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

In the past decades, the field of p-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, amongwhich we refer to [10, 11] for Riesz potentials [12–16], for p-adic pseudodifferential equations, and so forth. In the past few years, there is an increasing interest in the study of harmonic analysis onp-adic field and their various generalizations and the related theory of operators and spaces; see, for example [17–27]. For a prime number p, let Qp be the field of p-adic numbers. It is defined as the completion of the field of rational numbers Q with respect to the non-Archimedean p-adic norm | ⋅ |p. This norm is defined as follows: |0|p = 0; if any nonzero rational number x is represented as x = p(m/n), where γ is an integer and integers m, n are indivisible by p, then |x|p = p . It is easy to see that the norm satisfies the following properties:

For a prime number , let Q  be the field of -adic numbers.It is defined as the completion of the field of rational numbers Q with respect to the non-Archimedean -adic norm | ⋅ |  .This norm is defined as follows: |0|  = 0; 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: From the standard -adic analysis [7], we see that any nonzero -adic number  ∈ Q  can be uniquely represented in the canonical series as follows: where   are integers, 0 ≤   ≤  − 1, and  0 ̸ = 0.The series (2) converges in the -adic norm since |    |  =  − .
The space Q   consists of points  = ( 1 ,  2 , . . .,   ), where   ∈ Q  ,  = 1, 2, . . ., .The -adic norm on Q   is Denote by   () = { ∈ Q   : | − |  ≤   } the ball with center at  ∈ Q   and radius   and by   () := { ∈ Q   : | − |  =   } the sphere with center at  ∈ Q   and radius   ,  ∈ Z.It is clear that   () =   () \  −1 (), and We set   (0) =   and   (0) =   .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 positive constant multiple and is translation invariant.We normalize the measure  by the equality where for any  ∈ Q   .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 1 <  < ∞, where the classical Hardy operator is defined by for nonnegative integral function  on R + , and the constant /(−1) is the best possible.Thus the norm of Hardy operator on   (R + ) is Faris [29] introduced the following -dimensional Hardy operator, for nonnegative function  on R  , where Ω  is the volume of the unit ball in R  .Christ and Grafakos [30] obtained that the norm of H on   (R  ) 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 Q   , we define the -adic Hardy operator as follows: where (0, ||  ) is a ball in Q   with center at 0 ∈ Q   and radius ||  .
Morrey [36] introduced the  , (R  ) 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 1 ≤  < ∞ and let  ≥ −1/.The -adic Morrey space  , (Q   ) is defined by where ()      ) For some recent developments of Morrey spaces and their related function spaces on R  , 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  ∈ R and let 1 <  < ∞.The -adic central Morrey space Ḃ , (Q   ) is defined by where   =   (0).
Remark 5.It is clear that When  < −1/, the space Ḃ , (Q   ) reduces to {0}; therefore, we can only consider the case  ≥ −1/.If 1 ≤  1 <  2 < ∞, by Hölder's inequality, for  ∈ R. where for  ∈ R. By the standard proof as that in R  , we can see that Remark 8.The formulas (18) and (15) 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 () = (|| −1  ) 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.

Sharp Estimates of 𝑝-Adic Hardy Operator
We get the following precise norms of -adic Hardy operators on -adic central Morrey spaces and -adic -central BMO spaces.
Proof of Theorem 9.
When  > −1/, we first claim that the operator H  and its restriction to the subset of Ḃ , (Q   ), which consist of functions  satisfying () = (|| −1  ), have the same operator norm on Ḃ , (Q   ).In fact, for  ∈ Ḃ , (Q   ), set It is easy to see that  satisfies that () = (|| −1  ) and H   = H  .By Hölder's inequality, for  ∈ Z, we have Therefore, Consequently, which implies the claim.In the following, without loss of generality, we may assume that  ∈ Ḃ Thus, On the other hand, take  0 () = ||   .Then where the series converges due to  > −1/.Thus,  0 ∈ Ḃ , (Q   ) 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 H  and its restriction to the subset of CBMO , (Q   ) consisting of functions  with () = (|| −1  ) have the same operator norm on CBMO , (Q   ).In fact, set Then () = (|| −1  ) and H   = H  .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  ∈ CBMO Namely, On the other hand, take  0 () = ||   .By Remark 8 and (32), for  > −1/, we have Then by (33), we get We arrive at As a result, Then Theorem 10 follows from (43) and (47).     () On the other hand, as in the proof of Theorem 9, we take  0 () = ||   , and we only need to show that  0 ∈  , (Q   ).Consider the following. ( 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  0 ∈  , (Q   ).Then by (33), This completes the proof.

Boundedness for Commutators of 𝑝-Adic Hardy Operators on 𝑝-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 H  and -central BMO functions on -adic central Morrey spaces.
Definition 13.Let  ∈ CBMO , (Q   ).The commutator of H  is defined by for some suitable functions .
Theorem 14 is proved.