𝑛, (13) then we say that 𝛼(⋅) is log-Hölder continuous at the infinity. If 𝛼(⋅) is both locally log-Hölder continuous and log-Hölder continuous at the infinity, then we say 𝛼(⋅) is global log-Hölder continuous.

We denote by 𝒫₀^log(ℝ^𝑛) and 𝒫_∞^log(ℝ^𝑛) the class of all exponents 𝑝∈𝒫(ℝ^𝑛) which is log-Hölder continuous at the origin and at the infinity, respectively.

Next, we recall the definitions of Herz spaces and Herz-Morrey spaces with variable exponents. We use the following notations. For each 𝑘∈ℤ, we define 𝐵_𝑘∶=𝑥∈ℝ^𝑛∶|𝑥|≤2^𝑘,𝐷_𝑘∶=𝐵_𝑘⧵𝐵_𝑘−1,𝜒_𝑘∶=𝜒_𝐷𝑘. (14)

Definition 3 (see [29, Definition  2]). Let 𝑞∈(0,∞], 𝑝(⋅)∈𝒫(ℝ^𝑛), and 𝛼(⋅)∶ℝ^𝑛→ℝ with 𝛼∈𝐿^∞(ℝ^𝑛). The homogeneous Herz space 𝐾̇^𝛼(⋅)_{𝑞,𝑝(⋅)}(ℝ^𝑛) is defined as the set of all 𝑓∈𝐿^𝑝(⋅)_loc(ℝ^𝑛⧵{0}) such that 𝑓_̇𝐾^𝛼(⋅)_{𝑞,𝑝(⋅)}∶=_𝑘∈ℤ2^{𝑘𝛼(⋅)}𝑓𝜒_𝑘^𝑞_𝑝(⋅)^1/𝑞<∞, (15) with the usual modifications when 𝑞=∞.

Definition 4 (see [30, Definition  2]). Let 0<𝑞<∞, 𝑝(⋅)∈𝒫(ℝ^𝑛), 0≤𝜆<∞ and 𝛼(⋅)∶ℝ^𝑛→ℝ with 𝛼∈𝐿^∞(ℝ^𝑛). The Herz-Morrey space 𝑀𝐾̇^{𝛼(⋅),𝜆}_{𝑞,𝑝(⋅)}(ℝ^𝑛) with variable exponents is defined by 𝑀𝐾̇_(⋅)^{𝛼(⋅),𝜆}_𝑞,𝑝ℝ^𝑛∶=𝑓∈𝐿^𝑝(⋅)_locℝ^𝑛⧵{0}∶𝑓_𝑀̇𝐾^{𝛼(⋅),𝜆}_{𝑞,𝑝(⋅)(ℝ𝑛)}<∞, (16) where 𝑓_𝑀̇𝐾^{𝛼(⋅),𝜆}_{𝑞,𝑝(⋅)(ℝ𝑛)}∶=sup_𝐿∈ℤ2^{−𝐿𝜆𝐿}_𝑘=−∞2^{𝑘𝛼(⋅)}𝑓𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞. (17)

If 𝛼(⋅) is a constant, then 𝑀𝐾̇^{𝛼(⋅),𝜆}_{𝑞,𝑝(⋅)}(ℝ^𝑛)=𝑀𝐾̇^𝛼,𝜆_{𝑞,𝑝(⋅)}(ℝ^𝑛) was defined in [30]. If 𝜆=0, then 𝑀𝐾̇^{𝛼(⋅),𝜆}_{𝑞,𝑝(⋅)}(ℝ^𝑛)=𝐾̇^𝛼(⋅)_{𝑞,𝑝(⋅)}(ℝ^𝑛). If both 𝛼(⋅) and 𝑝(⋅) are constant and 𝜆=0, then 𝑀𝐾̇^{𝛼(⋅),𝜆}_{𝑞,𝑝(⋅)}(ℝ^𝑛)=𝐾̇^𝛼_𝑞,𝑝(ℝ^𝑛) is the classical Herz space in [31].

Lemma 5 (see [30, Lemma  1 and (10)]). If 𝑝(⋅)∈ℬ(ℝ^𝑛), then there exist constants 𝛿₁, 𝛿₂∈(0,1) and 𝐶>0 such that, for all balls 𝐵 in ℝ^𝑛 and all measurable subsets 𝑆⊂𝐵, 𝜒_{𝑆𝐿𝑝(⋅)(ℝ𝑛)}𝜒_{𝐵𝐿𝑝(⋅)(ℝ𝑛)}|𝑆|≤𝐶|𝐵|^𝛿1,𝜒_{𝑆𝐿𝑝′(⋅)(ℝ𝑛)}𝜒_{𝐵𝐿𝑝′(⋅)(ℝ𝑛)}|𝑆|≤𝐶|𝐵|^𝛿2. (18)

There is a position to state our result.

Theorem 6. Let 𝑏_𝑖∈𝐵𝑀𝑂(ℝ^𝑛) and 𝑝(⋅), 𝑝_𝑖(⋅)∈ℬ(ℝ^𝑛) satisfy 1/𝑝(𝑥)=1/𝑝₁(𝑥)+1/𝑝₂(𝑥)+⋯+1/𝑝_𝑚(𝑥), 1<𝑝⁻_𝑖≤𝑝⁺_𝑖<𝑛/𝜆_𝑖, (𝑝(⋅)/𝑝_∗)^′∈ℬ(ℝ^𝑛) for some 0<𝑝_∗<𝑝₋, 𝑖=1,2,…,𝑚. Let 0<𝑞_𝑖<∞, 0≤𝜆_𝑖<∞ and 𝛼_𝑖(⋅)∈𝐿^∞(ℝ^𝑛)∩𝒫₀^log(ℝ^𝑛)∩𝒫_∞^log(ℝ^𝑛) for 𝑖=1,2,…,𝑚 with 𝜆_𝑖−𝑛𝛿₁<𝛼⁻_𝑖<𝛼⁺_𝑖<𝑛𝛿₂, (19) where 𝛿₁,𝛿₂∈(0,1) are the constants appearing in (18). Suppose that 𝜆=∑^𝑚_𝑖=1𝜆_𝑖, 𝛼(⋅)=∑^𝑚_𝑖=1𝛼_𝑖(⋅), and 1/𝑞=∑^𝑚_𝑖=1(1/𝑞_𝑖). Then [^→𝑏,𝑇](𝑓^→)_{𝑀̇𝐾𝛼(⋅),𝜆𝑞,𝑝(⋅)(ℝ𝑛)}≤𝐶^𝑚_𝑖=1𝑏_𝑖∗𝑓_{𝑖𝑀̇𝐾}^𝛼_{𝑖𝑖𝑞𝑖𝑖}^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)} (20) with the constant 𝐶>0 independent of 𝑓^→=(𝑓₁,𝑓₂,…,𝑓_𝑚).

