#### Abstract

In this paper, we give characterization of a -adic version of Lipschitz spaces in terms of the boundedness of commutators of maximal function in the context of the -adic version of Lebesgue spaces and Morrey spaces, where the symbols of the commutators belong to the Lipschitz spaces. A useful tool is a Lipschitz norm involving the John-Nirenberg-type inequality for homogeneous Lipschitz functions, which is new in the -adic field context.

#### 1. Introduction and Statement of Main Results

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 and are integers which are not divisible by , and is an integer, then . It is not difficult to show that the norm satisfies the following properties:

It follows from the second property that when , then . From the standard -adic analysis [1], we see that any nonzero -adic number can be uniquely represented in the canonical serieswhere are integers, , . Series (2) converges in the -adic norm because .

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

Since is a locally compact commutative group under addition, it follows from the standard analysis that there exists a unique Harr measure on (up to positive constant multiple) which is translation invariant. We normalize the measure so thatwhere denotes the Harr measure of a measurable subset of . From this integral theory, it is easy to obtain that and for any .

In what follows, we say that a (real-valued) measurable function defined on is in , , if it satisfies

Here the integral in equation (4) is defined as follows:

If the limit exists. We now mention some of the previous works on harmonic analysis on the -adic field, see [2–8] and the references therein.

For a function , we defined the Hardy-Littlewood maximal function of on by the following equation:

In [9, 10], Kim proved boundedness of the version of maximal function and gave some properties similar to the Euclidean setting.

The maximal commutator of with a locally integrable function is defined by the following equation:

The first part of this paper is to study the boundedness of when the symbol belongs to a Lipschitz space (see in Section 2). Some characterizations of the Lipschitz space via such commutator are given. Our first result can be stated as follows:

Theorem 1. *Let be a locally integrable function and , then the following statements are equivalent.*(1)*;*(2)* is bounded from to for all with and ;*(3)* is bounded from to for some with and ;*(4)* satisfies the weak-type estimates, namely, there exists a positive constant such that for any ,*(5)* is bounded from to .*

We remark that the boundedness of the commutators of maximal function were pretty much unknown on the -adic vector space. However, in Euclidean case, the mapping properties of the maximal commutator have been studied intensively by many authors, see [11–21].

It is well-known that the Morrey space introduced by Morrey in 1938, is connected to certain problem in elliptic PDE [22]. Later the Morrey spaces were found to have many important applications to the Navier-Stokes equations, the Schrödinger equations, the elliptic equations with discontinuous coefficients and the potential analysis, see [23–27] and so on. Next, we introduce the -adic version of Morrey space on -adic vector space.

For and . The -adic version of Morrey space is defined by the following equation:where

It is well known that if , then and .

Theorem 2. *Let be a locally integrable function and . Assume that , and . Then, if and only if is bounded from to .*

Theorem 3. *Let be a locally integrable function and . Assume that , , , and . Then, if and only if is bounded from to .*

On the other hand, the classical commutator of the -adic version of maximal function with a locally integrable function can be defined by the following equation:

In the Euclidean setting, the commutator of maximal function as in (11) has attracted much more attention. For examples, Milman and Schonbek [28] established a commutator result by real interpolation techniques. As an application, they obtained the -boundedness of the commutator of maximal function when and . This operator can be used in studying the product of a function in and a function in BMO (see [29] for instance). Bastero et al. [30] studied the necessary and sufficient conditions for the boundedness of on spaces when . Zhang and Wu [31] extended their results to commutators of the fractional maximal function. The results in [30, 31] were extended to variable Lebesgue spaces in [32, 33]. Recently, Zhang [20] studied the commutator when belongs to Lipschitz spaces. Some necessary and sufficient conditions for the boundedness of on Lebesgue and Morrey spaces are given. Some of the results were extended to variable Lebesgue spaces in [21]. For more information about the characterization of the commutator of maximal operator, see also [19, 20] and the references therein.

We would like to remark that operators and essentially differ from each other. For example, is positive and sublinear, but is neither positive nor sublinear. Motivated by the papers mentioned above, in this paper, the second part of this paper aims to study the mapping properties of the commutator when belongs to some Lipschitz spaces. More precisely, we will give some new necessary and sufficient conditions for the boundedness of on -adic vector spaces, by which some new characterizations for certain subclasses of Lipschitz spaces are obtained.

Theorem 4. *Let be a locally integrable function and . Assume that and . Then, the following statements are equivalent.*(1)* and ;*(2)* is bounded from to ;*(3)*There exists a constant such that*

Where denote the maximal operator with respect to a -adic ball which is defined by the following equation:

Here, the supremum is taken over all the -adic with for a fixed -adic ball .

Theorem 5. *Let be a locally integrable function, and . Then, there exist a positive such that for any ,*

Theorem 6. *Let be a locally integrable function and . Assume that , , and . Then, the following statements are equivalent*(1)* and ;*(2)* is bounded from to .*

Theorem 7. *Let be a locally integrable function and . Assume that , , , and . Then, the following statements are equivalent.*(1)* and ;*(2)* is bounded from to .*

*Remark 1. *Theorem 6 is essentially the Adams-type result and Theorem 7 is the Spanne-type result. Also, Theorem 7 can be immediately proved via Theorem 6.

