Commutators of the Fractional Hardy Operator on Weighted Variable Herz-Morrey Spaces

Abstract

In the present paper, our aim is to establish the boundedness of commutators of the fractional Hardy operator and its adjoint operator on weighted Herz-Morrey spaces with variable exponents .

1. Introduction

Hardy operators and related commutators play an indispensable role in the theory of partial differential equations [1, 2] and the characterization of function spaces [3–5]. Without going into much details, let us first define the fractional Hardy operators [3] and related commutators:

It is important to note that taking in (1), we get multidimensional Hardy operator defined and studied in [6, 7]. Also, (1) reduces to the one dimensional Hardy operator [8] if we choose and . Here, we cite some important literature with regards to the study of Hardy-type operators on different function spaces which include [9–15].

The new development of variable exponent commenced with the work of Kov’aˇcik and R’akosn’ık in [16], where a class of function spaces having variable exponent was defined, and basic properties of variable exponent Lebesgue space were explored. Recently, the theory of variable exponent analysis is modeled in terms of the boundedness of the Hardy Littlewood maximal operator [17–21]:

Besides, Muckenhoupt theory [22] is generalized in the recent span of time with regard to variable exponent spaces ([23–28]). By taking into account the generalization of function spaces with variable exponents and the same with weights, many results like duality, boundedness of sublinear operators, the wavelet characterization, and commutators of fractional and singular integrals have been studied [29–38].

Recently, authors have studied generalized Herz space in terms of both Muckenhoupt weights and variable exponent [39–41]. Moreover, an idea of combining two function spaces to develop a new one is also an interesting problem in Harmonic analysis. One such problem is considered in [42] in which Herz-Morrey space was defined. Although, the weighted versions of Herz-Morrey spaces were introduced recently in [43, 44].

In this piece of work, our main focus is on establishing the boundedness of commutators of fractional Hardy operators on a class of function spaces called the weighted Herz-Morrey space with variable exponents. We seek to find the boundedness of these commutators with symbol functions in BMO (bounded mean oscillation) spaces. In establishing such a boundedness, we make use of the boundedness of the fractional integral operator on weighted Lebesgue space which was done in [39].

In the rest of this paper, the symbol expresses a constant whose value may differ at all of its occurrences. The Greek letter denotes the characteristics function of a sphere where is a measurable subset of and represents its Lebesgue measure. Before turning to our key results, let us first define the relevant variable exponent function spaces.

2. Preliminaries

Let us consider a measurable function on having range . The Lebesgue space with variable exponent is the set of all measurable function f such that

The space turns out to be Banach function space under the norm:

We denote by the set of all measurable functions such that where

Definition 1. Suppose is a real valued function on . We say that (i) is the set of all local log-Holder continuous functions satisfying(ii) is the set of all local log-Holder continuous function satisfying at the origin(iii) is the set of all log-Holder continuous functions satisfying at infinity(iv) denotes the set of all global log-Holder continuous functions .

It was proved in [21] that if , then Hardy-Littlewood maximal operator is bounded on .

Suppose is a weight function on , which is nonnegative and locally integrable on . Let be the space of all complex-valued functions f on such that. The space is a Banach function space equipped with the norm:

Benjamin Muckenhoupt introduced the theory of weights on in [22]. Recently, in [39, 40], Izuki and Noi generalized the Muckenhoupt class by taking as a variable.

Definition 2. Let . A weight is an weight if In [25], the authors proved that if and only if is bounded on the space .

Remark 3 (see [39]). Suppose and , then we have

Definition 4. Suppose and such that . A weight is said to be weight if

Definition 5 (see [39]). Suppose and such that . Then, if and only if .

Now, we define the variable exponent weighted Morrey-Herz space . Let , , and for .

Definition 6. Let w be a weight on and with . The space is the set of all measurable functions which is given by where Obviously, is the weighted Herz space with variable exponent (see [30]). Here, it is important to refer to some of the pioneering studies of the Herz space with constant exponents made in [45, 46].

