Variable <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 9.63935 12.533" width="9.63935pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-233" d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"/></g></svg>-</span>Central Morrey Space Estimates for the Fractional Hardy Operators and Commutators

Hussain, Amjad; Asim, Muhammad; Jarad, Fahd

doi:https://doi.org/10.1155/2022/5855068

Journal of Mathematics

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

Research Article | Open Access

Volume 2022 | Article ID 5855068 | https://doi.org/10.1155/2022/5855068

Variable -Central Morrey Space Estimates for the Fractional Hardy Operators and Commutators

Amjad Hussain,¹Muhammad Asim,¹and Fahd Jarad^2,3

Academic Editor: Valerii Obukhovskii

Received28 Oct 2021

Accepted01 Apr 2022

Published09 May 2022

Abstract

This paper aims to show that the fractional Hardy operator and its adjoint operator are bounded on central Morrey space with variable exponent. Similar results for their commutators are obtained when the symbol functions belong to -central bounded mean oscillation (-central BMO) space with variable exponent.

1. Introduction

The boundedness of operators on function spaces is one of the core issues in harmonic analysis [1–3]. It is mainly because many problems in the theory of partial differential equations, in their simplified form, are reduced to the boundedness of operators on function spaces. It stimulates the research community to embark on such problems in this field. In this paper, we mainly obtain the boundedness of fractional Hardy operators [4]:on variable exponent central Morrey spaces. In addition, commutators of these operatorswith symbol functions in variable -central BMO spaces are shown bounded on central Morrey spaces with variable exponent. However, before stating our main results, we need to introduce the reader to some basic definitions and preliminary results regarding variable exponent function spaces.

Notably, the function spaces with variable exponents have considerable importance in Harmonic analysis as well. Back in 1931, Orlicz [5] started the theory of variable exponent Lebesgue space. Musielak Orlicz spaces were defined and studied in [6]. The study of Sobolev and Lebesgue spaces with variable exponents in [7–11] further stimulated the subject. In the meantime, central Morrey space, central BMO space, and associated function spaces have attractive applications by exploring estimates for operators along with their commutators [12–20]. Mizuta et al. defined the variable exponent nonhomogeneous -central Morrey space in [21]. The central BMO space first appeared in [22]. Meanwhile, the authors in [23] gave the definition of variable exponent central Morrey and -central BMO space along with some important results regarding the estimation of some operators. Recently, some publications [24–26] discussing the continuity of multilinear integral operators on these function spaces have added substantially to the existing literature on this topic.

The one-dimensional Hardy operator was firstly defined by Hardy in [27] and is considered a classical operator in operator theory. Its mathematical form can be obtained from (1) by taking and . Later on, different authors extended the definition of the one-dimensional Hardy operator to multidimensions in [28, 29]. As stated earlier, the fractional Hardy operator and its adjoint operator were introduced first in [4]. Following these publications, a flux of new results emerged discussing the boundedness of Hardy-type operators and their commutators on different function spaces [30–35]. The commutator operator also enjoyed a lot of attention from different zones of the globe [4,20,36–40]. However, the continuity of Hardy-type operators and their commutators on variable exponent function spaces took less attention by the research community worldwide [41–44]. The same is the case with central Morrey space with variable exponent. The present article aims to fill this gap by proving the boundedness of the fractional Hardy operator and its adjoint operator in this space. In addition, this article also includes new results discussing the boundedness of commutators generated by (or ) and the -central BMO function on the variable central Morrey space.

Let us describe the framework of this paper. In Section 2, we will remind some lemmas and propositions related to variable exponent function spaces. In Section 3 of this article, we will demonstrate the boundedness for Hardy operators and their commutators on central Morrey space with variable exponent. In Section 4, we shall investigate the similar estimates for the adjoint fractional Hardy operator and its commutators.

2. Function Spaces with Variable Exponents

In this section, we are going to introduce some notations and definitions related to the variable exponent function spaces. Throughout this article, we denote by and the Lebesgue measure and characteristic function of a measurable set , respectively. Also, with and for . The notation implies that there exist two positive constants and such that . Furthermore, represents an open set and is a measurable function, and denotes the conjugate exponent of which satisfies

The set consists of all and such that

The space is a set of all measurable function on the open set , in such a way that, for positive ,which becomes a Banach function space when equipped with the Luxemburg-norm

Local version of variable exponent Lebesgue space is denoted by and is defined by

We use to denote a set containing satisfying the condition that the Hardy-Littlewood maximal operator : where is bounded on .

Proposition 1 (see [8,45]). Let E denote an open set and fulfill the following inequalities:then , where is a positive constant independent of and .

Lemma 1 (see [7]) (generalized Hölder inequality). Let .(a)If and , then we have where .(b)If , and , then we have where .

Lemma 2 (see [46]). If , then there exist constants and a positive constant such that for all balls in and all measurable subsets ,

Remark 1. Let and meet conditions (9) and (10) in Proposition 1, then so does . This implies that . Therefore, using Lemma 2, we have a constant such that the inequalityis satisfied for all balls and for . Similarly, if , then by Lemma 2, we have constants , such thatfor all balls and for .

Lemma 3 (see [46]). Assuming that , for all balls and for a positive constant , the following inequality holds:

Definition 1 (see [47]). Let , setwhere supremum is taken all over the ball and . The function is known as bounded mean oscillation if and consist of all with .

Lemma 4 (see [48]). Let , then for all and all with , we have

Definition 2 (see [23]). Let and . Then, the variable exponent central Morrey space is defined aswhere

Definition 3 (see [23]). Let and . Then, the variable exponent -central BMO space is defined aswhereWhile proving our main results, we control the boundedness of the fractional Hardy operator using the boundedness of the fractional integral operator :on variable Lebesgue space. In this regard, we need the following proposition.

Proposition 2 (see [49]). Let and define by

Then,

Proposition 2 is useful in establishing the following Lemma (see [50]).

Lemma 5. Suppose , be as defined in Proposition 2, thenfor all balls with .

3. Fractional Hardy Operator and Commutator

In this section, we present theorems on the boundedness of the fractional Hardy operator and commutators on central Morrey space with their proofs.

Theorem 1. Let and satisfy conditions (9) and (10) in Proposition 1. Define the variable exponent by

If and , where and are the same constants as appeared in inequalities (14) and (15), then

Proof. By definition of the fractional Hardy operator and Lemma 1, it is easy to see that

Taking the norm on both sides, we have

Through the use of Lemma 3 and the inequality (15), it is easy to see that

In view of the condition and Lemma 5, the last inequality reduces to the following inequality:

Sincetherefore, from the inequality (32), we infer that

Finally, inequality (14) helps us to have

Since , so we get

Theorem 2. Let and let and satisfying conditions (9) and (10) in Proposition 1 with and

Let and . If , with , where are the same constants as appeared in inequalities (14) and (15), and , then

Proof. We decompose the integral appearing in the commutator operator as

Let us first estimate . By taking the variable Lebesgue space norm on both sides, we get

Taking into consideration the condition , the generalized Hölder inequality gives us the following estimation of :where . Using the result of Theorem 1, we obtain

Here, it is easy to see that

Therefore, on account of the condition , for , we have

Next, we consider for approximation:which can be decomposed further aswhere

We define a new variable such that , then by the generalized Hölder inequality, we have

With the Lebesgue space with variable exponent norm on both sides, the above inequality takes the following form:

Hence,

Finally, considerThe factor in the above inequality needs to be dealt with first. So,Next, Lemma 3 helps us to write

In turn, satisfies the below inequality:

Ultimately, our last step would be applying the norm on both sides to get

Combining all the approximations of , , , , we obtained the required result

4. Adjoint Fractional Hardy Operator and Commutator

In this last section, we first establish the boundedness of adjoint fractional Hardy operator and then use it to prove the boundedness of commutator generated by this operator and -central BMO function . The first result is as follows.

Theorem 3. Let and satisfying conditions (9) and (10) in Proposition 1, define the variable exponent by

If and , where is the same constant as appeared in inequality (15), then

Proof. Sincefrom which we infer that

Lemma 3 guides us to have the following inequality:which by Lemma 5 reduces to the following one:where we made use of inequality (15) in the last step of the above result. Hence, we obtainUtilizing the condition , a result similar to (33) and Lemma 2, we obtain

Finally, the series is convergent due to the fact that , and hence the result.

Keeping in view the analysis made in the previous section, we only outline the proof of the following theorem without going into many details.

Theorem 4. Let and satisfying conditions (9) and (10) in Proposition 1 with andLet and . If , with , where is the same constant as appeared in inequality (14) and , thenProof. As in the previous section, we start from decomposing the integral:Following the steps taken to approximate in Theorem 2, we directly estimate as follows:where . Now, the result (43) and Theorem 3 help us to writeNext, comparing with of Theorem 2, we arrive at

and for the approximation of , we follow a procedure similar to the one followed in the estimation of . Hence, we get

Eventually, it is easy to see that

Next, by virtue of inequality (53), satisfies

To finish the estimation, we take norm on both sides of the above inequality to obtain

In the end, combining all the estimates of , , , , we arrive at the following conclusive inequality:which is as desired.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Higher Education Commission (HEC) of Pakistan through the National Research Program for Universities (NRPU), Project No. 7098/Federal/NRPU/R&D/HEC/2017, and the Quaid-I-Azam University Research Fund URF FY 2018-2019.

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