The rest of the present paper is organized as follows: in Section 2, we gave some definitions and lemmas. The Proof of Theorems 1–3 are presented in Section 3. In Section 4, we gave the proof of Theorems 4–7. By we mean that with some positive constant independent of appropriate quantities. The positive constants varies from one occurrence to another. For a real number , , is the conjugate number of , that is, .

#### 2. Preliminary Definitions and Lemmas

To prove our main results, we need the following definitions and lemmas.

*Definition 1. *Assume that . The -adic version of homogeneous Lipschitz spaces (see [34]) is the space of all measurable functions on such thatFor , the -adic version of Lipschitz spaces is the space of all measurable functions on such thatwhere denotes the average of over , i.e., . When , we use as .

We shall give the -adic version of John-Nirenberg inequality for homogeneous Lipschitz functions in the following lemma.

Lemma 1. *Let , then there exist some depending on and such that**for any and for any -adic , where denotes the family of all the -adic balls which is defined as follows:*

Kim in [9] observed several interesting properties on the family .

Lemma 2. *The family has the following properties:*(a)*If , then either or .*(b)* if and only if .*

*Proof of Lemma 1. *Let be a fixed -adic ball and be some positive real number which will be determined later. Applying the -adic version of the Calderón-Zygmund decomposition (see Corollary 3.4 in [9], or see also Theorem 5.16 in [9]) of for relative to to obtain a pair-wise disjoint family of -adic balls (by Lemma 2) which satisfies the following equation:andCombining and equation (20) gives the following equation:We denote . For any , we haveBy using equation (19), we can infer that for any Thus, equations (23) and (24) now implies.For any , setObviously, is a decreasing function and . Thus, we have the following equation:Hence, for any , we conclude thatTaking , then is also a fixed positive number and for any ,Using induction argument for any , we obtain the following equation:Thus, for , we have the following equation:Notice that this inequality is also true for , due to . Therefore, for any , we conclude thatThis finishes the proof of the lemma.

Kim in [9] gave the property of as in the Euclidean case.

Lemma 3. *For , the distribution function of on defined by the following equation:*

If for , then we have the following equation:

We also need the -adic version of Lebesgue differential theorem, which is due to Kim [10].

Lemma 4. *If , then we have that for a.e. *

The following lemma is about some properties of Lipschitz space .

Lemma 5. *If is given, then we have the following properties:*(a)*For , then there exists a constant such that .*(b)*The norm is equivalent to the norm on Lipschitz space.*(c)*For any with , where is the constant given in Lemma 1,*

*Proof. *.(a)For , using Lemmas 1 and 3 we give the following equation: where . Hence, we conclude that .(b)For , by Hölder’s inequality it is obvious that for any . Thus, we have that . Hence, by (a) we can obtain the equivalence of those two norms.(c)We first observe thatFor any . Then, it follows from Lemma 1 and the -adic version [1] of changing the order of integration thatfor any , provided that . Therefore, we have the following equation:This completes the proof of the lemma.

To prove the theorems, we need the following key lemma. In Euclidean case, the Lipschitz space coincides with some Morrey-Compana to space [35] and can be characterized by mean oscillation which is due to DeVore and Sharpley [36], Janson et al. [35] and Harboure et al. [37] (see also Paluszyński [38]).

Lemma 6. *For and , then the homogeneous Lipschitz space coincides with the space .*

*Proof. *Using the property (b) of Lemma 5, we only need to prove that for some constant ,Now, we give the proof of the right side of inequality (42). Assume that . Given and , set and . Then we have the following equation:We estimate only the first term of the right side, since the others follow similar lines. Letting for and , then by Lemma 4, we show thatThus, we obtain .

On the other hand, for and , we have the following equation:Since and (b) of Lemma 2 implies that . Hence, by using above inequality, we can infer thatfor any . Thus, we have . This concludes the proof of the lemma.

In [34], Taibleson introduced and studied the Riesz potentials on local fields. Let us recall the definition of Riesz potentials on -adic vector space as follows:where with .

Let us recall the Hardy-Littlewood-Sobolev inequality for the Riesz potentials on -adic vector space. More details, the interested reader may refer to the book [34].

Lemma 7. *Let be a complex number with and such that . Then, the following statements are held.*(a)*If , then is bounded from to .*(b)*If , then is bounded from to .**Let be a complex number with and be locally integrable function on , the -adic version of fractional maximal function of is given by the following equation:**Using the definition of the -adic version of Riesz potential in equation (47), we give the following inequality:**So inequality (49) gives the following equation:**Using inequality (50) and Lemma 7, we give the boundedness of -adic version of fractional maximal function on the -adic vector space.*

Lemma 8. *Let be a complex number with and such that . Then, the following statement are held.*(a)*If , then is bounded from to .*(b)*If , then is bounded from to .**We now state the boundedness of on Morrey space which will be useful in proving our results. This lemma is similar to the results of ones from [39–41]. In Euclidean setting, Adams [23] and Spanne [42] studied the corresponding results, and see also [43].*

Lemma 9. *Let , and .*(1)*If , then is bounded from to .*(2)*If and , then is bounded from to .*

*Proof. *(1) Let and . Then, we split the following equation:First, we estimate . Using the definition of , we obtain the following equation:On the other hand, by Hölder’s inequality, we have that Combining equations (52) and (53) gives the following equation:Takingthen equation (54) becomes as follows: