Characterization of Lipschitz Spaces via Commutators of Fractional Maximal Function on the $p$-Adic Variable Exponent Lebesgue Spaces

In this paper, the main aim is to give some characterizations of the boundedness of the maximal or nonlinear commutator of the $p$-adic fractional maximal operator $ \mathcal{M}_{\alpha}^{p}$ with the symbols belong to the $p$-adic Lipschitz spaces in the context of the $p$-adic version of variable Lebesgue spaces, by which some new characterizations of the Lipschitz spaces and nonnegative Lipschitz functions are obtained in the $p$-adic field context. Meanwhile, Some equivalent relations between the $p$-adic Lipschitz norm and the $p$-adic variable Lebesgue norm are also given.


Introduction and main results
During the last several decades, the p-adic analysis has attracted extensive attention due to its important applications in mathematical physics, science and technology, for instance, p-adic harmonic analysis, p-adic pseudo-differential equations, p-adic wavelet theory, etc (see [2,19,26,29]).Moreover, the theory of variable exponent function spaces has been intensely investigated in the past twenty years since some elementary properties were established by Kováčik and Rákosník in [21].
It is worthwhile to note that the fractional maximal operator plays an important role in real and harmonic analysis and applications, such as potential theory and partial differential equations (PDEs), since it is intimately related to the Riesz potential operator, which is a powerful tool in the study of the smooth function spaces (see [4,6,13]).
On the other hand, there are two major reasons why the study of the commutators has got widespread attention.The first one is that the boundedness of commutators can produce some characterizations of function spaces [18,24].The other one is that the theory of commutators is intimately related to the regularity properties of the solutions of certain PDEs [5,9,12,25].
Let T be the classical singular integral operator.The Coifman-Rochberg-Weiss type commutator [b, T ] generated by T and a suitable function b is defined by [b, T ]f = bT (f ) − T (bf ). (1.1) A well-known result shows that [b, T ] is bounded on L s (R n ) for 1 < s < ∞ if and only if b ∈ BMO(R n ) (the space of bounded mean oscillation functions).The sufficiency was provided by (author?) [10] and the necessity was obtained by (author?) [18].Furthermore, (author?) [18] also established some characterizations of the Lipschitz space Λ β (R n ) via commutator (1.1) and proved that [b, T ] is bounded from L s (R n ) to L q (R n ) for 1 < s < n/β and 1/s − 1/q = β/n with 0 < β < 1 if and only if b ∈ Λ β (R n ) (see also (author?) [24]).Denote by N, Z, Q and R the sets of positive integers, integers, rational numbers and real numbers, respectively.For γ ∈ Z and a prime number p, let Q n p be a vector space over the p-adic field Q p , B γ (x) denote a p-adic ball with center x ∈ Q n p and radius p γ ( for the notations and notions, see Section 2 below).Let 0 ≤ α < n, for a locally integrable function f , the p-adic fractional maximal function of f is defined by |f (y)|dy, where the supremum is taken over all p-adic balls B γ (x) ⊂ Q n p and |E| h represents the Haar measure of a measurable set E ⊂ Q n p .When α = 0, we simply write M p instead of M p 0 , which is the p-adic Hardy-Littlewood maximal function defined as |f (y)|dy.
The reader can refer to Stein [27] for the definition on the Euclidean case.Similar to (1.1), we can define two different kinds of commutator of the fractional maximal function as follows.
Definition 1.1 Let 0 ≤ α < n and b be a locally integrable function on Q n p .
(1) The maximal commutator of M p α with b is given by where the supremum is taken over all p-adic balls B γ (x) ⊂ Q n p .
(2) The nonlinear commutators generated by M p α and b is defined by When α = 0, we simply denote by [b, the nonlinear commutator because it is not even a sublinear operator, although the commutator [b, T ] is a linear one.It is worth noting that the nonlinear commutator [b, M p α ] and the maximal commutator M p α,b essentially differ from each other.For example, M p α,b is positive and sublinear, but [b, M p α ] is neither positive nor sublinear Denote by M and M α the classical Hardy-Littlewood maximal function and the fractional maximal function in R n respectively.In fact, the nonlinear commutator [b, M ] and [b, M α ] have been studied by many authors in the Euclidean spaces, for instance, [1,3,14,23,32,33,[35][36][37][38] etc.When the symbol b belongs to BMO(R n ), (author?) [3] studied the necessary and sufficient conditions for the boundedness of [b, M ] in L q (R n ) for 1 < q < ∞.Zhang and Wu obtained similar results for the fractional maximal function in [35] and extended the mentioned results to variable exponent Lebesgue spaces in [36,37].When the symbol b belongs to Lipschitz spaces, (author?) [34,38] gave the necessary and sufficient conditions for the boundedness of [b, M α ] on Orlicz spaces and variable Lebesgue spaces respectively.And recently, (author?) [31] considered some new characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of fractional maximal functions or fractional maximal commutators in the context of the variable Lebesgue spaces.
On the other hand, (author?) [15] gave the characterization of p-adic Lipschitz spaces in terms of the boundedness of commutators of maximal function M p in the context of the p-adic Lebesgue spaces and Morrey spaces when the symbols b belong to p-adic Lipschitz spaces Λ β (Q n p ).And (author?) [8] prove the boundedness of the fractional maximal and the fractional integral operator in the p-adic variable exponent Lebesgue spaces.
Inspired by the above literature, we focus on the case of p-adic fields Q p , in some sense it can also be pointed out that our work was motivated by the standard harmonic analysis on the Euclidean space, the purpose of this paper is to study the boundedness of the p-adic fractional maximal commutator M p α,b or the nonlinear commutator [b, M p α ] generated by p-adic fractional maximal function M p α over p-adic variable exponent Lebesgue spaces, where the symbols b belong to the p-adic Lipschitz spaces, by which some new characterizations of the p-adic version of Lipschitz spaces are given.
Let α ≥ 0, for a fixed p-adic ball B * , the fractional maximal function with respect to B * of a locally integrable function f is given by where the supremum is taken over all the p-adic ball B γ (x) with B γ (x) ⊂ B * for a fixed p-adic ball B * .When α = 0, we simply write M p B * instead of M p 0,B * .Our main results can be stated as follows, which are to study the boundedness of M p α,b and [b, M p α ] in the context of p-adic variable exponent Lebesgue spaces when the symbol belongs to a p-adic version of Lipschitz spaces Λ β (Q n p ) (see Section 2 below).And some new characterizations of the Lipschitz spaces via such commutators are given.Theorem 1.1 Let 0 < β < 1, 0 < α < α + β < n and b be a locally integrable function on Q n p .Then the following assertions are equivalent: For the case of r(•) and q(•) being constants, we have the following result from Theorem 1.1, which is new even for this case.
(ii) Moreover, it was proved in Theorem 4 of [15], see also Lemma 2.7 below, that b ∈ Λ β (Q n p ) and b ≥ 0 if and only if Compared with (1.4), (1.3) gives a new characterization for nonnegative Lipschitz functions.
In particular, when α = 0, the results are also true come from Theorem 1.1 and Corollary 1.1.Now we only give the case in the context of the p-adic version of variable exponent Lebesgue spaces, and it is new.Corollary 1.2.Let 0 < β < 1 and b be a locally integrable function on Q n p .Then the following statements are equivalent: Then the following assertions are equivalent: When r(•) and q(•) are constants, we get the following result from Theorem 1.2.
Finally, we give the follows result, which is valid and new, from Theorem 1.2 with α = 0.
Corollary 1.4.Let 0 < β < 1 and b be a locally integrable function on Q n p .Then the following statements are equivalent: (C.4)There exists some r(•), q(•) Throughout this paper, the letter C always stands for a constant independent of the main parameters involved and whose value may differ from line to line.In addition, we give some notations.Here and hereafter |E| h will always denote the Haar measure of a measurable set E on Q n p and by χ E denotes the characteristic function of a measurable set E ⊂ Q n p .

Preliminaries and lemmas
To prove the main results of this paper, we first recall some necessary notions and remarks.

p-adic field Q p
Firstly, we introduce some basic and necessary notations for the p-adic field.Let p ≥ 2 be a fixed prime number in Z and G p = {0, 1, . . ., p − 1}.For every non-zero rational number x, by the unique factorization theorem, there is a unique γ = γ(x) ∈ Z, such that x = p γ m n , where m, n ∈ Z are not divisible by p (i.e.p is coprime to m, n).Define the mapping | • | p : Q → R + as follows: The p-adic absolute value Denote by Q p the p-adic field which is defined as the completion of the field of rational numbers Q with respect to the p-adic absolute value | • | p .From the standard p-adic analysis, any non-zero element x ∈ Q p can be uniquely represented as a canonical series form where a j ∈ G p and a 0 0, and γ = γ(x) ∈ Z is called as the p-adic valuation of x.The series converges in the p-adic absolute value since the inequality |a j p j | p ≤ p −j holds for all j ∈ N.
Moreover, the n-dimensional p-adic vector space , consists of all points x = (x 1 , . . ., x n ), where x i ∈ Q p (i = 1, . . ., n), equipped with the following absolute value the closed ball with the center at a and radius p γ and by the corresponding sphere.For a = 0, we write B γ (0) = B γ , and and It is easy to see that the equalities hold for all a 0 ∈ Q n p and γ ∈ Z.It follows from non-Archimedean triangle inequality that two balls B γ (x) and B γ ′ (y) either do not intersect or one of them is contained in the other, which differ from those of the Euclidean case.And note that in the second case under conditions γ = γ ′ these balls are equal.The above properties can also be found in [20] (see Lemma 3.1).
The p-adic balls have the following properties: Since Q n p is a locally compact commutative group with respect to addition, there exists a unique Haar measure dx on Q n p (up to positive constant multiple) which is translation invariant (i.e., d(x + a) = dx), such that where |E| h denotes the Haar measure of measurable subset E of Q n p .Furthermore, from this integral theory, it is easy to obtain that and hold for all a ∈ Q n p and γ ∈ Z.For more information about the p-adic field, we refer readers to [28,30].

p-adic function spaces
In what follows, we say that a real-valued measurable function f defined on and denote by L ∞ (Q n p ) the set of all measurable real-valued functions f on Here, the integral in equation (2.2) is defined as follows: |f (x)| q dx, if the limit exists.
Some often used computational principles are worth noting.In particular, if f ∈ L 1 (Q n p ), then where t ∈ Q p \ {0}, tx = (tx 1 , . . ., tx n ) and d(tx) = |t| n p dx.We now introduce the notion of p-adic variable exponent Lebesgue spaces and give some properties needed in the sequel (see [7] for the respective proofs).
We say that a measurable function q(•) is a variable exponent if q(•) : Definition 2.1 Given a measurable function q(•) defined on Q n p , we denote by q − := ess inf x∈Q n p q(x), q + := ess sup (1) q ′ − = ess inf (2) Denote by P 0 (Q n p ) the set of all measurable functions q(•) : (3) Denote by P 1 (Q n p ) the set of all measurable functions q(• ) : (4) Denote by P(Q n p ) the set of all measurable functions q(•) : (5) The set B(Q n p ) consists of all measurable functions q(•) ∈ P(Q n p ) satisfying that the Hardy-Littlewood maximal operator M p is bounded on L q(•) (Q n p ).

Definition 2.2 (p-adic variable exponent Lebesgue spaces) Let q(•) ∈ P(Q n p ). Define the p-adic variable exponent Lebesgue spaces
where ) is a Banach function space with respect to the Luxemburg norm dx ≤ 1 .
(1) Denote by C log 0 (Q n p ) the set of all q(• ) which satisfies for all γ ∈ Z and any x ∈ Q n p , where C denotes a universal positive constant.
(2) The set C log ∞ (Q n p ) consists of all q(•) which satisfies for any x, y ∈ Q n p , where C is a positive constant.

the set of all global log-Hölder continuous functions q(• ).
In what follows, we denote C (Q n p ) P(Q n p ) by P log (Q n p ).And for a function b defined on (2) For 1 ≤ q < ∞, the p-adic version of Lipschitz spaces Lip q β (Q n p ) is the set of all measurable functions f on Q n p such that f (y)dy.In particular, (ii) Since Λ β (R n ) and BMO(R n ) are Campanato space when 0 < β < 1 and β = 0,respectively.Thus, in this sense, the space BMO(Q n p ) can be seen as a limit case of Λ β (Q n p ) as β → 0.

Auxiliary propositions and lemmas
The first part of Lemma 2.2 may be found in [7] (see Theorem 5.2).By elementary calculations, the second of Lemma 2.2 can be obtained as well.
Lemma 2.2 Let q(•) ∈ P(Q n p ). (1 (2) The following conditions are equivalent: where r ′ stand for the conjugate exponent of r, viz.,

for the Euclidean case see Remark 2.13 of [11] for more details).
We now present the following result related to the Hölder's inequality.The part ( 1) is known as the Hölder's inequality on Lebesgue spaces over p-adic vector space Q n p .And similar to the Euclidean case, the part (2) can be deduced by simple calculations (or see Lemma 3.8 in [7]).

Lemma 2.3 (Generalized Hölder's inequality on Q n
p ) Let Q n p be an n-dimensional p-adic vector space.
(1) Suppose that 1 ≤ q ≤ ∞ with 1 q + 1 q ′ = 1, and measurable functions f ∈ L q (Q n p ) and g ∈ L q ′ (Q n p ). Then there exists a positive constant C such that (2) Suppose that q 1 (•), almost everywhere.Then there exists a positive constant C such that the inequality (3) When r(•) = 1 in (2) as mentioned above, we have q 1 (•), q 2 (•) ∈ P(Q n p ) and 1 q1(•) + 1 q2(•) = 1 almost everywhere.Then there exists a positive constant C such that the inequality The following results for the characteristic function χ Bγ (x) are required as well.By elementary calculations, the first part may be obtain from the p-adic integral theory (or refer to (2.1)).The second part may be founded in [8] (see Lemma 7), and the part (4) follows from (2) of Lemma 2.3.Moreover, according to Lemma 2.2 and (2) of Lemma 2.4, the third part can also be deduced by simple calculations.So, we omit the proofs.
Lemma 2.4 (Norms of characteristic functions) Let Q n p be an n-dimensional p-adic vector space.
(1) If 1 ≤ q < ∞.Then there exist a constant C > 0 such that where Then there exist a constant C > 0 such that holds for all p-adic ball B γ (x) ⊂ Q n p .
(1) Then the norm From the proof of Theorem 4 in [15], we can obtain the following characterization of nonnegative Lipschitz functions.
Now we recall the Hardy-Littlewood-Sobolev inequality for the fractional maximal function M p α on padic vector space.The first part of Lemma 2.8 can be founded in [15] (see Lemma 8), and the second comes from Theorem 4 of [8].
Lemma 2.8 Let 0 < α < n and Q n p be an n-dimensional p-adic vector space.
(1) If By (1) of Lemma 2.8, if 0 < α < n, 1 < r < n/α and f ∈ L r (Q n p ), then M p α (f )(x) < ∞ almost everywhere.A similar result is also valid in variable Lebesgue spaces.And the method of proof can refer to Lemma 2.6 in [34], so we omit its proof.
Now, we give the following pointwise estimate for [b, Finally, we also need the following results.Similar to the proof of Lemma 2.3 in [35], and referring to the course of the proof of Theorem 1.4 in [16], using Lemma 2.1 and the properties of p-adic ball, through elementary calculations and derivations, the following assertions can be obtained.Hence, we omit the proofs.Lemma 2.11 Let b be a locally integrable function and Q n p be an n-dimensional p-adic vector space.For any fixed p-adic ball B γ (x) ⊂ Q n p .

Proofs of the main results
Now we give the proofs of the Theorem 1.1 and Theorem 1.2.

Proof of Theorem 1.1
To prove Theorem 1.1, we first prove the following lemma.
Then, for any y ∈ B γ (x), we have

By using assertion (A.3) and (4) of Lemma 2.4 (norms of characteristic functions), we have
(A.4) = =⇒ (A.1):By Lemma 2.7, it suffices to prove For any fixed p-adic ball B γ (x) ⊂ Q n p , we have where the constant C is independent of B γ (x).Now we consider I 2 .For all y ∈ B γ (x), it follows from Lemma 2.11 that Then, for any y ∈ B γ (x), we get This shows that b ∈ Λ β (Q n p ) by Lemma 2.6 and Definition 2.4 since the constant C is independent of B γ (x).
The proof of Theorem 1.2 is finished.

Definition 2 . 4 (
p-adic Lipschitz space) (1) Assume that 0 < β < 1.The p-adic version of homogeneous Lipschitz spaces Λ β (Q n p ) is the set of all measurable functions f on Q n p with the finite norm f Λ β (Q n p ) = sup x,y∈Q n p x y |f (x) − f (y)| |x − y| β p .

Lemma 3 . 1
Let 0 < α < n, 0 < β < 1 and b be a locally integrable function on Q n p .If there exists a positive constant C such that sup γ∈Z x∈Q n p 1