Commutators of Multilinear Calderón-Zygmund Operator on Weighted Herz-Morrey Spaces with Variable Exponents

The space of all Schwartz functions on Rn was denoted by SðRnÞ, and the space of all tempered distributions on Rn was denoted by S ′ðRnÞ. The space of compactly supported bounded functions denoted by LC ðRnÞ, and the support set of function f was denoted by supp ð f Þ. On the m-fold of the Schwartz function space SðRnÞ, we also set T as an m -linear operator originally defined and m ≥ 2, and its value belongs to S ′ðRnÞ:


Introduction
The space of all Schwartz functions on ℝ n was denoted by Sðℝ n Þ, and the space of all tempered distributions on ℝ n was denoted by S ′ ðℝ n Þ. The space of compactly supported bounded functions denoted by L ∞ C ðℝ n Þ, and the support set of function f was denoted by supp ðf Þ. On the m-fold of the Schwartz function space Sðℝ n Þ, we also set T as an m -linear operator originally defined and m ≥ 2, and its value belongs to S ′ ðℝ n Þ: We say that T is an m-linear Calderón-Zygmund operator, if for some p 1 , ⋯, p m ∈ ½1, ∞Þ, it extends to a bounded multilinear operator from L p 1 × L p 2 × ⋯× L p m to L p with 1/ p 1 + 1/p 2 + ⋯ + 1/p m = 1/p, and for f 1 where kernel K is a function in ðℝ n Þ m+1 away from the diagonal x = y 1 = y 2 = ⋯ = y m and there exist positive constants ε, A satisfies the following: whenever jx − x ′ j ≤ 1/2 max fjx − y 1 j, jx − y 2 j,⋯,jx − y m jg, and for all 1 ≤ i ≤ m, K x, y 1 ,⋯,y i ,⋯,y m ð Þ j j − K x, y 1 ,⋯,y i ′ ,⋯,y m where jy i − y i ′j ≤ 1/2 max fjx − y 1 j, jx − y 2 j,⋯,jx − y m jg.
where b B = ð1/jBjÞ Ð B bðyÞdy and the supremum is taken over all B ⊂ ℝ n , and what follows |B | is the Lebesgue measure of measurable set B in ℝ n : A function b is called bounded mean oscillation if kbk * < ∞: Denote by BMOðℝ n Þ the set of all bounded mean oscillation functions on ℝ n : Although our method suits any multilinear operator, only the bilinear Calderón-Zygmund operator will be considered here for the sake of simplicity. Specifically, we will discuss the commutator of a bilinear Calderón-Zygmund operator T, BMO functions b 1 and b 2 , and suitable functions f 1 and f 2 , Many analyses of linear commutators have been extended to other fields, such as weighted space, homogeneous space, multiparameter, and multilinear settings. Huang and Xu [1] obtained boundedness of multilinear singular integrals and their commutators from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. Huet al. [2] proved the boundedness of commutators generated by fractional integrals and BMO on generalized Herz spaces with general Muckenhoupt weights. Tang et al. [3] obtained the boundedness of a commutator generated by the multilinear Calderón-Zygmund operator and BMO functions in Herz-Morrey spaces with variable exponents. Chen et al. [4] studied multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. Wang et al. [5] proved the boundedness for a class of multisublinear singular integral operators on the product central Morrey spaces with variable exponents.
Motivated by the mentioned works, we will consider the boundedness of commutators generated by multilinear Calderón-Zygmund operator and BMO functions on products of weighted Herz-Morrey spaces with variable exponents.

Notations and Main Result
In this section, we recall some notations and definitions; then, we describe our results. Assume pð·Þ be a measurable function on ℝ n and take values in ½1, ∞Þ, the Lebesgue space with variable exopnent L pð·Þ ðℝ n Þ is acquired by The norm is defined by On a Banach function space, the Lebesgue space L pð·Þ ðℝ n Þ is equipped with the norm k f k L pð·Þ The space L pð·Þ loc ðℝ n Þ is defined by where and what follows, χ A denotes the characteristic function of a measurable set A ⊂ ℝ n : Let pð·Þ: ℝ n ⟶ ð0,∞Þ, we denote The set P ðℝ n Þ consists of all pð·Þ satisfying p − > 1 and p + < ∞; P 0 ðℝ n Þ consists of all pð·Þ satisfying p − > 0 and p + < ∞. L pð·Þ can be equally defined as above for pð·Þ ∈ P 0 ðℝ n Þ. q′ ð·Þ is the conjugate exponent of pð·Þ, defined pointwise by 1/p ð·Þ + 1/p′ð·Þ = 1.
Let pð·Þ ∈ P ðℝ n Þ and w be a weight which is a nonnegative measurable function on ℝ n . Then, the weighted variable exponent Lebesgue space L pð·Þ ðwÞ is the set of all complexvalued measurable function f such that f w ∈ L pð·Þ . The space L pð·Þ ðwÞ is a Banach space equipped with the norm Let f ∈ L 1 loc ðℝ n Þ. Then, the standard Hardy-Littlewood maximal function of f is defined by where the supremum is taken over all balls containing x in ℝ n . Generally speaking, on weighted variable Lebesgue spaces, the Hardy-Littlewood maximal operator is not bounded. But if it meets certain conditions, it will be established. Namely, let pð·Þ ∈ P ðℝ n Þ and meet the following global log-Hölder continuous and w ∈ A pð·Þ such that M is bounded on L pð·Þ ðwÞ, see [6].
Definition 1. Assume αð·Þ be a real-valued measurable function on ℝ n .
(i) We say that αð·Þ satisfies the local log-Hölder continuity condition if there exists a constant C 1 such that Denote by P log 0 ðℝ n Þ the set of all log-Hölder continuous functions at the origin.
(iii) We say that αð·Þ satisfies the log-Hölder continuous at the infinity if there exists α ∞ ∈ ℝ and a constant C 3 such that Denote by P log ∞ ðℝ n Þ the set of all log-Hölder continuous functions at infinity.
(iv) We say that αð·Þ satisfies the global log-Hölder continuous if αð·Þ is both log-Hölder continuous and locally log-Hölder continuous at infinity. We denote by P log ðℝ n Þ the set of all global log-Hölder continuous functions Definition 2. Given pð·Þ ∈ P ðℝ n Þ and a positive measurable function w, we say that w ∈ A pð·Þ if there exists a positive constant C for all balls B in ℝ n such that Remark 3. In [7], Cruz-Uribe et al. obtained that if pð·Þ ∈ P ðℝ n Þ and w ∈ A pð·Þ , then w −1 ∈ A p ′ ð·Þ .
Lemma 4 (see [7,Theorem 1.5]). If pð·Þ ∈ P log ðℝ n Þ ∩ P ðℝ n Þ and w ∈ A pð·Þ , then there is a positive constant C such that for each f ∈ L pð·Þ ðwÞ, Next, we define the weighted Herz-Morrey space with variable exponents, and we use the following concepts. Let k ∈ ℤ, we define and where Let B and C be two real numbers. If there exists a constant K > 0 such that B ≤ KC, we denote B ≲ C: If B ≲ C and C ≲ B, we denote B ≈ C.

Lemma 7.
If pð·Þ ∈ P log ðℝ n Þ ∩ P ðℝ n Þ and w ∈ A pð·Þ , then there exist constants δ 1 , δ 2 ∈ ð0, 1Þ and C > 0 such that for all balls B in ℝ n and all measurable subsets S ⊂ B, The following is the main result.

Journal of Function Spaces
Proof of Theorem 8. Assume f 1 and f 2 are bounded functions with compact support and write By Proposition 6, we have where Since the estimates of E and F are essentially analogical, we only need to obtain E and G bounded in the Herz-Morrey space with variable exponents. It is easy to see that where We shall use the following estimates. If l ≤ k − 1, then pass Hölder's inequality, we have By Lemmas 7 and 10, Hölder's inequality, and Definition 2, we acquire that 5 Journal of Function Spaces If l = k, then If l ≥ k + 1, then pass Hölder's inequality, we have By Lemmas 7 and 10, Hölder's inequality, and Definition 2, we acquire that By the interchange of f 1 and f 2 , we see that the estimates of E 2 , E 3 , and E 6 are similar to E 4 , E 7 , and E 8 , respectively. Thus, we only to estimate E 1 , E 2 , E 3 , E 5 , E 6 , and E 9 .
Combining all the estimates of E i , i = 1, 2, ⋯, 9, we obtain that In order to continue, we need further preparation. If l < 0, since Proposition 6, we obtain that conclusively, we estimate G: according to the interchange of f 1 and f 2 , we see that the estimates of G 2 , G 3 , and G 6 are similar to G 4 , G 7 , and G 8 , respectively. Thus, it was only necessary to estimate G 1 ,G 2 ,G 3 ,G 5 ,G 6 , and G 9 .