Remark 7. Let 𝜆_𝑖=0, and then the commutator [^→𝑏,𝑇] is bounded from the product of variable exponents Herz spaces 𝐾̇^𝛼1_𝑞^(⋅)_1,𝑝1(⋅)(ℝ^𝑛)×𝐾̇^𝛼2_𝑞^(⋅)_2,𝑝2(⋅)(ℝ^𝑛)×⋯×𝐾̇^𝛼𝑚_𝑞^(⋅)_{𝑚,𝑝𝑚(⋅)}(ℝ^𝑛) to variable exponents Herz space 𝐾̇^𝛼(⋅)_{𝑞,𝑝(⋅)}(ℝ^𝑛) when −𝑛𝛿₁<𝛼⁻_𝑖<𝛼⁺_𝑖<𝑛𝛿₂.
Finally, we point out that 𝐶 denotes a positive constant which may be different at different occurrences.

2. Proof of the Main Result

To give our proof, we need some lemmas.

Lemma 8 (see [32, Proposition  2]). If 𝛼(⋅)∈𝐿^∞(ℝ^𝑛)∩𝒫₀^log(ℝ^𝑛)∩𝒫_∞^log(ℝ^𝑛), 0≤𝜆<∞, 𝑝(⋅)∈𝒫(ℝ^𝑛), and 𝑞∈(0,∞), then 𝑓_𝑀̇𝐾^{𝛼(⋅),𝜆}_{𝑞,𝑝(⋅)(ℝ𝑛)}≈maxsup_{𝐿≤0,𝐿∈ℤ}2^{−𝐿𝜆𝐿}_𝑘=−∞2^{𝑘𝛼(0)𝑞}𝑓𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,sup_{𝐿>0,𝐿∈ℤ}2^{−𝐿𝜆−1}_𝑘=−∞2^{𝑘𝛼(0)𝑞}𝑓𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞+2^{−𝐿𝜆𝐿}_𝑘=02^{𝑘𝛼∞𝑞}𝑓𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞. (21)

Lemma 9 (see [30, Lemma  2]). If 𝑝(⋅)∈ℬ(ℝ^𝑛), then there exists a constant 𝐶>0 such that, for all balls 𝐵 in ℝ^𝑛, 𝐶⁻¹≤1|𝐵|𝜒_{𝐵𝐿𝑝(⋅)(ℝ𝑛)}𝜒_{𝐵𝐿𝑝′(⋅)(ℝ𝑛)}≤𝐶. (22)

Lemma 10 10 (see [29, Lemma  3]). Let 𝑘 be a positive integer. Then one has that, for all 𝑏∈𝐵𝑀𝑂(ℝ^𝑛) and all 𝑖, 𝑗∈ℤ with 𝑗>𝑖, 𝐶⁻¹‖𝑏‖^𝑘_∗≤sup_{𝐵∶𝑏𝑎𝑙𝑙}1𝜒_{𝐵𝐿𝑝(⋅)(ℝ𝑛)}𝑏−𝑏_𝐵^𝑘𝜒_{𝐵𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶‖𝑏‖^𝑘_∗,𝑏−𝑏_𝐵𝑖^𝑘𝜒_{𝐵𝑗𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶𝑗−𝑖^𝑘‖𝑏‖^𝑘_∗𝜒_{𝐵𝑗𝐿𝑝(⋅)(ℝ𝑛)}. (23)

In fact, if 𝑘=1, then, from inequalities (23), for all balls 𝐵 and 𝑖, 𝑗∈ℤ with 𝑗>𝑖, we have 𝑏−𝑏_𝐵𝜒_{𝐵𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶‖𝑏‖_∗𝜒_{𝐵𝐿𝑝(⋅)(ℝ𝑛)},𝑏−𝑏_𝐵𝑖𝜒_{𝐵𝑗𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶𝑗−𝑖‖𝑏‖_∗𝜒_{𝐵𝑗𝐿𝑝(⋅)(ℝ𝑛)}. (24)

Lemma 11 (see [1, Theorem  2.1] ). If 𝑝(⋅)∈𝒫(ℝ^𝑛), then, for all 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛) and all 𝑔∈𝐿^𝑝′(⋅)(ℝ^𝑛), one has _ℝ𝑛𝑓(𝑥)𝑔(𝑥)𝑑𝑥≤𝑟_𝑝𝑓_{𝐿𝑝(⋅)(ℝ𝑛)}𝑔_{𝐿𝑝′(⋅)(ℝ𝑛)}, (25) where 𝑟_𝑝∶=1+1/𝑝₊−1/𝑝₋.

Lemma 12 (see [14, Theorem 2.3]). Let 𝑝,𝑝₁,𝑝₂∈𝒫⁰(ℝ^𝑛) such that 1/𝑝(𝑥)=1/𝑝₁(𝑥)+1/𝑝₂(𝑥). Then there exists a constant 𝐶_𝑝,𝑝1 independent of the functions 𝑓 and 𝑔 such that 𝑓𝑔_{𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶_𝑝,𝑝1𝑓_{𝐿𝑝1(⋅)(ℝ𝑛)}𝑔_{𝐿𝑝2(⋅)(ℝ𝑛)} (26) holds for every 𝑓∈𝐿^𝑝1(⋅)(ℝ^𝑛) and 𝑔∈𝐿^𝑝2(⋅)(ℝ^𝑛).

