Abstract

In this paper, we obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of Herz spaces and Herz-Morrey spaces with variable exponents.

1. Introduction

Denote by the set of all complex-valued locally integrable functions on The Hardy operator was first considered in [1] as follows: for In 1976, Faris [2] generalized it to the dimensional Euclidean space as where and is the volume of the unit ball in . In [3], Christ and Grafakos proved that the Hardy operator is bounded on .

For , the -linear Hardy operator was defined in [4] as for in The 2-linear operator will be referred to as bilinear operator.

A function belongs to (bounded mean oscillation), if and where the supremum is taken over all balls in , is the mean of on , and what follows is the Lebesgue measure of measurable set in

Let for Then the commutator generated by the -linear Hardy operator and is defined by where When , the operator is

Since the fundamental paper [5] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent have attracted a great attention. Many constant exponent function spaces are generalized to variable exponent setting, such as Bessel potential spaces with a variable exponent [6], Besov and Trieble-Lizorkin spaces with variable exponents [7–12], Hardy spaces with variable exponents [13], Morrey spaces with variable exponents [14], Herz spaces with variable exponents [15], and Herz-Morrey spaces with variable exponents [16, 17]. These spaces have many applications, such as in the electrorheological fluid [18]; image restoration [19–23]; the Black-Scholes equation [24]; ordinary and partial differential equations [25–27] and references therein.

The boundedness of the Hardy operator was considered in many function spaces, such as in variable Lebesgue spaces [28, 29], variable exponent Sobolev spaces [30]. The boundeness of commutators of the Hardy operator was obtained in -central BMO spaces [31], Herz spaces with variable exponent [32]. In [33], Wu proved the boundedness of multilinear commutators of fractional Hardy operators on Herz-Morrey spaces . In [34], Wu considered the boundedness for fractional Hardy type operator on Herz-Morrey spaces with variable exponent but fixed and Wu and Zhang obtained the boundedness of multilinear Hardy type operators on the product of Herz-Morrey spaces with variable exponent and the boundedness of fractional Hardy type operators on Herz-Morrey spaces with variable exponent in [35, 36]. Wu and Zhang considered the boundedness of commutators of the fractional Hardy operators on Herz-Morrey spaces with variable exponent in [17]. Shu, Wang and Meng obtained the boundedness of commutators of Hardy type operators on Herz spaces with variable exponents and in [37]. Wu and Zhao proved the boundedness for variable fractional Hardy type operator on variable exponent Herz-Morrey spaces in [38]. In [39], Xu and Yang introduced the Herz-Morrey-Hardy spaces with variable exponents and established their characterization in terms of atom. Moreover, applying the characterization, they obtained the boundedness of some singular integral operators on these spaces.

Motivated by the mentioned works, in this paper, we will consider the boundedness of -linear commutators generated by the -linear Hardy operator and BMO functions on Herz spaces and Herz-Morrey spaces with variable exponents.

2. Main Results

To state our results, let us first recall some definitions and notations. Let be a positive measurable subset of , given a measurable function The Lebesgue space with variable exponent is defined by The Lebesgue space becomes a Banach function space equipped with the norm The space is the collection of all functions such that for each compact subset Here and what follows, denotes the characteristic function of a measurable set Let ; we denote , The set consists of all such that and ; consists of all such that and . If , then the space is similarly defined as above. means that the conjugate exponent of that means .

The standard Hardy-Littlewood maximal operator is defined for function by where is a ball. We denote by the set of such that is bounded on The common sufficient conditions for variable exponent to be in are the following well known log-Hölder continuity, which introduced in [40–42].

Definition 1. Let be a real-valued measurable function on
(i) If there exists a constant such that then the function is called locally log-Hölder continuous.
(ii) If there exists a constant such that then the function is called log-Hölder continuous at the origin and denoted by ;
(iii) If there exist and a positive constant such that then the function is called log-Hölder continuous at infinity and denoted by .
(iv) If are both locally log-Hölder continuous and log-Hölder continuous at the infinity, then the function is called global log-Hölder continuous and denoted by

To state the definitions of Herz space and Herz-Morrey space with variable exponents, we use the following notations. For each we denote

Definition 2. Let , , and with
(i) The homogeneous Herz space is defined by where (ii) The inhomogeneous Herz space is defined by where

Remark 3. Obviously, if , then . If both and are constants, then is the classical Herz spaces in [43].

To generalize the above spaces to variable exponent , we need the notation of the variable mixed sequence space , which is defined as follows. Given a sequence of functions , define the modular where If or , the above can be written as The norm is

Now, the space is the collection of all functions such that and the space is the collection of all functions in such that

In [44], Drihem and Seghiri proved the following result.

Lemma 4 (see [44, Proposition 1]). Let If and are -Hölder continuous at infinity, then Additionally, if and have a decay at the origin, then

Definition 5. Let , , and with .
(i) The homogeneous Herz-Morrey space is defined by where (ii) The inhomogeneous Herz space is defined by where

Remark 6. If is constant, then was defined in [45]. If , then . If both and are constants and , then is the classical Herz space in [43].

Definition 7. Let Let be a bounded real-valued measurable function on The homogeneous Herz-Morrey space and nonhomogeneous Herz-Morrey space are defined, respectively, by and where and

By the previous definition, we obtain the following proposition.

Proposition 8. Let , , and
(i) If , , then for any , (ii) If , , then

Proof. Obviously, When , from Lemma 4 we know that When , from Lemma 4 again we also obtain that Thus we proved (i). The proof of (ii) is similar.

Lemma 9 (see [46, Proposition 2]). Let , , and . If is log-Hölder continuous both at the origin and at infinity, then

Lemma 10 (see [45, Lemma 1 and ]). Suppose , then there exist constants , , and such that for all balls in and all measurable subsets ,

The following is our main results. Although the results are also hold for all multilinear cases, for brevity, we only consider the bilinear case.

Theorem 11. Let be a bilinear Hardy operator; , satisfying , , , , , , , , , are the constants in Lemma 10 for exponents If for , with , then is bounded from to , where

Corollary 12. Let be a bilinear Hardy operator; , satisfying , , , , , , are the constants in Lemma 10 for exponents . If for with , then is bounded from to , where

Corollary 13. Let be a bilinear Hardy operator; , satisfying , , , , , , , are the constants in Lemma 10 for exponents . If for with , then is bounded from to , where

Corollary 14. Let be a bilinear Hardy operator; , satisfying , , , , , , , , are the constants in Lemma 10 for exponents . If for with , then is bounded from to , where

3. Proof of Theorem 11

Since Corollaries 12–14 are special case of Theorem 11, we only need to prove Theorem 11.

To proceed, we need the following lemmas.

Lemma 15 (see [45, Lemma 2]). If , then there exists a constant such that, for all balls in ,

Lemma 16 (see [5, Theorem 2.1]). Let Then for every and every , where .

Lemma 17 (see [47, Theorem 2.3]). Let such that . Then there exists a constant depending only on such that holds for every and .

Lemma 18 (see [32, Theorem 3]). Let , and , then there exists a positive constant such that, for ,

Indeed, if , then for all balls and , with , we have From the definition of BMO, it is easy to know that

Lemma 19 (see [48, Proposition 1.2]). Let Then we have for nonnegative sequences Here, when , it is understood that (46) stands for

Proof of Theorem 11. Let , Let express the average of on the ball for , and . If , then Since , and . Using Hölder’s inequality, we have By Lemma 18, Hölder’s inequality (Lemma 17), and Lemmas 10 and 15, we have Then we consider the term . By Lemma 18 and Hölder’s inequality (Lemma 16), we have By Lemma 18 and Hölder’s inequality (Lemma 16), we have By Hölder’s inequality (Lemma 16) and Lemmas 10 and 15, we have By Hölder’s inequality (Lemma 16) and Lemmas 10 and 15, we get Therefore, we have By Proposition 8, we obtain that where Since , , and , by (56) and Hölder’s inequality, we have Since , by Lemma 19 we obtain that where we wrote for some
Since , again by Lemma 19 we obtain that Thus, we obtain Since , , and , by (56) and Hölder’s inequality, we have Here Since , by Lemma 19 we obtain that where we wrote for some
Since , by Lemma 19 we obtain that Therefore, we get Now we turn to estimate Since , , by (56) and Hölder’s inequality, we have where To go on, we need further preparation.
If , by Proposition 8, we have Similarly, if , we have If , we have Similarly, if , we have Since and by (70) and (72), we obtain that Similarly, since , by (71) and (73), we obtain that Thus, we obtain Combining all the estimates for together, we obtain Similarly, we have Therefore, the proof of Theorem 11 is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The second author is partially supported by the Hainan Province Natural Science Foundation of China (2018CXTD338) and National Natural Science Foundation of China (Grant no. 11761026).