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

Abstract

In this paper, we acquire the boundedness of commutators generated by multilinear Calderón-Zygmund operator and BMO functions on products of weighted Herz-Morrey spaces with variable exponents.

1. Introduction

The space of all Schwartz functions on was denoted by , and the space of all tempered distributions on was denoted by . The space of compactly supported bounded functions denoted by , and the support set of function was denoted by . On the -fold of the Schwartz function space , we also set as an -linear operator originally defined and , and its value belongs to :

We say that is an -linear Calderón-Zygmund operator, if for some , it extends to a bounded multilinear operator from to with , and for , where kernel is a function in away from the diagonal and there exist positive constants satisfies the following:whenever , and for all ,where .

If , setwhere and the supremum is taken over all , and what follows is the Lebesgue measure of measurable set in A function is called bounded mean oscillation if Denote by the set of all bounded mean oscillation functions on

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 , BMO functions and , and suitable functions and ,

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.

2. Notations and Main Result

In this section, we recall some notations and definitions; then, we describe our results. Assume be a measurable function on and take values in , the Lebesgue space with variable exopnent is acquired by

The norm is defined by

On a Banach function space, the Lebesgue space is equipped with the norm The space is defined bywhere and what follows, denotes the characteristic function of a measurable set

Let , we denote

The set consists of all satisfying and ; consists of all satisfying and . can be equally defined as above for . is the conjugate exponent of , defined pointwise by .

Let and be a weight which is a nonnegative measurable function on . Then, the weighted variable exponent Lebesgue space is the set of all complex-valued measurable function such that . The space is a Banach space equipped with the norm

Let . Then, the standard Hardy-Littlewood maximal function of is defined bywhere the supremum is taken over all balls containing in . 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 and meet the following global -Hölder continuous and such that is bounded on , see [6].

Definition 1. Assume be a real-valued measurable function on .(i)We say that satisfies the local log-Hölder continuity condition if there exists a constant such that(ii)We say that satisfies the log-Hölder continuous at the origin if there exists a constant such that Denote by the set of all log-Hölder continuous functions at the origin.(iii)We say that satisfies the log-Hölder continuous at the infinity if there exists and a constant such that Denote by the set of all log-Hölder continuous functions at infinity.(iv)We say that satisfies the global log-Hölder continuous if is both log-Hölder continuous and locally log-Hölder continuous at infinity. We denote by the set of all global log-Hölder continuous functions

Definition 2. Given and a positive measurable function , we say that if there exists a positive constant C for all balls in such that

Remark 3. In [7], Cruz-Uribe et al. obtained that if and , then .

The Muckenhoupt class with constant exponent was firstly proposed by Muckenhoupt in [8]. The variable Muckenhoupt was considered in [7, 9–12].

Lemma 4 (see [7, Theorem 1.5]). If and , then there is a positive constant such that for each ,

Next, we define the weighted Herz-Morrey space with variable exponents, and we use the following concepts. Let , we define

Definition 5. Let , , Let be a bounded real-valued measurable function on The nonhomogeneous weighted Herz-Morrey space and homogeneous weighted Herz-Morrey space are defined, respectively, byand where

Let and be two real numbers. If there exists a constant such that , we denote If and , we denote .

Proposition 6 (see [13, Proposition 1]). Let , , be a weight, , and (i)If then for all where and hereafter, (ii)If , then

Lemma 7 has been proved by Noi and Izuki in [14, 15].

Lemma 7. If and , then there exist constants , and such that for all balls in and all measurable subsets

The following is the main result.

Theorem 8. Let be a bilinear Calderón-Zygmund operator and let and be BMO functions. Given satisfying for . Let be weights, , , . Assume that , , , , , , , , are the constants in Lemma 7 for exponents and weights , . If , then

3. Proof of Theorem 8

Before we prove Theorem 8, we need to introduce some lemmas.

Lemma 9 (see [1, Theorem 2.3]). Let , , such that for . Then, there exists a constant independent of functions and such thatholds for every and .

If , with , then by the Hölder inequality, we have

Lemma 10 (see [16, Corollary 3.11]). Let , , , , and such that , then one has

Lemma 11 (see [17, Theorem 2.6]). If and , then there exists a positive constant such thatfor nonnegative sequences

Lemma 12 (see [10, Theorems 2.23 and 2.24]). Assume that for some , , and every ,where is a pair of nonnegative functions. Given , assume that there exists such that and is bounded on . Then,

Lemma 13. Let , , , and satisfying for , . Let and be BMO functions. Let , , and . If is a bilinear Calderón-Zygmund operator, then

Proof. We assume that and are bounded functions. Let and be bounded functions with compact support. By the same argument in the proof of [1, Theorem 2.2], we haveBy Lemma 12, we haveBy Lemmas 9 and 4, we have☐

Proof of Theorem 8. Assume and are bounded functions with compact support and writeBy Proposition 6, we havewhereSince the estimates of and are essentially analogical, we only need to obtain and bounded in the Herz-Morrey space with variable exponents. It is easy to see thatwhereWe shall use the following estimates. If , then pass Hölder’s inequality, we haveBy Lemmas 7 and 10, Hölder’s inequality, and Definition 2, we acquire thatIf , thenIf , then pass Hölder’s inequality, we haveBy Lemmas 7 and 10, Hölder’s inequality, and Definition 2, we acquire thatBy the interchange of and , we see that the estimates of , , and are similar to , , and , respectively. Thus, we only to estimate , , , , , and .
To estimate , due to , , we infer that for ,Therefore, for , we haveThus, and , we haveThus, according to Hölder’s inequality, we haveSince , , according to Hölder’s inequality, we will havewhereSince , by (41) and Lemma 11, we acquire thatThus, we obtain thatTo estimate , due to for , then, we haveSo, according to Hölder’s inequality, we haveSince , , by Hölder’s inequality, we acquire thatIt is obviously thatNow we estimate . Taking (41), (42), and (44) together, we havewhere we used and , for (41) and (44), respectively. Therefore, we acquire thatTo estimate , since , then, we haveTherefore, , we getSo, according to Hölder’s inequality, we acquire thatSince , according to Hölder’s inequality, we acquire thatIt is obviously thatBy (44), we acquire thatWe consider . By Lemma 11, we havewhere for
We consider . Since , we obtain thatWe consider . Since and , we obtainThus, we haveTo estimate , according to Lemma 13 and Hölder’s inequality, we haveTo estimate , due to and , then we haveThus, and , we obtain thatSo, according to Hölder’s inequality, we acquire thatSince , according to Hölder’s inequality, we haveBy the interchange of and , we acquire that the estimate of is analogical to the estimate of and .
To estimate , since , then. we getTherefore, , we haveSo, according to Hölder’s inequality, we obtain thatSince , , according to Hölder’s inequality, we haveclearly, the estimate is analogical to the estimated for .
Combining all the estimates of , , we obtain thatIn order to continue, we need further preparation. If , since Proposition 6, we obtain thatconclusively, we estimate according to the interchange of and , we see that the estimates of , , and are similar to , , and , respectively. Thus, it was only necessary to estimate and .
To estimate , due to , , , by (48) and Hölder’s inequality, we obtain thatwhereWe consider . By (41), we obtain thatIf , since and , then by the Minkowski inequality and (79), we haveIf , since and , then by (79), we haveWe consider . Since , then by Lemma 11, we havewhere we wrote for
Thus, we getTo estimate , due to , , , by (54) and Hölder’s inequality, we obtain thatIt is obviously thatNow, we estimate . Combing (41), (42), and (44), we havewhere we used and for (41) and (44), respectively.
Thus, we acquire thatTo estimate , since , using (61) and Hölder’s inequality, we obtain thatIt is obviously thatSince , by (44) and Lemma 11, we obtain thatWe consider . By Lemma 11, we obtain thatwhere wrote for
We consider . Since , we haveThus, we getTo estimate , according to Lemma 13 and Hölder’s inequality, we haveTo estimate , due to and , using (72) and Hölder’s inequality, we obtain thatSince the symmetry of and , we can know that the estimate is analogical to the estimated and .
To estimate , due to , using (76) and Hölder’s inequality, we obtain thatclearly, the estimate is analogical to the estimated for .
Combining all the estimates ofr together, , we obtain thatCombining the above estimates for , , and , the proof of Theorem 8 is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgments

The work is partially supported by the National Natural Science Foundation of China (Grant No. 11871184).