Lemma 13 13 (see [14, Corollary  2.2]). Let 𝑇 be a 2-linear Calderón-Zygmund operator and let 𝑏₁ and 𝑏₂ be 𝐵𝑀𝑂 functions. Let 𝑝(⋅)∈𝒫⁰(ℝ^𝑛) such that there exists 0<𝑝_∗<𝑝₋ with (𝑝(⋅)/𝑝_∗)^′∈ℬ(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 functions 𝑓_𝑖∈𝐿^{𝑝𝑖(⋅)}(ℝ^𝑛) for 𝑖=1,2 such that [𝑏₁,𝑏₂,𝑇](𝑓₁,𝑓₂)_{𝐿𝑝(⋅)}≤𝐶𝑏_1∗𝑏_2∗𝑓_{1𝐿𝑝1(⋅)}𝑓_{2𝐿𝑝2(⋅)}. (27)

Proof of Theorem 6. Although our method is suitable for any multilinear operator, for simplicity, we only consider 2-linear operators. Let 𝑏₁ and 𝑏₂ be BMO functions. Since the set of all bounded compactly supported functions is dense in Herz-Morrey spaces with variable exponents, we let 𝑓₁ and 𝑓₂ be bounded compactly supported functions; then, for 𝑥∈ℝ^𝑛, we write 𝑏₁,𝑏₂,𝑇𝑓₁,𝑓₂(𝑥)=𝑏₁(𝑥)𝑏₂(𝑥)𝑇𝑓₁,𝑓₂(𝑥)−𝑏₁(𝑥)𝑇𝑓₁,𝑏₂𝑓₂(𝑥)−𝑏₂(𝑥)𝑇𝑏₁𝑓₁,𝑓₂(𝑥)+𝑇𝑏₁𝑓₁,𝑏₂𝑓₂(𝑥). (28) We write 𝑓_𝑖(𝑥)=^∞_{𝑙𝑖=−∞}𝑓_𝑖(𝑥)𝜒_𝑙𝑖(𝑥)=∶^∞_{𝑙𝑖=−∞}𝑓_𝑙𝑖(𝑥),𝑖=1,2. (29) Then, for each 𝑘∈ℤ, if 𝑙_𝑖≤𝑘−1, from Lemma 11, we obtain _ℝ𝑛𝑏_𝑖(𝑥)−𝑏_𝑖𝑦_𝑖𝑓_𝑙𝑖𝑦_𝑖𝑑𝑦_𝑖≤𝐶_ℝ𝑛𝑏_𝑖(𝑥)−𝑏_𝐵𝑙𝑖𝑓_𝑙𝑖𝑦_𝑖𝑑𝑦_𝑖+_ℝ𝑛𝑏_𝐵𝑙𝑖−𝑏_𝑖𝑦_𝑖𝑓_𝑙𝑖𝑦_𝑖𝑑𝑦_𝑖≤𝐶𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑏_𝑖(𝑥)−𝑏_𝐵𝑙𝑖⋅𝜒_{𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}+𝑏_𝐵𝑙𝑖−𝑏_𝑖⋅𝜒_{𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}, (30) and by using inequalities (18), (22), and (24), it follows that _ℝ𝑛𝑏_𝑖(⋅)−𝑏_𝑖𝑦_𝑖𝑓_𝑙𝑖𝑦_𝑖𝑑𝑦_𝑖𝜒_{𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}≤𝐶𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑏_𝑖−𝑏_𝐵𝑙𝑖𝜒_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}⋅𝜒_{𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}+𝑏_𝐵𝑙𝑖−𝑏_𝑖𝜒_{𝐵𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}⋅𝜒_{𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}≤𝐶𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑘−𝑙_𝑖𝑏_𝑖∗𝜒_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝜒_{𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}+𝑏_𝑖∗𝜒_{𝐵𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}𝜒_{𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}≤𝐶𝑏_𝑖∗𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑘−𝑙_𝑖𝜒_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝜒_{𝐵𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}≤𝐶𝑏_𝑖∗𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑘−𝑙_𝑖𝜒_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}×𝜒_{𝐵𝑘𝐿𝑝′𝑖(⋅)(ℝ𝑛)}𝜒_{𝐵𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}𝜒_{𝐵𝑘𝐿𝑝′𝑖(⋅)(ℝ𝑛)}≤𝐶𝑘−𝑙_𝑖2^{(𝑙𝑖−𝑘)𝑛𝛿2}2^𝑘𝑛𝑏_𝑖∗𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}. (31) Similarly, if 𝑙_𝑖≥𝑘+1, we have _ℝ𝑛[𝑏_𝑖(⋅)−𝑏_𝑖(𝑦_𝑖)]𝑓_𝑙𝑖(𝑦_𝑖)𝑑𝑦_𝑖𝜒_{𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}≤𝐶𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑏_𝑖−𝑏_𝐵𝑘𝜒_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}⋅𝜒_{𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}+𝑏_𝐵𝑘−𝑏_𝑖𝜒_{𝐵𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}⋅𝜒_{𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}≤𝐶𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑏_𝑖∗𝜒_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝜒_{𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}+𝑙_𝑖−𝑘𝑏_𝑖∗𝜒_{𝐵𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}𝜒_{𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}≤𝐶𝑏_𝑖∗𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑙_𝑖−𝑘𝜒_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝜒_{𝐵𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}≤𝐶𝑏_𝑖∗𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑙_𝑖×𝜒−𝑘_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝜒_{𝐵𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝜒_{𝐵𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝜒_{𝐵𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}≤𝐶𝑙_𝑖−𝑘2^{(𝑘−𝑙𝑖)𝑛𝛿1}2^𝑙𝑖𝑛𝑏_𝑖∗𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}. (32) As for the case 𝑙_𝑖=𝑘, we get _ℝ𝑛𝑏_𝑖(⋅)−𝑏_𝑖𝑦_𝑖𝑓_𝑙𝑖(𝑦_𝑖)𝑑𝑦_𝑖𝜒_{𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}≤𝐶𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝑏_𝑖−𝑏_𝐵𝑘𝜒_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}⋅𝜒_{𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}+𝑏_𝐵𝑘−𝑏_𝑖𝜒_{𝐵𝑙𝑖𝐿𝑝′𝑖(⋅)(ℝ𝑛)}⋅𝜒_{𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}≤𝐶𝑏_𝑖∗𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝜒_{𝐵𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}𝜒_{𝐵𝑘𝐿𝑝′𝑖(⋅)(ℝ𝑛)}≤𝐶2^𝑘𝑛𝑏_𝑖∗𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}. (33)
By Lemma 8, we know [𝑏₁,𝑏₂,𝑇](𝑓₁,𝑓₂)_𝑀̇𝐾^{𝛼(⋅),𝜆}_{𝑞,𝑝(⋅)(ℝ𝑛)}≈max{𝐸,𝐹}, (34) where 𝐸∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞∞}_𝑙1^∞_{=−∞𝑙2=−∞}𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_𝑙2𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,𝐹∶=sup_{𝐿>0,𝐿∈ℤ}2^{−𝐿𝜆−1}_𝑘=−∞2^{𝑘𝛼(0)𝑞}𝑏₁,𝑏₂,𝑇𝑓₁,𝑓₂𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞+2^{−𝐿𝜆𝐿}_𝑘=02^{𝑘𝛼∞𝑞}×𝑏₁,𝑏₂,𝑇𝑓₁,𝑓₂𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞. (35) Since the estimate of 𝐹 is essentially similar to that of 𝐸, so it suffices to prove that 𝐸 is bounded in Herz-Morrey spaces with variable exponents. It is easy to see that 𝐸≤𝐶⁹_𝑖=1𝐼_𝑖, (36) where 𝐼₁∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘−2}_𝑙1=−∞^𝑘−2_𝑙2=−∞𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_𝑙2𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,𝐼₂∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘−2}_𝑙1=−∞^𝑘+1_{𝑙2=𝑘−1}𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_𝑙2𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,𝐼₃∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘−2}_𝑙1^∞_{=−∞𝑙2=𝑘+2}𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_𝑙2𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,𝐼₄∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘+1}_{𝑙1=𝑘−1}^𝑘−2_𝑙2=−∞𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_𝑙2𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