Academic Editor:Yoshihiro Sawano
Department of Mathematics, Linyi University, Linyi, Shandong 276005, China
Received 21 October 2014; Accepted 15 December 2014; 22 April 2015
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.
1. Introduction
Let [figure omitted; refer to PDF] . The Riesz potential operator [figure omitted; refer to PDF] is defined by setting, for all locally integrable functions [figure omitted; refer to PDF] on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . It is closely related to the Laplacian operator of fractional degree. When [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is a solution of Poisson equation [figure omitted; refer to PDF] . The importance of Riesz potentials is owing to the fact that they are smooth operators and have been extensively used in various areas such as potential analysis, harmonic analysis, and partial differential equations. For more details about Riesz potentials one can refer to [1].
This paper focuses on the Riesz potentials on [figure omitted; refer to PDF] -adic field. In the last 20 years, the field of [figure omitted; refer to PDF] -adic numbers [figure omitted; refer to PDF] has been intensively used in theoretical and mathematical physics (cf. [2-12]). And it has already penetrated intensively into several areas of mathematics and its applications, among which harmonic analysis on [figure omitted; refer to PDF] -adic field has been drawing more and more concern (see [13-22] and references therein).
For a prime number [figure omitted; refer to PDF] , the field of [figure omitted; refer to PDF] -adic numbers [figure omitted; refer to PDF] is defined as the completion of the field of rational numbers [figure omitted; refer to PDF] with respect to the non-Archimedean [figure omitted; refer to PDF] -adic norm [figure omitted; refer to PDF] , which satisfies [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] . Moreover, if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] . It is well-known that [figure omitted; refer to PDF] is a typical model of non-Archimedean local fields. If any nonzero rational number [figure omitted; refer to PDF] is represented as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and integers [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are indivisible by [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] .
The space [figure omitted; refer to PDF] consists of points [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The [figure omitted; refer to PDF] -adic norm on [figure omitted; refer to PDF] is [figure omitted; refer to PDF] Denote by [figure omitted; refer to PDF] the ball of radius [figure omitted; refer to PDF] with center at [figure omitted; refer to PDF] and by [figure omitted; refer to PDF] the sphere of radius [figure omitted; refer to PDF] with center at [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . It is clear that [figure omitted; refer to PDF]
It is well-known that [figure omitted; refer to PDF] is a classical kind of locally compact Vilenkin groups. A locally compact Vilenkin group [figure omitted; refer to PDF] is a locally compact Abelian group containing a strictly decreasing sequence of compact open subgroups [figure omitted; refer to PDF] such that (1) [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and (2) [figure omitted; refer to PDF] . For several decades, parallel to the [figure omitted; refer to PDF] -adic harmonic analysis, a development was under way of the harmonic analysis on locally compact Vilenkin groups (cf. [23-25] and references therein).
Since [figure omitted; refer to PDF] is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure [figure omitted; refer to PDF] on [figure omitted; refer to PDF] , which is unique up to a positive constant factor and is translation invariant. We normalize the measure [figure omitted; refer to PDF] by the equality [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the Haar measure of a measurable subset [figure omitted; refer to PDF] of [figure omitted; refer to PDF] . By simple calculation, we can obtain that [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] . We should mention that the Haar measure takes value in [figure omitted; refer to PDF] ; there also exist [figure omitted; refer to PDF] -adic valued measures (cf. [26, 27]). For a more complete introduction to the [figure omitted; refer to PDF] -adic field, one can refer to [22] or [10].
On [figure omitted; refer to PDF] -adic field, the [figure omitted; refer to PDF] -adic Riesz potential [figure omitted; refer to PDF] [22] is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] , Haran [4, 28] obtained the explicit formula of Riesz potentials on [figure omitted; refer to PDF] and developed analytical potential theory on [figure omitted; refer to PDF] . Taibleson [22] gave the fundamental analytic properties of the Riesz potentials on local fields including [figure omitted; refer to PDF] , as well as the classical Hardy-Littlewood-Sobolev inequalities. Kim [18] gave a simple proof of these inequalities by using the [figure omitted; refer to PDF] -adic version of the Calderon-Zygmund decomposition technique. Volosivets [29] investigated the boundedness for Riesz potentials on generalized Morrey spaces. Like on Euclidean spaces, using the Riesz potential with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , one can introduce the [figure omitted; refer to PDF] -adic Laplacians [13].
In this paper, we will consider the Riesz potentials and their commutators with [figure omitted; refer to PDF] -adic central BMO functions on [figure omitted; refer to PDF] -adic central Morrey spaces. Alvarez et al. [30] studied the relationship between central BMO spaces and Morrey spaces. Furthermore, they introduced [figure omitted; refer to PDF] -central BMO spaces and central Morrey spaces, respectively. In [31], we introduce their [figure omitted; refer to PDF] -adic versions.
Definition 1.
Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The [figure omitted; refer to PDF] -adic central Morrey space [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .
Remark 2.
It is clear that [figure omitted; refer to PDF] When [figure omitted; refer to PDF] , the space [figure omitted; refer to PDF] reduces to [figure omitted; refer to PDF] ; therefore, we can only consider the case [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , by Hölder's inequality, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .
Definition 3.
Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The space [figure omitted; refer to PDF] is defined by the condition [figure omitted; refer to PDF]
Remark 4.
When [figure omitted; refer to PDF] , the space [figure omitted; refer to PDF] is just [figure omitted; refer to PDF] , which is defined in [32]. If [figure omitted; refer to PDF] , by Hölder's inequality, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . By the standard proof as that in [figure omitted; refer to PDF] , we can see that [figure omitted; refer to PDF]
Remark 5.
Formulas 9 and 12 yield that [figure omitted; refer to PDF] is a Banach space continuously included in [figure omitted; refer to PDF] .
Here we introduce the [figure omitted; refer to PDF] -adic weak central Morrey spaces.
Definition 6.
Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The [figure omitted; refer to PDF] -adic weak central Morrey space [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .
In Section 2, we will get the Hardy-Littlewood-Sobolev inequalities on [figure omitted; refer to PDF] -adic central Morrey spaces. Namely, under some conditions for indexes, [figure omitted; refer to PDF] is bounded from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] and is also bounded from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] . In Section 3, we establish the boundedness for commutators generated by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] -central BMO functions on [figure omitted; refer to PDF] -adic central Morrey spaces.
Throughout this paper the letter [figure omitted; refer to PDF] will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.
2. Hardy-Littlewood-Sobolev Inequalities
We get the following Hardy-Littlewood-Sobolev inequalities on [figure omitted; refer to PDF] -adic central Morrey spaces.
Theorem 7.
Let [figure omitted; refer to PDF] be a complex number with [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .
(i) If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is bounded from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] .
(ii) If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is bounded from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] .
In order to give the proof of this theorem, we need the following result.
Lemma 8 (see [22]).
Let [figure omitted; refer to PDF] be a complex number with [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] satisfy [figure omitted; refer to PDF] .
(i) If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is independent of [figure omitted; refer to PDF] .
(ii) If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is independent of [figure omitted; refer to PDF] .
Proof of Theorem 7.
Let [figure omitted; refer to PDF] be a function in [figure omitted; refer to PDF] . For fixed [figure omitted; refer to PDF] , denote [figure omitted; refer to PDF] by [figure omitted; refer to PDF] .
(i) If [figure omitted; refer to PDF] , write [figure omitted; refer to PDF]
For [figure omitted; refer to PDF] , since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , by Lemma 8, [figure omitted; refer to PDF]
For [figure omitted; refer to PDF] , we firstly give the following estimate. For [figure omitted; refer to PDF] , by Hölder's inequality, we have [figure omitted; refer to PDF] The last inequality is due to the fact that [figure omitted; refer to PDF] . Consequently, [figure omitted; refer to PDF]
The above estimates imply that [figure omitted; refer to PDF]
(ii) If [figure omitted; refer to PDF] , set [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; by Lemma 8, we have [figure omitted; refer to PDF]
On the other hand, by the same estimate as 30, we have [figure omitted; refer to PDF] Then using Chebyshev's inequality, we obtain [figure omitted; refer to PDF]
Since [figure omitted; refer to PDF] we get [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . This completes the proof.
