Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 359193, 10 pages

http://dx.doi.org/10.1155/2013/359193

## Boundedness of -Adic Hardy Operators and Their Commutators on -Adic Central Morrey and BMO Spaces

Department of Mathematics, Linyi University, Linyi, Shandong 276005, China

Received 22 May 2013; Accepted 10 August 2013

Academic Editor: Dachun Yang

Copyright © 2013 Qing Yan Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We obtain the sharp bounds of *p*-adic Hardy operators on *p*-adic central Morrey spaces and *p*-adic -central BMO spaces, respectively. We also establish the -central BMO estimates for commutators of *p*-adic Hardy operators on *p*-adic central Morrey spaces.

#### 1. Introduction

In the past decades, the field of -adic numbers has been intensively used in *theoretical and mathematical physics* (see [1–9] and references therein). As a consequence, new mathematical problems have emerged, among which we refer to [10, 11] for Riesz potentials [12–16], for -adic pseudodifferential equations, and so forth. In the past few years, there is an increasing interest in the study of *harmonic analysis on **-adic field* and their various generalizations and the related theory of operators and spaces; see, for example [17–27].

For a prime number , let be the field of -adic numbers. It is defined as the completion of the field of rational numbers with respect to the non-Archimedean -adic norm . This norm is defined as follows: ; if any nonzero rational number is represented as , where is an integer and integers , are indivisible by , then . It is easy to see that the norm satisfies the following properties: Moreover, if , then . It is well known that is a typical model of non-Archimedean local fields. From the standard -adic analysis [7], we see that any nonzero -adic number can be uniquely represented in the canonical series as follows: where are integers, , and . The series (2) converges in the -adic norm since .

The space consists of points , where , . The -adic norm on is Denote by the ball with center at and radius and by the sphere with center at and radius , . It is clear that , and We set and .

Since is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure on , which is unique up to positive constant multiple and is translation invariant. We normalize the measure by the equality where denotes the Haar measure of a measurable subset of . By simple calculation, we can obtain that for any . For a more complete introduction to the -adic field, see [27] or [7].

The well-known Hardy’s integral inequality [28] tells us that, for , where the classical Hardy operator is defined by for nonnegative integral function on , and the constant is the best possible. Thus the norm of Hardy operator on is

Faris [29] introduced the following -dimensional Hardy operator, for nonnegative function on , where is the volume of the unit ball in . Christ and Grafakos [30] obtained that the norm of on is which is the same as that of the 1-dimensional Hardy operator.

In [31], Fu et al. obtained the precise norm of -linear Hardy operators on weighted Lebesgue spaces and central Morrey spaces. Fu et al. [32] introduced -adic Hardy operators and got the sharp estimates of -adic Hardy operators on -adic weighted Lebesgue spaces. Moreover, they proved that the commutators generated by the -adic Hardy operators and the central BMO functions are bounded on -adic weighted Lebesgue spaces and -adic Herz spaces see; [33] for more information about Herz spaces. Ren and Tao [34] Yu and Lu [35] studied the boundedness of commutators of Hardy type on some spaces.

Inspired by these results, in this paper we will establish the sharp estimates of -adic Hardy operators on -adic central Morrey and -central BMO spaces. Furthermore, we will discuss the boundedness for commutators of -adic Hardy operators and -central BMO functions on -adic central Morrey spaces.

*Definition 1. *For a function on , we define the *-adic Hardy operator* as follows:
where is a ball in with center at and radius .

Morrey [36] introduced the spaces to study the local behavior of solutions to second-order elliptic partial differential equations. The -adic Morrey space is defined as follows.

*Definition 2. *Let and let . The -adic Morrey space is defined by
where

*Remark 3. *It is clear that , .

For some recent developments of Morrey spaces and their related function spaces on , we refer the reader to [37]. In 2000, Alvarez et al. [38] studied the relationship between central BMO spaces and Morrey spaces. Furthermore, they introduced -central bounded mean oscillation spaces and central Morrey spaces, respectively. Next, we introduce their -adic versions.

*Definition 4. *Let and let . The -adic central Morrey space is defined by
where .

*Remark 5. *It is clear that
When , the space reduces to ; therefore, we can only consider the case . If , by Hölder’s inequality,
for .