,𝐼₅∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘+1}_{𝑙1=𝑘−1}^𝑘+1_{𝑙2=𝑘−1}𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_𝑙2𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,𝐼₆∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘+1}_𝑙1^∞_{=𝑘−1𝑙2=𝑘+2}𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_𝑙2𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,𝐼₇∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞∞}_𝑙1=𝑘+2^𝑘−2_𝑙2=−∞𝑏₁,𝑏₂,𝑇×(𝑓_𝑙1,𝑓_𝑙2)𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,𝐼₈∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞∞}_𝑙1=𝑘+2^𝑘+1_{𝑙2=𝑘−1}𝑏₁,𝑏₂,𝑇×(𝑓_𝑙1,𝑓_𝑙2)𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞,𝐼₉∶=sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞∞}_𝑙1^∞_{=𝑘+2𝑙2=𝑘+2}𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_𝑙2𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞. (37) Using the symmetry of 𝑓₁ and 𝑓₂, we only need to estimate 𝐼₁, 𝐼₂, 𝐼₃, 𝐼₅, 𝐼₆, 𝐼₉ because the estimates of 𝐼₂, 𝐼₃, and 𝐼₆ are analogous to those of 𝐼₄, 𝐼₇, and 𝐼₈, respectively. In what follows, we divide it into 6 steps.
Step 1. To estimate the term of 𝐼₁, we note that 𝑙_𝑖≤𝑘−2 for 𝑖=1,2. Thus, for 𝑥∈𝐷_𝑘, 𝑦_𝑖∈𝐷_𝑙𝑖, 𝑥−𝑦_𝑖≥|𝑥|−𝑦_𝑖>2^𝑘−1−2^𝑙𝑖≥2^𝑘−2. (38) Then, for 𝑥∈𝐷_𝑘, we get 𝐾𝑥,𝑦₁,𝑦₂≤𝐶𝑥−𝑦₁+𝑥−𝑦₂^−2𝑛≤𝐶2^−2𝑘𝑛. (39) Therefore, 𝑏₁,𝑏₂,𝑇𝑓_𝑙1,𝑓_𝑙2(𝑥)≤𝐶2^{2−2𝑘𝑛}_{𝑖=1ℝ𝑛}𝑏_𝑖(𝑥)−𝑏_𝑖𝑦_𝑖𝑓_𝑙𝑖𝑦_𝑖𝑑𝑦_𝑖. (40) By Lemma 12 and inequality (31), we obtain ^𝑘−2_𝑙1=−∞^𝑘−2_𝑙2=−∞[𝑏₁,𝑏₂,𝑇](𝑓_𝑙1,𝑓_𝑙2)𝜒_{𝑘𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶2^{2−2𝑘𝑛}_𝑖=1^𝑘−2_{𝑙𝑖=−∞ℝ𝑛}𝑏_𝑖(⋅)−𝑏_𝑖𝑦_𝑖𝑓_𝑙𝑖𝑦_𝑖𝑑𝑦_𝑖𝜒_{𝑘𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶2^{2−2𝑘𝑛}_𝑖=1^𝑘−2_{𝑙𝑖=−∞ℝ𝑛}[𝑏_𝑖(⋅)−𝑏_𝑖(𝑦_𝑖)]𝑓_𝑙𝑖(𝑦_𝑖)𝑑𝑦_𝑖𝜒_{𝑘𝐿𝑝𝑖(⋅)(ℝ𝑛)}≤𝐶²_𝑖=1^𝑘−2_{𝑙𝑖=−∞}𝑘−𝑙_𝑖2^{(𝑙𝑖−𝑘)𝑛𝛿2}𝑏_𝑖∗𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}. (41)
Since 1/𝑞=1/𝑞₁+1/𝑞₂, it follows that 𝐼₁≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞2}_𝑖=1^𝑘−2_{𝑙𝑖=−∞}𝑘−𝑙_𝑖2^{(𝑙𝑖−𝑘)𝑛𝛿2}𝑏_𝑖∗×𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}^1/𝑞≤𝐶sup²_{𝐿≤0,𝐿∈ℤ𝑖=1}2^{−𝐿𝜆𝑖}𝑏_𝑖∗×^𝐿_𝑘=−∞^𝑘−2_{𝑙𝑖=−∞}𝑘−𝑙_𝑖2^{(𝑙𝑖−𝑘)𝑛𝛿2+𝑘𝛼𝑖(0)}×𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}^{𝑞𝑖1/𝑞𝑖}=∶𝐶sup_{𝐿≤0,𝐿∈ℤ}𝐼₁₁(𝐿)𝐼₁₂(𝐿), (42) where 𝐼_1𝑖(𝐿)∶=2^{−𝐿𝜆𝑖}𝑏_𝑖∗×^𝐿_𝑘=−∞^𝑘−2_{𝑙𝑖=−∞}𝑘−𝑙_𝑖2^{(𝑙𝑖−𝑘)𝜇𝑖}2^{𝑙𝑖𝛼𝑖(0)}×𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}^{𝑞𝑖1/𝑞𝑖}. (43) Here, we denote 𝜇_𝑖=𝑛𝛿₂−𝛼_𝑖(0) for short. It follows from condition (19) that 𝜇_𝑖>0. To continue calculations for (43), we consider the two cases 0<𝑞_𝑖≤1 and 1<𝑞_𝑖<∞.
