-Adic Riesz Potential and Its Commutators on Morrey-Herz Spaces

Abstract

In this paper, we establish the boundedness of -adic Riesz potential on Morrey-Herz spaces, as well as the -central BMO estimates for multilinear commutators of -adic Riesz potential on Morrey-Herz spaces.

1. Introduction

For a prime number , the field of -adic numbers is defined as the completion of the field of rational numbers with respect to the non-Archimedean -adic normal , which is defined as follows: if , ; if is an arbitrary rational number with the unique representation , where , are not divisible by , , then, . This norm satisfies the following properties: (i), , and (ii), (iii), and when , we have

The space consists of all points , where . The -adic norm of is defined by

Denote by the ball of radius with center at and by the sphere of radius with center at , where . For convenience, we denote and , it is easy to see that the equalities , , and hold for any .

Since the space is a locally compact commutative group under addition, there exists the Haar measure on the additive group of normalized by

where denotes the Haar measure of a measurable set . Then, by a simple calculation, we can obtain that for any . We should mention that the Haar measure takes value in , there also exist -adic valued measures [1, 2]. For a complete introduction to the -adic analysis, one can refer to [3] or [4] and references therein.

In the last several decades, there has been growing interest in the -adic models appearing in various branches of science. The -adic analysis has cemented its role in the field of mathematical physics (see, for example, [4, 5]). Many researchers have also paid relentless attention to harmonic analysis on the -adic fields [6–18]. It is well known that the Riesz potential operator in harmonic analysis is one of the most important operators and plays an important role in many areas such as Sobolev spaces, potential theory, and PDE, to name a few.

On the -adic field, let with , then, the Riesz potential is defined by

When , Haran [6, 7] obtained the explicit formula of the Riesz potential on . Taibleson [3] gave the fundamental analytic properties of on local fields. In particular, Kim [8] gave the -adic Hardy-Littlewood-Sobolev theorem on , which implies is bounded from to when and is of weak type when , where and . Later on, Volosivets [11, 12] extended the theorem to -adic generalized Morrey spaces and showed analogs of the results in Euclidean space given by Nakai [19]. Also, Wu and Fu [9] established counterparts of the theorem on -adic central Morrey spaces. Additionally, Lu and Xu [20] extended Morrey spaces and introduced Morrey-Herz spaces in the setting of , and they also gave the Hardy-Littlewood-Sobolev inequalities on the spaces, which led many scholars to study the boundedness of on Morrey-Herz spaces, for details, see [21, 22]. The first aim of the current paper is to study the boundedness of the Riesz potential on Morrey-Herz spaces over the -adic fields.

Theorem 1. Let be a complex number with , with , , , . If , then, there exists a constant such that Moreover, if , then, there exists a constant such that

Remark 2. The definitions of the homogeneous Morrey-Herz spaces and weak homogeneous Morrey-Herz spaces will be given in Section 2, here, we point out that for . We observe that Theorem 1 is counterparts of famous Hardy-Littlewood-Sobolev theorem on -adic Morrey-Herz spaces. In addition, Theorem 1 coordinates with those on Morrey-Herz spaces in the setting of , which was proved by Lu and Xu [20].
Moreover, let , where for , . Then, multilinear commutator generated by and can be defined as follows: When , Wu and Fu [9] obtained the -central BMO estimates for commutator on -adic central Morrey spaces. Furthermore, Mo et al. [13] considered multilinear case for and established the boundedness of with symbols in Campanato spaces on -adic generalized Morrey spaces. Motivated by the works, our second goal is to show the -central BMO estimates for the multilinear commutator on Morrey-Herz spaces over the -adic fields.

Theorem 3. Let be a complex number with , , where and . , , with , , . If , then, there exists a constant such that We notice that the important particular case of -central BMO spaces is defined in [14] when . Hence, if let in Theorem 3, we obtain the following conclusion.

Theorem 4. Let be a complex number with , , where and . , , , . If , then, there exists a constant such that On the other hand, it is eminent that and . Therefore, by letting in Theorem 1, 3, and 4, respectively, we will get counterparts on -adic Herz spaces.

Corollary 5. Let be a complex number with , with , , . If , then, there exists a constant such that Moreover, if , then, there exists a constant such that

Corollary 6. Let be a complex number with , , where and . , with , , . If , then, there exists a constant such that

Corollary 7. Let be a complex number with , , where and . , , . If , then, there exists a constant such that Throughout this paper, for and , we always assume that takes the principal branch, that is Thus, we have The letter will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.

2. Some Notations and Lemmas

We start with some notations and definitions. Here and in what follows, denote by the characteristic function of the sphere .

Definition 8 (see [15]). Let and . The homogeneous Herz space is defined by where for , and the usual modifications should be made when .

Definition 9 (see [15]). Let , , and be a nonnegative real number. Then, the homogeneous Morrey-Herz space is defined by where for , and the usual modifications should be made when .
Here, we introduce the -adic weak Herz spaces and -adic weak Morrey-Herz spaces.

Definition 10. Let , . The weak homogeneous Herz space defined by where for , and the usual modifications should be made when .

Definition 11. Let , , and be a nonnegative real number. The weak homogeneous Morrey-Herz space defined by where for , and the usual modifications should be made when .

Definition 12 (see [9]). Given , , the -central bounded mean oscillation space is defined as the set of all functions such that where

Lemma 13 (see [10]). Suppose that , , and . Then

Lemma 14 (see [8]). Let be a complex number with , with .If , then, there exists a constant such that Moreover, if , there is a constant such that

Lemma 15. Let be a complex number with , with . Then. there exists a constant such that and for any integers and .

Proof. Obviously, by the properties of , it is not difficult to check that for any , . Thus, by applying the Hölder’s inequality, we have Therefore, Lemma 15 is completely proved.

3. Proof of Theorems

Proof of Theorem 16. Suppose that , we decompose in the following way First, we prove that for . By the decomposition of above and the fact , we get If , by the -boundedness of (see Lemma 14), we can obtain the estimates of by If , applying Lemma 15, Jensen’s inequality, Hölder’s inequality, and the fact , we can get the estimates for by If , Lemma 15 shows the estimates for by For , using similar methods as that for , by the fact , Jensen’s inequality, and Hölder’s inequality, we obtain For , one can see from the definition of Morrey-Herz space that Here, by the estimates (43) and the fact , we obtain Therefore, by a combination for the estimates of , and , we can conclude that .
Next, for , we will consider the weak type boundedness of . Note that for , so we only need to prove that in the case . For any , the decomposition of shows that For , applying the weak type property of the operator (see Lemma 14) and the estimates (43). Notice that when , , so we can conclude that For , by the Chebychev inequality, Lemma 15, the estimates (43), and the fact , we have Hence, by a combination for the estimates of , and , we can get the desired inequality .
Therefore, we complete the proof of Theorem 16.

Proof of Theorem 17. Similarly to the proof of Theorem 16, let and decompose into When , denote by , we consider Let us first estimate , note that Suppose that , , then, it is easy to see for and . Applying the fact , Hölder’s inequality, Minkowski inequality, the boundedness of from to and from to , we have Therefore, we get Now, let us turn to the estimates of and . If , we can easily deduce that Then, using the facts , , , Lemma 13, and Hölder’s inequality, we can get Thus, the fact and (43) imply Combining the estimates for , and , which completes the proof for Theorem 3 of the case .
Now, we consider the case . In order to simplify the proof, for positive integer and , we denote by the family of all finite subsets of of different elements, let for any . For , let , and denote the integral average of the function over the set , then where and .
We write Let us first estimate , we can obtain For , taking , then, . Applying Hölder’s inequality, the boundedness of from to and the fact , we obtain For , taking , then, for . By Hölder’s inequality, the boundedness of from to and the fact , we get For , we denote by and , then . Using Hölder’s inequality and the -boundedness of , we have Then, similarly to the method estimating for , it is not difficult for us to get Next, we will estimate and . Let Therefore, by Minkowski’s inequality, Hölder’s inequality, and Lemma 13, it follows that Thus, similarly to the method estimating for and , we get So far, by a combination for the estimates of , , and , we will finish the proof for Theorem 3 of the case . This completes the proof of Theorem 17.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is partially supported by the MOE (Ministry of Education in China) Liberal arts and Social Sciences Foundation (Grant no. 20YJC790111).