*Definition 6. *Let and let . The space is defined by the condition
where .

*Remark 7. *When , the space is just , which is defined in [32]. If , by Hölder’s inequality,
for . By the standard proof as that in , we can see that

*Remark 8. *The formulas (18) and (15) yield that is a Banach space continuously included in .

In Section 2, we obtain the sharp estimates of -adic Hardy operators on -adic central Morrey spaces and -adic -central BMO spaces. Analogous result is also established for -adic Morrey spaces. In Section 3, we discuss the boundedness of commutators generated by -adic Hardy operators and -adic -central BMO functions on -adic central Morrey spaces.

We should note that in Euclidean space, when estimating the Hardy operator, one usually discusses its restriction on radical functions. However, on -adic field, we will consider its restriction on the functions with instead.

Throughout this paper the letter will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.

#### 2. Sharp Estimates of -Adic Hardy Operator

We get the following precise norms of -adic Hardy operators on -adic central Morrey spaces and -adic -central BMO spaces.

Theorem 9. *Let and let . Then
*

Theorem 10. *Let and let . Then
*

Corollary 11. *Let . Then
*

Let denote the subspace of consisting of all functions with . We obtain the sharp estimate of -adic Hardy operator from to .

Theorem 12. *Suppose that , . Then maps to with norm
*

*Proof of Theorem 9. *When , , by Corollary 2.2 in [32],

When , we first claim that the operator and its restriction to the subset of , which consist of functions satisfying , have the same operator norm on .

In fact, for , set
It is easy to see that satisfies that and . By Hölder’s inequality, for , we have
Therefore,
Consequently,
which implies the claim. In the following, without loss of generality, we may assume that satisfies . Then by Minkowski’s inequality, we have
Thus,

On the other hand, take . Then
where the series converges due to . Thus, since
Therefore,
Then (31) and (34) imply that

*Proof of Theorem 10. *As in the proof of Theorem 9, we first show that the operator and its restriction to the subset of consisting of functions with have the same operator norm on .

In fact, set
Then and . By change of variable, we get
Using Minkowski’s inequality and (37), we have
Therefore,
We conclude that

In the following, without loss of generality, we may assume that with . By Fubini theorem, we have
Then by Minkowski's inequality, we get
Namely,

On the other hand, take . By Remark 8 and (32), for , we have . Then by (33), we get
Therefore,
We arrive at

As a result,
Then Theorem 10 follows from (43) and (47).

*Proof of Theorem 12. *Let . Then . Using Minkowski’s inequality, we have

On the other hand, as in the proof of Theorem 9, we take , and we only need to show that . Consider the following.

(I) If and , then . Since , we have

(II) If and , then ; therefore, . Recall that two balls in are either disjoint or one is contained in the other (cf. page 21 in [39]). So we have ; thus,
From the previous discussion, we can see that . Then by (33),
This completes the proof.

#### 3. Boundedness for Commutators of -Adic Hardy Operators on -Adic Central Morrey Spaces

The boundedness of commutators is an active topic in harmonic analysis due to its important applications. For example, it can be applied to characterizing some function spaces [40]. In this section, we consider the boundedness for commutators generated by and -central BMO functions on -adic central Morrey spaces.

*Definition 13. *Let . The commutator of is defined by
for some suitable functions .

Theorem 14. *Let , , , , , and . If ; then is bounded from to and satisfies
*

Before the proof of this theorem, we need the following calculations.

Lemma 15. *Suppose that and , . Then,
*

*Proof. *Without loss of generality, we may assume that . Recall that . By Hölder's inequality, we have

Therefore,

*Proof of Theorem 14. *Assume that . Fix , and by Minkowski’s inequality, we have

In the following, we will estimate and , respectively. For , since is bounded from to , [32], then, by Hölder's inequality (), we get

Next, let us estimate as follows:

For , by Hölder’s inequality () and the fact that , , we have
For , by Lemma 15 and Hölder’s inequality, we obtain
Notice that since , then and .

The above estimates imply that
Consequently,
Theorem 14 is proved.

#### Acknowledgment

This work was partially supported by NSF of China (Grant nos. 11271175, 11301248, 11171345, and 10901076), NSF of Shandong Province (Grant no. ZR2010AL006) and AMEP of Linyi University.

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