If 0<𝑞_𝑖≤1, since, for 𝑎₁, 𝑎₂,…≥0, ^∞_ℎ=1𝑎_ℎ^𝑞𝑖≤^∞_ℎ=1𝑎^𝑞𝑖_ℎ, (44) then 𝐼_1𝑖(𝐿)≤𝐶2^{−𝐿𝜆𝑖}𝑏_𝑖∗^𝐿_𝑘=−∞^𝑘−2_{𝑙𝑖=−∞}𝑘−𝑙_𝑖^𝑞𝑖2^{(𝑙𝑖−𝑘)𝜇𝑖𝑞𝑖}2^{𝑙𝑖𝛼𝑖(0)𝑞𝑖}×𝑓_𝑙𝑖^𝑞𝑖_{𝐿𝑝𝑖(⋅)(ℝ𝑛)}^1/𝑞𝑖=𝐶2^{−𝐿𝜆𝑖}𝑏_𝑖∗^𝐿−2_{𝑙𝑖=−∞}2^{𝑙𝑖𝛼𝑖(0)𝑞𝑖}𝑓_𝑙𝑖^𝑞𝑖_{𝐿𝑝𝑖(⋅)(ℝ𝑛)}×^𝐿_{𝑘=𝑙𝑖+2}𝑘−𝑙_𝑖^𝑞𝑖2^{(𝑙𝑖−𝑘)𝜇𝑖𝑞𝑖1/𝑞𝑖}≤𝐶𝑏_𝑖∗2^{−𝐿𝜆𝑖𝐿−2}_{𝑙𝑖=−∞}2^{𝑙𝑖𝛼𝑖(0)𝑞𝑖}𝑓_𝑙𝑖^𝑞𝑖_{𝐿𝑝𝑖(⋅)(ℝ𝑛)}^1/𝑞𝑖≤𝐶𝑏_𝑖∗𝑓_{𝑙𝑖𝑀̇𝐾𝛼𝑖𝑖𝑞𝑖𝑖(⋅),𝜆,𝑝(⋅)(ℝ𝑛)}. (45) If 1<𝑞_𝑖<∞, by the Hölder inequality, we obtain 𝐼_1𝑖(𝐿)=2^{−𝐿𝜆𝑖}𝑏_𝑖∗×^𝐿_𝑘=−∞^𝑘−2_{𝑙𝑖=−∞}𝑘−𝑙_𝑖2^{(𝑙𝑖−𝑘)𝜇𝑖}2^{𝑙𝑖𝛼𝑖(0)}×𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}^{𝑞𝑖1/𝑞𝑖}≤𝐶2^{−𝐿𝜆𝑖}𝑏_𝑖∗×^𝐿_𝑘=−∞^𝑘−2_{𝑙𝑖=−∞}2^{(𝑙𝑖−𝑘)𝜇𝑖(𝑞𝑖/2)}2^{𝑙𝑖𝛼𝑖(0)𝑞𝑖}𝑓_𝑙𝑖^𝑞𝑖_{𝐿𝑝𝑖(⋅)(ℝ𝑛)}×^𝑘−2_{𝑙𝑖=−∞}𝑘−𝑙_𝑖^𝑞′𝑖2^{(𝑙𝑖−𝑘)𝜇𝑖(𝑞′𝑖/2)𝑞𝑖/𝑞′𝑖1/𝑞𝑖}≤𝐶2^{−𝐿𝜆𝑖}𝑏_𝑖∗×^𝐿_𝑘=−∞^𝑘−2_{𝑙𝑖=−∞}2^{(𝑙𝑖−𝑘)𝜇𝑖(𝑞𝑖/2)}2^{𝑙𝑖𝛼𝑖(0)𝑞𝑖}𝑓_𝑙𝑖^𝑞𝑖_{𝐿𝑝𝑖(⋅)(ℝ𝑛)}^1/𝑞𝑖=𝐶2^{−𝐿𝜆𝑖}𝑏_𝑖∗^𝐿−2_{𝑙𝑖=−∞}2^{𝑙𝑖𝛼𝑖(0)𝑞𝑖}×𝑓_𝑙𝑖^𝑞𝑖_{𝐿𝑝𝑖(⋅)(ℝ𝑛)}^𝐿_{𝑘=𝑙𝑖+2}2^{(𝑙𝑖−𝑘)𝜇𝑖(𝑞𝑖/2)1/𝑞𝑖}≤𝐶𝑏_𝑖∗2^{−𝐿𝜆𝑖𝐿−2}_{𝑙𝑖=−∞}2^{𝑙𝑖𝛼𝑖(0)𝑞𝑖}𝑓_𝑙𝑖^𝑞𝑖_{𝐿𝑝𝑖(⋅)(ℝ𝑛)}^1/𝑞𝑖≤𝐶𝑏_𝑖∗𝑓_{𝑖𝑀̇𝐾}^𝛼_{𝑖𝑖𝑞𝑖𝑖}^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}. (46) Thus, for any 0<𝑞_𝑖<∞, 𝐼₁≤𝐶sup_{𝐿≤0,𝐿∈ℤ}𝐼₁₁(𝐿)𝐼₁₂(𝐿)≤𝐶𝑏_1∗𝑏_2∗𝑓_1𝑀̇𝐾^𝛼_11𝑞11^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}𝑓_2𝑀̇𝐾^𝛼_22𝑞22^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}. (47)
Step 2. To estimate 𝐼₂, for 𝑥∈𝐷_𝑘, 𝑦_𝑖∈𝐷_𝑙𝑖, 𝑖=1,2 and 𝑙₁≤𝑘−2, 𝑘−1≤𝑙₂≤𝑘+1, we have 𝑥−𝑦₂≥𝑥−𝑦₁≥|𝑥|−𝑦₁>2^𝑘−2. (48) Since 𝑙₁≤𝑘−2, it follows from inequality (31) that ^𝑘−2_𝑙1=−∞^𝑘+1_{𝑙2=𝑘−1}𝑏₁,𝑏₂,𝑇𝑓_𝑙1,𝑓_𝑙2𝜒_{𝑘𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶2^{−2𝑘𝑛𝑘−2}_{𝑙1=−∞ℝ𝑛}𝑏₁(⋅)−𝑏₁𝑦₁𝑓_𝑙1(𝑦₁)𝑑𝑦_{1𝐿𝑝1(⋅)(ℝ𝑛)}×^𝑘+1_{𝑙2=𝑘−1ℝ𝑛}[𝑏₂(⋅)−𝑏₂(𝑦₂)]𝑓_𝑙2(𝑦₂)𝑑𝑦₂𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶^𝑘−2_𝑙1=−∞𝑘−𝑙₁2^{(𝑙1−𝑘)𝑛𝛿2}𝑏_1∗𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}×2^{−𝑘𝑛𝑘+1}_{𝑙2=𝑘−1ℝ𝑛}[𝑏₂(⋅)−𝑏₂(𝑦₂)]𝑓_𝑙2(𝑦₂)𝑑𝑦₂𝜒_{𝑘𝐿𝑝2(⋅)(ℝ𝑛)}. (49) For 𝑘−1≤𝑙₂≤𝑘+1, combining the above term and inequalities (32) and (33), we have 𝐼₂≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆𝑏_1∗𝑏_2∗×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘−2}_𝑙1=−∞^𝑘+1_{𝑙2=𝑘−1}𝑘−𝑙₁2^{(𝑙1−𝑘)𝑛𝛿2}×𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}2^{(𝑙2−𝑘)𝑛}𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1/𝑞≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆1𝑏_1∗×^𝐿_𝑘=−∞^𝑘−2_𝑙1=−∞𝑘−𝑙₁2^{(𝑙1−𝑘)𝜇1}2^{𝑙1𝛼1(0)}×𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}^𝑞11/𝑞1×2^−𝐿𝜆2𝑏_2∗^𝐿_𝑘=−∞^𝑘+1_{𝑙2=𝑘−1}2^𝑘𝛼2(0)2^{(𝑙2−𝑘)𝑛}×𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞21/𝑞2=∶𝐶sup_{𝐿≤0,𝐿∈ℤ}𝐼₂₁(𝐿)𝐼₂₂(𝐿). (50) Here, we used 2^−𝑛𝛿2<1 and 2^{(𝑙2−𝑘)𝑛(1−𝛿1)}<2^{(𝑙2−𝑘)𝑛} in the first inequality.
Obviously, 𝐼₂₁(𝐿)=𝐼₁₁(𝐿)≤𝐶𝑏_1∗𝑓_1𝑀̇𝐾^𝛼_11𝑞11^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}. (51) Therefore, we only need to estimate 𝐼₂₂(𝐿): 𝐼₂₂(𝐿)=2^−𝐿𝜆2𝑏_2∗^𝐿_𝑘=−∞2^{𝑘𝛼2(0)𝑞2}×^𝑘+1_{𝑙2=𝑘−1}2^{(𝑙2−𝑘)𝑛}𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞21/𝑞2≤𝐶𝑏_2∗2^{−𝐿𝜆2𝐿}_𝑘=−∞2^{𝑘𝛼2(0)𝑞2}𝑓₂𝜒_𝑘^𝑞2_{𝐿𝑝2(⋅)(ℝ𝑛)}^1/𝑞2≤𝐶𝑏_2∗𝑓_2𝑀̇𝐾^𝛼_22𝑞22^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}. (52)