3. Some Useful Lemmas

We start this section with some useful lemmas that will be helpful in proving our main results.

Lemma 7 (see [47]). If is Banach function space, then (i)The associated space is also Banach function space(ii) and are equivalent(iii)If and , thenis the generalized Hölder inequality.

Lemma 8 (see [39]). Suppose is a Banach function space. Then, we have that for all balls ,

Lemma 9 (see [28, 39]). Let be a Banach function space. Suppose that the Hardy Littlewood maximal operator is weakly bounded on ; that is, is true for and for all . Then, we have

Lemma 10 (see [39, 48]). (1) is Banach function space equipped with the normwhere (2)The associate space is also a Banach function space

Lemma 11 (see [39]). Let be a Banach function space. Assume that is bounded on , then there exists a constant for all and , The paper [16] shows that is a Banach function space and the associated space with equivalent norm.

Remark 12. Let , and by comparing the Lebesgue space and with the definition of , we have (1)If we take and , then we get (2)If we consider and , then we have By virtue of Lemma 10, we get . Next, in view of Lemma 11 and Remark 12, we have the following Lemma.

Lemma 13 (see [41]). Let be a Log Hölder continuous function both at infinity and at origin, if implies . Thus, the Hardy Littlewood operator is bounded on , and there exist constants such that for all balls and all measurable sets .

Lemma 14 (see [39]). Let and and . If , then is bounded from to .

4. Main Results and their Proofs

Definition 15. Let and set where the supremum is taken all over the balls and . The function is a bounded mean oscillation if and consist of all with . For a comprehensive review of the space, we suggest the reader to follow the books [49, 50].

Lemma 16. Let and be an weight. Then, for all and all with , we have

Proof. First part of this lemma is a consequence of [[41], Theorem 18]. Next, we will prove (28), for all with In the view of (27), we have Also, it is easy to see that Combining (29), (30), and (31), we get (28).

Proposition 17. Let , , and . If , then

Proof. The proof is similar to the proof of Proposition 17 in [44]. So, we omit the details.

Theorem 18. Let , , and be such that .
Also, let , , , and be log Hölder continuous at the origin, with , where , then

Proof. For any , if we denote , and for each , then it is not difficult to see that The generalized Hölder inequality (Lemma 7) yields the following inequality for : Applying the norm on both sides and using Lemma 16, we get Now, we turn to estimate . For this, we have Similar to the estimation for , we take the norm on both sides of above inequality and use Lemma 16 to obtain Hence, from inequalities (35), (37), and (39), one has which by virtue of Lemma 9 reduces to Now using Lemma 13, we learn In the definition of the fraction integral , we replace by to obtain from which we infer that Taking the norm on both sides and using Lemmas 14 and 9, respectively, we get In view of Lemmas 8 and 9, the use of (44) into (41) results in the following inequality: Now, by virtue of the condition and Proposition 17, we have where To estimate , , and , we make use of the conditions on , such that for , we have and for , we obtain In order to estimate , we need to use . The result of is similar to that of . Next, we will estimate below Finally, we combine the estimates for , to have the desired result.

Theorem 19. Let and be as in Theorem 18. In addition, if , where , then

Proof. We write We estimate and separately. A use of generalized inequality results in the following: Applying the weighted Lebesgue space norm on both sides and using Lemma 16, we obtain Similarly, In view of the weighted Lebesgue norm and Lemma 16, we get Hence, from (53), (55), and (57), we obtain Using the condition of weights given in the Definition 4, the above inequality reduces to Next, the condition and Proposition 17 help us to write where Lastly, in view of the condition , we estimate , as we estimated , in Theorem 18. Hence, we finish the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank the referees for careful reading of the paper andvaluable suggestions. Amjad Hussain is supported by 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) Project. The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia for funding this work through research groups program under grant number R.G. P-2/29/42.