Abstract
We establish the Hardy-Littlewood-Sobolev inequalities on -adic central Morrey spaces. Furthermore, we obtain the -central BMO estimates for commutators of -adic Riesz potential on -adic central Morrey spaces.
1. Introduction
Let . The Riesz potential operator is defined by setting, for all locally integrable functions on , where . It is closely related to the Laplacian operator of fractional degree. When and , is a solution of Poisson equation . 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 -adic field. In the last 20 years, the field of -adic numbers 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 -adic field has been drawing more and more concern (see [13–22] and references therein).
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 norm , which satisfies if and only if ; ; . Moreover, if , then . It is well-known that is a typical model of non-Archimedean local fields. If any nonzero rational number is represented as , where and integers , are indivisible by , then .
The space consists of points , where , . The -adic norm on is Denote by the ball of radius with center at and by the sphere of radius with center at , where . It is clear that
It is well-known that is a classical kind of locally compact Vilenkin groups. A locally compact Vilenkin group is a locally compact Abelian group containing a strictly decreasing sequence of compact open subgroups such that (1) and and (2) . For several decades, parallel to the -adic harmonic analysis, a development was under way of the harmonic analysis on locally compact Vilenkin groups (cf. [23–25] and references therein).
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 a positive constant factor 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 . We should mention that the Haar measure takes value in ; there also exist -adic valued measures (cf. [26, 27]). For a more complete introduction to the -adic field, one can refer to [22] or [10].
On -adic field, the -adic Riesz potential [22] is defined by where , , . When , Haran [4, 28] obtained the explicit formula of Riesz potentials on and developed analytical potential theory on . Taibleson [22] gave the fundamental analytic properties of the Riesz potentials on local fields including , as well as the classical Hardy-Littlewood-Sobolev inequalities. Kim [18] gave a simple proof of these inequalities by using the -adic version of the Calderón-Zygmund decomposition technique. Volosivets [29] investigated the boundedness for Riesz potentials on generalized Morrey spaces. Like on Euclidean spaces, using the Riesz potential with and , one can introduce the -adic Laplacians [13].
In this paper, we will consider the Riesz potentials and their commutators with -adic central BMO functions on -adic central Morrey spaces. Alvarez et al. [30] studied the relationship between central BMO spaces and Morrey spaces. Furthermore, they introduced -central BMO spaces and central Morrey spaces, respectively. In [31], we introduce their -adic versions.
Definition 1. Let and . The -adic central Morrey space is defined by where .
Remark 2. It is clear that When , the space reduces to ; therefore, we can only consider the case . If , by Hölder’s inequality, for .
Definition 3. Let and . The space is defined by the condition
Remark 4. 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 5. Formulas 9 and 12 yield that is a Banach space continuously included in .
Here we introduce the -adic weak central Morrey spaces.
Definition 6. Let and . The -adic weak central Morrey space is defined by where .
In Section 2, we will get the Hardy-Littlewood-Sobolev inequalities on -adic central Morrey spaces. Namely, under some conditions for indexes, is bounded from to and is also bounded from to . In Section 3, we establish the boundedness for commutators generated by and -central BMO functions on -adic central Morrey spaces.
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. Hardy-Littlewood-Sobolev Inequalities
We get the following Hardy-Littlewood-Sobolev inequalities on -adic central Morrey spaces.
Theorem 7. Let be a complex number with and let , , , and .(i)If , then is bounded from to .(ii)If , then is bounded from to .
In order to give the proof of this theorem, we need the following result.
Lemma 8 (see [22]). Let be a complex number with and let satisfy .(i)If , , then where is independent of .(ii)If , , then where is independent of .
Proof of Theorem 7. Let be a function in . For fixed , denote by .(i)If , write
For , since and , by Lemma 8,
For , we firstly give the following estimate. For , by Hölder’s inequality, we have
The last inequality is due to the fact that . Consequently,
The above estimates imply that
(ii)If , set and ; by Lemma 8, we have
On the other hand, by the same estimate as 30, we have
Then using Chebyshev’s inequality, we obtain
Since
we get
Therefore,
for any and . This completes the proof.
For application, we now introduce a pseudo-differential operator defined by Vladimirov in [33].
The operator is defined as convolution of generalized functions and : where and .
Let us consider the equation where is the space of linear continuous functionals on and here denotes the set of locally constant functions on . A complex-valued function defined on is called locally constant if for any point there exists an integer such that
The following lemma (page 154 in [10]) gives solutions of 30.
Lemma 9. For any solution of 30 is expressed by the formula where is an arbitrary constant; for a solution of 30 is unique and it is expressed by formula 32 for .
Combining with Theorem 7, we obtain the following regular property of the solution.
Corollary 10. Let and let , , , and . If , then(i)when , 30 has a solution in ,(ii)when , 30 has a solution in .
3. Commutators of -Adic Riesz Potential
In this section, we will establish the -central BMO estimates for commutators of -adic Riesz potential which is defined by for some suitable functions .
Theorem 11. Suppose , , , and . Let , satisfies , and . If , then is bounded from to , and the following inequality holds:
Before proving this theorem, we need the following result.
Lemma 12 (see [31]). Suppose that and , . Then
Proof of Theorem 11. Suppose that is a function in . For fixed , denote by . We write Set ; then ; by Lemma 8 and Hölder’s inequality, we have Similarly, denote ; then , and by Hölder’s inequality and Lemma 8, we get To estimate and , we firstly give the following estimates. For , by Hölder’s inequality, we obtain where the penultimate “” is due to the fact that . Similarly, Since , by Lemma 12, we have Thus Now by 39 and Hölder’s inequality, we obtain It follows from 42 that The above estimates imply that This completes the proof of the theorem.
Remark 13. Since -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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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).