Step 3. To estimate 𝐼₃, for 𝑥∈𝐷_𝑘, 𝑦_𝑖∈𝐷_𝑙𝑖, 𝑖=1,2, and 𝑙₁≤𝑘−2,𝑙₂≥𝑘+2, then we have 𝑥−𝑦₁≥|𝑥|−𝑦₁>2^𝑘−2,𝑥−𝑦₂≥𝑦₂−|𝑥|>2^𝑙2−2. (53) Thus, for 𝑥∈𝐶_𝑘, we get 𝑏₁,𝑏₂,𝑇𝑓_𝑙1,𝑓_𝑙2(𝑥)≤𝐶2^−𝑘𝑛_ℝ𝑛𝑏₁(𝑥)−𝑏₁𝑦₁𝑓_𝑙1𝑦₁𝑑𝑦₁2^−𝑙2𝑛×_ℝ𝑛𝑏₂(𝑥)−𝑏₂𝑦₂𝑓_𝑙2𝑦₂𝑑𝑦₂. (54) From inequalities (31) and (32), we obtain ^𝑘−2_𝑙1^∞_{=−∞𝑙2=𝑘+2}[𝑏₁,𝑏₂,𝑇](𝑓_𝑙1,𝑓_𝑙2)𝜒_{𝑘𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶^𝑘−2_𝑙1=−∞𝑘−𝑙₁2^{(𝑙1−𝑘)𝑛𝛿2}𝑏_1∗𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}×^∞_𝑙2=𝑘+2𝑙₂−𝑘2^{(𝑘−𝑙2)𝑛𝛿1}𝑏_2∗𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}. (55) Consequently, it follows that 𝐼₃≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆𝑏_1∗𝑏_2∗×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘−2}_𝑙1=−∞𝑘−𝑙₁2^{(𝑙1−𝑘)𝑛𝛿2}𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}^𝑞×^∞_𝑙2=𝑘+2𝑙₂−𝑘2^{(𝑘−𝑙2)𝑛𝛿1}𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1/𝑞≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆1𝑏_1∗×^𝐿_𝑘=−∞^𝑘−2_𝑙1=−∞𝑘−𝑙₁2^{(𝑙1−𝑘)𝜇1}2^{𝑙1𝛼1(0)}×𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}^𝑞11/𝑞1×2^−𝐿𝜆2𝑏_2∗×^𝐿_𝑘=−∞^∞_𝑙2=𝑘+2𝑙₂−𝑘×2^{(𝑘−𝑙2)(𝑛𝛿1+𝛼2(0))}2^{𝑙2𝛼2(0)}𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞21/𝑞2=∶𝐶sup_{𝐿≤0,𝐿∈ℤ}𝐼₃₁(𝐿)𝐼₃₂(𝐿). (56) Note that 𝐼₃₁(𝐿)=𝐼₂₁(𝐿)≤𝐶𝑏_1∗𝑓_1𝑀̇𝐾^𝛼_11𝑞11^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}, (57) so we only compute 𝐼₃₂(𝐿). From inequality (19) and 𝑛𝛿₁+𝛼₂(0)−𝜆₂>0, we obtain 𝐼₃₂(𝐿)=2^−𝐿𝜆2𝑏_2∗^𝐿_𝑘=−∞^∞_𝑙2=𝑘+2𝑙₂−𝑘2^{(𝑘−𝑙2)(𝑛𝛿1+𝛼2(0))}2^{𝑙2𝛼2(0)}×𝑓₂𝜒_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞21/𝑞2≤𝐶2^−𝐿𝜆2𝑏_2∗×^𝐿_𝑘=−∞^∞_𝑙2=𝑘+2𝑙₂−𝑘2^{(𝑘−𝑙2)(𝑛𝛿1+𝛼2(0))}2^𝑙2𝜆2×2^{−𝑙2𝜆2𝑙2}_𝑚=−∞2^{𝑚𝛼2(0)𝑞2}×𝑓₂𝜒_𝑚^𝑞2_{𝐿𝑝2(⋅)(ℝ𝑛)}^{1/𝑞2𝑞21/𝑞2}≤𝐶2^−𝐿𝜆2𝑏_2∗^𝐿_𝑘=−∞2^{𝑘𝜆2𝑞2}×^∞_𝑙2=𝑘+2𝑙₂−𝑘2^{(𝑘−𝑙2)(𝑛𝛿1+𝛼2(0)−𝜆2)}×𝑓_2𝑀̇𝐾^𝛼_22𝑞22^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}^𝑞21/𝑞2≤𝐶𝑏_2∗2^{−𝐿𝜆2𝐿}_𝑘=−∞2^{𝑘𝜆2𝑞21/𝑞2}𝑓_2𝑀̇𝐾^𝛼_22𝑞22^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}≤𝐶𝑏_2∗𝑓_2𝑀̇𝐾^𝛼_22𝑞22^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}. (58) Therefore, 𝐼₃≤𝐶sup_{𝐿≤0,𝐿∈ℤ}𝐼₃₁(𝐿)𝐼₃₂(𝐿)≤𝐶𝑏_1∗𝑏_2∗𝑓_1𝑀̇𝐾^𝛼_11𝑞11^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}𝑓_2𝑀̇𝐾^𝛼_22𝑞22^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}. (59)
Step 4. It turns to estimate the term 𝐼₅. Applying the Hölder inequality and Lemma 13, we have 𝐼₅≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘+1}_{𝑙1=𝑘−1}^𝑘+1_{𝑙2=𝑘−1}𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_{𝑙2𝐿𝑝(⋅)(ℝ𝑛)}^𝑞1/𝑞≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^{−𝐿𝜆𝐿}_𝑘=−∞2^{𝑘𝛼(0)𝑞}×^𝑘+1_{𝑙1=𝑘−1}^𝑘+1_{𝑙2=𝑘−1}𝑏_1∗𝑏_2∗×𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1/𝑞≤𝐶²_𝑖=1𝑏_𝑖∗sup_{𝐿≤0,𝐿∈ℤ}2^{−𝐿𝜆𝑖}×^𝐿_𝑘=−∞2^{𝑘𝛼𝑖(0)𝑞𝑖𝑘+1}_{𝑙𝑖=𝑘−1}𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}^{𝑞𝑖1/𝑞𝑖}≤𝐶²_𝑖=1𝑏_𝑖∗sup_{𝐿≤0,𝐿∈ℤ}2^{−𝐿𝜆𝑖}×^𝐿_𝑘=−∞2^{𝑘𝛼𝑖(0)𝑞𝑖}𝑓_𝑖𝜒_𝑘^𝑞𝑖_{𝐿𝑝𝑖(⋅)(ℝ𝑛)}^1/𝑞𝑖≤𝐶𝑏_1∗𝑏_2∗𝑓_1𝑀̇𝐾^𝛼_11𝑞11^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}×𝑓_2𝑀̇𝐾^𝛼_22𝑞22^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}. (60)
Step 5. Now it goes to the estimate of 𝐼₆.
It is clear that, for 𝑥∈𝐷_𝑘 and 𝑘−1≤𝑙₁≤𝑘+1 and 𝑙₂≥𝑘+2, 𝑏₁,𝑏₂,𝑇𝑓₁,𝑓₂(𝑥)≤𝐶2^−𝑘𝑛_ℝ𝑛𝑏₁(𝑥)−𝑏₁𝑦₁𝑓₁𝑦₁𝑑𝑦₁2^−𝑙2𝑛×_ℝ𝑛𝑏₂(𝑥)−𝑏₂𝑦₂𝑓₂𝑦₂𝑑𝑦₂. (61) Therefore, ^𝑘+1_𝑙1^∞_{=𝑘−1𝑙2=𝑘+2}𝑏₁,𝑏₂,𝑇𝑓_𝑙1,𝑓_𝑙2𝜒_{𝑘𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶𝑏_1∗𝑏_2∗×^𝑘+1_𝑙1^∞_{=𝑘−1𝑙2=𝑘+2}2^{(𝑙1−𝑘)𝑛}𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}×𝑙₂−𝑘2^{(𝑘−𝑙2)𝑛𝛿1}𝑏_2∗𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)},𝐼₆≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘+1}_𝑙1^∞_{=𝑘−1𝑙2=𝑘+2}𝑏₁,𝑏₂,𝑇×(𝑓_𝑙1,𝑓_𝑙2)_{𝐿𝑝(⋅)(ℝ𝑛)}^𝑞1/𝑞≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆𝑏_1∗𝑏_2∗×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞𝑘+1}_𝑙1^∞_{=𝑘−1𝑙2=𝑘+2}2^{(𝑙1−𝑘)𝑛}𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}×𝑙₂−𝑘2^{(𝑘−𝑙2)𝑛𝛿1}𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞1/𝑞≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆1𝑏_1∗×^𝐿_𝑘=−∞^𝑘+1_{𝑙1=𝑘−1}2^𝑘𝛼1(0)2^{(𝑙1−𝑘)𝑛}𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}^𝑞11/𝑞1×2^−𝐿𝜆2𝑏_2∗^𝐿_𝑘=−∞^∞_𝑙2=𝑘+2𝑙₂−𝑘2^{(𝑘−𝑙2)(𝑛𝛿1+𝛼2(0))}×2^{𝑙2𝛼2(0)}𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}^𝑞21/𝑞2=∶𝐶sup_{𝐿≤0,𝐿∈ℤ}𝐼₆₁(𝐿)𝐼₆₂(𝐿). (62) Here, the estimate of 𝐼₆₁(𝐿) is similar to that of 𝐼₂₂(𝐿) and 𝐼₆₂(𝐿)=𝐼₃₂(𝐿).
Step 6. Finally, we will finish the estimation of the last term 𝐼₉. Note that 𝑙₂≥𝑘+2 and |𝑥−𝑦_𝑖|>2^𝑙𝑖−2 for 𝑥∈𝐷_𝑘, 𝑦_𝑖∈𝐷_𝑙𝑖, 𝑖=1,2: we get 𝑏₁,𝑏₂,𝑇𝑓_𝑙1,𝑓_𝑙2(𝑥)≤𝐶2^−𝑙1𝑛_ℝ𝑛𝑏₁(𝑥)−𝑏₁𝑦₁𝑓_𝑙1𝑦₁𝑑𝑦₁⋅2^−𝑙2𝑛×_ℝ𝑛𝑏₂(𝑥)−𝑏₂𝑦₂𝑓_𝑙2𝑦₂𝑑𝑦₂. (63) Applying the Hölder inequality to the last integral, we obtain ^∞_𝑙1^∞_{=𝑘+2𝑙2=𝑘+2}[𝑏₁,𝑏₂,𝑇](𝑓_𝑙1,𝑓_𝑙2)𝜒_{𝑘𝐿𝑝(⋅)(ℝ𝑛)}≤𝐶𝑏_1∗𝑏_2∗×^∞_𝑙1^∞_{=𝑘+2𝑙2=𝑘+2}𝑙₁−𝑘2^{(𝑘−𝑙1)𝑛𝛿1}𝑓_{𝑙1𝐿𝑝1(⋅)(ℝ𝑛)}×𝑙₂−𝑘2^{(𝑘−𝑙2)𝑛𝛿1}𝑓_{𝑙2𝐿𝑝2(⋅)(ℝ𝑛)}≤𝐶²_𝑖=1𝑏_𝑖∗^∞_{𝑙𝑖=𝑘+2}𝑙_𝑖−𝑘2^{(𝑘−𝑙𝑖)𝑛𝛿1}𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}. (64) Thus, 𝐼₉≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞∞}_𝑙1^∞_{=𝑘+2𝑙2=𝑘+2}𝑏₁,𝑏₂×𝑓,𝑇_𝑙1,𝑓_{𝑙2𝐿𝑝(⋅)(ℝ𝑛)}^𝑞1/𝑞≤𝐶sup_{𝐿≤0,𝐿∈ℤ}2^−𝐿𝜆×^𝐿_𝑘=−∞2^{𝑘𝛼(0)𝑞2}_𝑖=1𝑏_𝑖∗^∞_{𝑙𝑖=𝑘+2}𝑙_𝑖−𝑘×2^{(𝑘−𝑙𝑖)𝑛𝛿1}𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}^𝑞1/𝑞≤𝐶sup²_{𝐿≤0,𝐿∈ℤ𝑖=1}2^{−𝐿𝜆𝑖}𝑏_𝑖∗×^𝐿_𝑘=−∞^∞_{𝑙𝑖=𝑘+2}𝑙_𝑖−𝑘2^{(𝑘−𝑙𝑖)(𝑛𝛿1+𝛼𝑖(0))}×2^{𝑙𝑖𝛼𝑖(0)}𝑓_{𝑙𝑖𝐿𝑝𝑖(⋅)(ℝ𝑛)}^{𝑞𝑖1/𝑞𝑖}=∶𝐶sup_{𝐿≤0,𝐿∈ℤ}𝐼₉₁(𝐿)𝐼₉₂(𝐿)≤𝐶𝑏_1∗𝑏_2∗𝑓_1𝑀̇𝐾^𝛼_11𝑞11^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}×𝑓_2𝑀̇𝐾^𝛼_22𝑞22^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}. (65) Here, the estimate of 𝐼_9𝑖(𝐿), 𝑖=1,2, is similar to that of 𝐼₃₂(𝐿).
Combing all estimates for 𝐼_𝑖, 𝑖=1,2,…,9 together, we get 𝐸≤𝐶𝑏_1∗𝑏_2∗𝑓_1𝑀̇𝐾^𝛼_11𝑞11^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}𝑓_2𝑀̇𝐾^𝛼_22𝑞22^(⋅),𝜆_{,𝑝(⋅)(ℝ𝑛)}. (66)
This finishes the proof of Theorem 6.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referee for his suggestion. The first author was supported by the National Natural Science Foundation of China (Grant no. 11021043) and the Fundamental Research Funds for the Central Universities. The third author was supported by the National Natural Science Foundation of China (Grants nos. 11071064 and 11361020) and the Natural Science Foundation of Hainan Province (no. 113004).