For application, we now introduce a pseudo-differential operator [figure omitted; refer to PDF] defined by Vladimirov in [33].
The operator [figure omitted; refer to PDF] is defined as convolution of generalized functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .
Let us consider the equation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the space of linear continuous functionals on [figure omitted; refer to PDF] and here [figure omitted; refer to PDF] denotes the set of locally constant functions on [figure omitted; refer to PDF] . A complex-valued function [figure omitted; refer to PDF] defined on [figure omitted; refer to PDF] is called locally constant if for any point [figure omitted; refer to PDF] there exists an integer [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]
The following lemma (page 154 in [10]) gives solutions of 30.
Lemma 9.
For [figure omitted; refer to PDF] any solution of 30 is expressed by the formula [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an arbitrary constant; for [figure omitted; refer to PDF] a solution of 30 is unique and it is expressed by formula 32 for [figure omitted; refer to PDF] .
Combining with Theorem 7, we obtain the following regular property of the solution.
Corollary 10.
Let [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then
(i) when [figure omitted; refer to PDF] , 30 has a solution in [figure omitted; refer to PDF] ,
(ii) when [figure omitted; refer to PDF] , 30 has a solution in [figure omitted; refer to PDF] .
3. Commutators of [figure omitted; refer to PDF] -Adic Riesz Potential
In this section, we will establish the [figure omitted; refer to PDF] -central BMO estimates for commutators [figure omitted; refer to PDF] of [figure omitted; refer to PDF] -adic Riesz potential which is defined by [figure omitted; refer to PDF] for some suitable functions [figure omitted; refer to PDF] .
Theorem 11.
Suppose [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is bounded from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , and the following inequality holds: [figure omitted; refer to PDF]
Before proving this theorem, we need the following result.
Lemma 12 (see [31]).
Suppose that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF]
Proof of Theorem 11.
Suppose that [figure omitted; refer to PDF] is a function in [figure omitted; refer to PDF] . For fixed [figure omitted; refer to PDF] , denote [figure omitted; refer to PDF] by [figure omitted; refer to PDF] . We write [figure omitted; refer to PDF]
Set [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] ; by Lemma 8 and Hölder's inequality, we have [figure omitted; refer to PDF]
Similarly, denote [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] , and by Hölder's inequality and Lemma 8, we get [figure omitted; refer to PDF]
To estimate [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we firstly give the following estimates. For [figure omitted; refer to PDF] , by Hölder's inequality, we obtain [figure omitted; refer to PDF] where the penultimate " [figure omitted; refer to PDF] " is due to the fact that [figure omitted; refer to PDF] . Similarly, [figure omitted; refer to PDF]
Since [figure omitted; refer to PDF] , by Lemma 12, we have [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF]
Now by 39 and Hölder's inequality, we obtain [figure omitted; refer to PDF]
It follows from 42 that [figure omitted; refer to PDF]
The above estimates imply that [figure omitted; refer to PDF] This completes the proof of the theorem.
Remark 13.
Since [figure omitted; refer to PDF] -adic field is a kind of locally compact Vilenkin groups, we can further consider the Hardy-Littlewood-Sobolev inequalities on such groups, which is more complicated and will appear elsewhere.
Acknowledgments
This work was partially supported by NSF of China (Grant nos. 11271175, 11171345, and 11301248) and AMEP (DYSP) of Linyi University and Macao Science and Technology Development Fund, MSAR (Ref. 018/2014/A1).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2015 Qing Yan Wu and Zun Wei Fu. 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 establish the Hardy-Littlewood-Sobolev inequalities on p -adic central Morrey spaces. Furthermore, we obtain the λ -central BMO estimates for commutators of p -adic Riesz potential on p -adic central Morrey spaces.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer