Hardy-Littlewood-Sobolev Inequalities on p-Adic Central Morrey Spaces

where γ n (α) = π n/2 2 α Γ(α/2)/Γ((n − α)/2). It is closely related to the Laplacian operator of fractional degree. When n > 2 and α = 2, I α f is a solution of Poisson equation −Δu = f. 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 p-adic field. In the last 20 years, the field of p-adic numbers Q p

Since Q   is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure  on Q   , which is unique up to a positive constant factor and is translation invariant.We normalize the measure  by the equality where for any  ∈ Q   .We should mention that the Haar measure takes value in R; 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 Γ  () = (1 −  − )/(1 −  − ),  ∈ C,  ̸ = 0.When  = 1, Haran [4,28] obtained the explicit formula of Riesz potentials on Q  and developed analytical potential theory on Q  .Taibleson [22] gave the fundamental analytic properties of the Riesz potentials on local fields including Q   , 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  > 2 and  = 2, 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.
for  ∈ R. By the standard proof as that in R  , we can see that where   =   (0).
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 Ḃ , (Q   ) to Ḃ , (Q   ) and is also bounded from Ḃ 1, (Q   ) to  Ḃ , (Q   ).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.
In order to give the proof of this theorem, we need the following result.
Lemma 8 (see [22]).Let  be a complex number with 0 < Re  <  and let where   is independent of .
where   > 0 is independent of .
Proof of Theorem 7. Let  be a function in For , since 1/ = The last inequality is due to the fact that  < −Re /.Consequently, The above estimates imply that ) for any  > 0 and  ∈ Z.This completes the proof.
For application, we now introduce a pseudo-differential operator   defined by Vladimirov in [33].
Let us consider the equation where E  is the space of linear continuous functionals on E and here E denotes the set of locally constant functions on Q  .A complex-valued function () defined on Q  is called locally constant if for any point  ∈ Q  there exists an integer () ∈ Z such that The following lemma (page 154 in [10]) gives solutions of (30).Lemma 9.For  > 0 any solution of ( 30) is expressed by the formula where  is an arbitrary constant; for  < 0 a solution of ( 30) is unique and it is expressed by formula (32) for  = 0.
Combining with Theorem 7, we obtain the following regular property of the solution.

Commutators of 𝑝-Adic Riesz Potential
In this section, we will establish the -central BMO estimates for commutators  ,

𝛼
of -adic Riesz potential which is defined by for some suitable functions .
, and the following inequality holds: Before proving this theorem, we need the following result.
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.