Parametric Marcinkiewicz Integral and Its Higher-Order Commutators on Variable Exponents Morrey-Herz Spaces

Omer, Omer Abdalrhman; Saibi, Khedoudj; Abidin, Muhammad Zainul; Osman, Mawia

doi:https://doi.org/10.1155/2022/7209977

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 7209977 | https://doi.org/10.1155/2022/7209977

Parametric Marcinkiewicz Integral and Its Higher-Order Commutators on Variable Exponents Morrey-Herz Spaces

Omer Abdalrhman Omer,¹Khedoudj Saibi,¹Muhammad Zainul Abidin,^1,2and Mawia Osman¹

Academic Editor: Andrea Scapellato

Received07 Jun 2022

Accepted18 Aug 2022

Published14 Sept 2022

Abstract

In this article, we prove the boundedness of the parametric Marcinkiewicz integral and its higher-order commutators generated by BMO spaces on the variable Morrey-Herz space. All the results are new even when is a constant.

1. Introduction

Throughout the entirety of this article, we assume that , is the -dimensional Euclidean space, and is the unit sphere in equipped with the normalized Lebesgue measure . The function is assumed to be homogeneous of degree zero on with and where for any . For , the parametric Marcinkiewicz integral of higher dimensions is defined as follows:

Let be a ball with a radius , and a center . A locally integrable function is said to be in the space, if it satisfies where and denotes the Lebesgue measure of the set in . For the -order commutator for the parametric Marcinkiewicz integral is defined as follows:

If in (2), then the operator is equivalent to the classical Marcinkiewicz function , which was initially introduced by Stein [1] in 1958. When , Stein [1] demonstrated that is bounded on for . Subsequently, the authors of [2] established the -boundedness of for every when . On the other hand, Calderón [3] proved that the commutator the Hilbert transform generated by , defined by , is bounded on . Coifman et al. [4] arrived at the conclusion that the commutator, which was generated by the Calderón-Zygmund operator and the , is bounded on for . Since then, the commutators of the Calderón-Zygmund operator have played an essential role in the study of the regularity of solutions to second-order elliptic, parabolic, and ultraparabolic partial differential equations, see for example [5–11]. Moreover, the boundedness of the commutators of various operators generated by a BMO function has been widely studied. Particularly, Torchinsky and Wang [12] studied the weighted -boundedness of , where is the -order commutator of Marcinkiewicz integral. The authors of [13] studied the behaviour of the Hardy-Littlewood maximal operator and the action of commutators in generalized local Morrey spaces and generalized Morrey spaces. For further research works studying the commutators on different function spaces, we refer to [9, 14–21] and references therein.

The parametric Marcinkiewicz integral was originally introduced by Hörmander in [22] where the author established the boundeness of on for under the condition and . Shi and Jiang [23] investigated the weighted -boundedness of and . Since that time, the boundedness of the parametric Marcinkiewicz integral, as well as its related commutator, in several types of function spaces have attracted the attention of many researchers. Deringoz and Hasanov [24] considered the boundedness of the operator on generalized Orlicz-Morrey spaces. On generalized weighted Morrey spaces, Deringoz [25] investigated the boundedness of rough parametric Marcinkiewicz integral and its higher-order commutator . For more applications and recent developments on the research of the parametric Marcinkiewicz function, see [26–31].

In the last decades, the variable Lebesgue spaces have been intensively studied since the pioneering work of [32] by Kovácık and Rákosnık. Additionally, different studies on variable function spaces, such as variable exponents Fourier-Besov-Morrey spaces [33–35], variable exponents Fourier-Besov spaces [36, 37], variable exponent Morrey spaces [38], variable Bessel potential spaces [39, 40], and variable exponent Hardy spaces [41, 42], were developed due to their applications in the modeling of electro-rheological fluids, PDEs with nonstandard growth, and image restoration. Recently, Izuki studied the Herz spaces in [43, 44]. As a generalization, Izuki [45] introduced the variable Morrey-Herz spaces . In fact, the author of [45] found that vector-valued sublinear operators which satisfy a certain size condition are bounded on the variable Morrey-Herz spaces. Furthermore, Almeida and Drihem [46] enhanced the variable case of the Herz spaces and established the boundedness results for a class of sublinear operators. Lu and Zhu [47] generalized Izuki’s result for the . For further information and applications, consult [48–54].

Inspired by the research mentioned above, the main goal of this article is to prove the boundedness of the rough parametric Marcinkiewicz integral and its higher-order commutators on the variable exponents Morrey-Herz spaces.

Henceforth, wherever the symbol appears, it represents a positive constant whose value may vary but is independent of the basic variables. The expression denotes the existence of constant such that , and means that . If no further instructions are provided, the symbol for any space denoted by is represented by . For instance, is abbreviated as .

2. Definitions and Preliminaries

In this section, we review some notations, definitions, and properties related to our work.

A variable exponent is a measurable function . For any variable exponent , we set and . Define the sets by

Let . The variable Lebesgue space consists of all measurable functions on such that where

It is obvious that the variable exponent Lebesgue norm has the following property .

Define the set by where stands for the Hardy-Littlewood maximal function, which is defined as follows:

Definition 1 (see [46]). Let be real function on . (i)If there exists a constant such thatthen the function is said to be a log-Hölder continuous at the origin (or has a log decay at the origin). (ii)If there exist and a constant such thatthen the function is said to be a log-Hölder continuous at the infinity (or has a log decay at the infinity).
If , then the following expression of Hölder’s inequality is valid: See [55].
Here and hereafter, denotes the conjugate exponent of , i.e., . It is well-known that if belongs to , then (see [56]).
For any , let the characteristic function of .

Definition 2 (see [46]). Let and let . The homogeneous variable Herz space is defined as the set of all functions such that for , and the usual modification should be made when .

Definition 3 (see [47]). Let and let . The homogeneous variable Morrey-Herz space is defined as the set of all functions such that for , and the usual modification should be made when .

Lemma 4 (see [44]). Let Then, for any ball in ,

Lemma 5 (see [44]). Let . Then, there are positive constants such that for any ball in and any measurable subset ,

Proposition 6 (see [47]). Let , and let . If the function is log-Hölder continuous function both at origin and at infinity, then the following inequalities hold:

Lemma 7 (see [46]). Let and . If the function is log-Hölder continuous both at origin and infinity, then the following inequality holds: for every and .

Lemma 8 (see [56]). Let , and let such that . Then, for any measurable functions and ,

Lemma 9 (see [57]). Let be a positive integer, and let . Then, there exists a positive such that for all , and The main results of this article are as follows.

Theorem 10. Suppose that and with satisfying (1). Let , and be log-Hölder continuous both at the origin and at infinity, such that where are the constants mentioned in Lemma 5. Then, the operator is bounded on .

Theorem 11. Suppose that and with satisfies (1). Let , be log-Hölder continuous both at the origin and at infinity, such that where are the constants mentioned in Lemma 5. Then, the operator is bounded on .

It is worth noting that if , then the variable Morrey-Herz space dates back to the variable Herz space . Thus, by letting in Theorems 10 and 11, we will get the following results on the variable exponents Herz spaces.

Corollary 12. Suppose that and with satisfies (1). Let , and be log-Hölder continuous both at the origin and at infinity, such that where are the constants mentioned in Lemma 5. Then, the operator is bounded on .

Corollary 13. Suppose that and with satisfies (1). Let and and be log-Hölder continuous both at the origin and at infinity, such that where are the constants mentioned in Lemma 5. Then, the operator is bounded on .

Remark 14. If is a constant function, i.e., , then the results of Corollaries 12 and 13 can be founded in [30].

3. Proofs of Theorems 10 and 11

Proof of Theorem 10. Let . For any , let , then we have

Using the definition of , we have

First, we estimate . By Proposition 6, we obtain

Using the fact that , and the boundedness of on (see [30]), we deduce

For the term , we firstly estimate .

It is clear that if , then . Thus, for , by the mean value theorem, we have

By (31) and the Minkowski’s inequality, it follows that

By Lemma 7, we deduce

It follows from Hölder’s inequality (12) that

Since , we can find a variable exponent such that , then by Lemma 8, it follows that where the last inequality (35) is based on the fact that ; see [18]. From (34) and (35), Lemmas 4 and 5, we can deduce

To estimate , we need to consider two cases: and .

Case 1. , by the fact that , the Hölder’s inequality, and the inequality (36), it follows that

Case 2. , we use the inequality and obtain

For the term , by applying Proposition 6, we can get

For , it is clear that if , then . By (31), (12), and the Minkowski’s inequality, we deduce that

Similar to (35), we conclude

From (42), Lemmas 4 and 5, it follows that

To estimate , we need to consider two cases below: and .

Case 3. , combining the above inequalities and using (38), we can obtain

For , in view of , we get

Now, let us deal with , noting that , it follows that

Case 4. , we have

For , by the fact that and using Hölder inequality, we infer that

For , from the inequality (36) and using the method as for , we obtain

By combining , , , and estimates, we arrive at

By the similar method used in the estimate for , it is not difficult to show that

Thus, we have

The proof for Theorem 10 is finished.

Proof of Theorem 11. Let . For any , let , then we have

Using the definition of , we have

Let us first estimate . From Proposition 6 and the boundedness of on (see [30]), and using the similar methods as that for , it is not difficult to see that

Now, let us turn to the estimates of . We consider

It is clear that if , then . Thus, for , we use the Minkowski’s inequality to get

Using Lemma 7 and inequality (12), it follows that

Applying Hölder’s inequality (12), the inequality (35), and Lemmas 4–8, we obtain

Hence, combining the above estimate and using the same approach as the one used for estimating , we conclude that

Finally, we estimate . It is clear that if , then . By (31), the Minkowski’s inequality, and the inequality (12), we deduce, for ,

From this, Lemmas 4–8 and (38), we deduce

Thus, combining the above estimates and using the same approach as for the estimate, we deduce that

Summing up the estimates of , and , we conclude that

The proof for Theorem 11 is finished.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the Zhejiang Normal University Postdoctoral Research fund under grant (no. ZC304021907)

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