Commutators of Hardy-Littlewood operators on p -adic function spaces with variable exponents

: In this article, we obtain some su ﬃ cient conditions for the boundedness of commutators of p -adic Hardy-Littlewood operators with symbols in central bounded mean oscillation space and Lipschitz space on the p -adic function spaces with variable exponents such as the p -adic local central Morrey, p -adic Morrey-Herz, and p -adic local block spaces with variable exponents.


Introduction
The theory of functions from p n into plays a significant role in p-adic probability, p-adic quantum mechanics, p-adic harmonic analysis, and p-adic partial differential equations [1][2][3][4][5][6][7][8][9]. Furthermore, there has been a lot of interest in the theory of p-adic operators. For instance, the p-adic Hardy-Littlewood operator studied by Rim and Lee [10]  We now discuss the commutator of p-adic Hardy-Littlewood operator as follows: Here, b and f are continuous functions, u is a solution, and the Vladimirov operator D α [1,3,5,6] is defined by In case of = b 1, the readers can find equation (1) in [16]. Moreover, a generalization of equation (1) is the fractional p-adic Brownian motion equation [6,Section 5.3]. It is well known that the equation (1) has a radial solution u as follows: 1 . Variable exponent spaces have some essential applications in image processing, harmonic analysis, and differential equations [17][18][19][20][21][22][23]. In 2009, Izuki [24] introduced the variable exponent Morrey-Herz spaces. Next, the nonhomogeneous central Morrey spaces of variable exponent are defined in [25]. Besides, Yee et al. [26] stated the local block space with variable exponent ⋅ u p n ,

LB
( ) ( ) as follows: , defined by the central variable exponent BMO space on the p-adic above spaces is also discussed. Section 2 introduces the p-adic Lebesgue, p-adic local central Morrey, p-adic Morrey-Herz, p-adic local block, p-adic central bounded mean oscillation spaces with variable exponents, and the p-adic Lipschitz spaces. In Section 3, our main results are presented.

Preliminaries
Let us fix a prime number p. On the field of rational numbers , we set the p-adic norm ⋅ p | | as follows: if = x 0, The completion of the field with the p-adic norm ⋅ p | | leads to the field p of p-adic numbers. Then x y x y , , p n ( ) be the set of all nonnegative weighted functions on p n , and be the norm of between two normed vector spaces X and Y . Let B a k ( ) and S a k ( ) define, respectively, the ball and sphere of p n with a center at ∈ a p n and radius p k : a n d : .
Besides, χ k is the characteristic function of the sphere S k . As is known, there exists a Haar measure x d on p n , which is unique up to positive constant multiple and is translation invariant. We normalize the measure x d , The space For more details, the readers can find the book [8].
The norm is given by For ∈ q p n ( ), we define ′ ⋅ q ( ) as follows: is given by satisfying at the origin, Denote by ∞ C p n log ( ) the set of all log-Hölder continuous functions ⋅ → α : p n ( ) satisfying at infinity, , for all , x p ( ) Next, we would like to introduce the p-adic local central Morrey and p-adic Morrey-Herz spaces with variable exponents (see [21,22] for the real field).
The p-adic variable exponent Morrey-Herz space By Theorem 2.4, we obtain the following lemma.
Following [26], we state the p-adic variable exponent local block space.
The p-adic variable exponent local block space Because of the proof of [26,Theorem 4], we have the following result.  (see [28] for the real field) and the p-adic Lipschitz space Lip β p n ( ) are introduced as follows.
By Lemmas 1 and 2 in [29], we obtain the following lemma.

Main results and their proofs
For simplicity, we denote , for almost everywhere supp . , on the p-adic local central Morrey space with variable exponent.
Next, by putting = t m log p p | | , Moreover, we immediately have By ∈ r p n 1 B ( ) and Lemma 2.13.(ii), for any ∈ k , Thus, by ≤ m 0 and ∈ m , By the formula for change of variables and Next, by considering . Consequently, by the inequalities (10)- (14), for any ∈ k , Hence, for any ≤ k 0 and ∈ k , By the relations (4) and (9) This completes our proof. □ Proof. For any ∈ k , by the Minkowski inequality, For ∈ ∞ η 0, ( ) and . Combining this with (16), From this, for any ≤ k 0 and ∈ k , . Hence, Proof. By (15) and (4), for any ∈ k , From this, by (20), Here, By Theorem 2.4, we have Similarly,