The boundedness of commutators of rough p-adic fractional Hardy type operators on Herz-type spaces

In this paper, we obtain some inequalities about commutators of a rough p-adic fractional Hardy-type operator on Herz-type spaces when the symbol functions belong to two different function spaces.

For every non-zero rational number x there is a unique γ = γ (x) ∈ Z such that x = p γ m/n, where p ≥ 2 is a fixed prime number which is coprime to m, n ∈ Z. We define a mapping | · | p : Q → R + as follows: (1.1) The p-adic absolute value | · | p has many properties of the usual real norm | · | with an additional non-Archimedean property, |x + y| p ≤ max |x| p , |y| p .
The n-dimensional vector space Q n p , n ≥ 1, consists of the vectors x = (x 1 , x 2 , . . . , x n ), where x j ∈ Q p and j = 1, 2, . . . , n, with the following absolute value: For γ ∈ Z and a = (a 1 , a 2 , . . . , a n ) ∈ Q n p , we denote by the closed ball with the center a and radius p γ and by S γ (a) = x ∈ Q n p : |x -a| p = p γ the corresponding sphere. For a = 0, we write B γ (0) = B γ , and S γ (0) = S γ . It is easy to see that the equalities hold for all a 0 ∈ Q n p and γ ∈ Z. Since Q n p is a locally compact commutative group under addition, there exists a unique Haar measure dx on Q n p , such that where |B| h denotes the Haar measure of measurable subset B of Q n p . Furthermore, a simple calculation shows that B γ (a) h = p nγ and S γ (a) h = p nγ 1p -n hold for all a ∈ Q n p and γ ∈ Z. The one-dimensional Hardy operator where f : R + → R + is a measurable functions, was introduced by Hardy in [13]. This operator satisfies the inequality: where the constant q/(q -1) is sharp. In [7], Faris proposed an extension of the operator H on higher dimensional Euclidean space R n which is given by for x = (x 1 , . . . , x n ). In addition, Christ and Grafakos [4] obtained the exact value of the norm of operator H defined by (1.5). For boundedness results for these operators on function spaces we refer to some recent publications including [8,10,16,17,28,29,38].
On the other hand, the n-dimensional fractional p-adic Hardy operator |y| p ≤|x| p f (y) dy was defined and studied for f ∈ L loc 1 (Q n p ) and 0 ≤ α < n in [36]. When α = 0, the operator H p α transfers to the p-adic Hardy-type operator (see [10] for more details). Fu et al. in [9], fixed the optimal bounds of p-adic Hardy operator on L q (Q n p ). On the central Morrey space the p-adic Hardy-type operators and their commutators were discussed in [37]. In this connection see also [19,21,25].
There is still zero attention towards the rough Hardy operators on the p-adic linear spaces. Motivated by papers cited above and results of Fu et al. in [8], we define the special kind of p-adic rough fractional Hardy operator H p ,α and its commutators as follows.

Remark 1.2 Obviously
holds for every integer n ≥ 1 and prime p ≥ 2. Since the inclusion holds and Q n p is a linear space over field Q p , the product |y| p y is well defined. Moreover, if a non-zero y ∈ Q n p has the form y = (y 1 , . . . , y n ) and whenever y i = 0. Using (1.3) we obtain |y| p = p -γ i 0 . Now from (1.10) and (1.11) it follows that Thus, for every non-zero y ∈ Q n p , the vector |y| p y belongs to the sphere S 0 (0) = y ∈ Q n p : |y| p = 1 .
From (1.8) it directly follows that H p ,α ∈ R for every non-zero x ∈ Q n p and using (1.8), (1.9) we have ,α are well defined.
The aim of the current paper is to study the boundedness of H p,b ,α on p-adic Herz-type spaces by considering the symbol function b belonging to the p-adic CMO and Lipschitz spaces. In Euclidean space R n , Herz spaces and Morrey-Herz spaces were firstly introduced in [14] and [26], respectively. For more recent developments in the said spaces we mention the articles [15,27,39] and the references therein. Also, some operators with rough kernels defined on Euclidian space were recently studied on function spaces; see for example [11,12]. Before turning to our main results, let us recall the definitions of p-adic function spaces first. where
Obviously, the equalitiesK

CBMO estimates for commutators of p-adic rough fractional Hardy operator
The present section discusses the boundedness of p-adic rough fractional Hardy operator on p-adic Herz-type spaces. We begin this section with the following useful lemma.

Lemma 2.1 ([36]) Suppose b is a CMO 1 (Q n p ) function and suppose i, j ∈ Z. Then the inequality
holds.
Remark 2.2 From now on the letter C indicates a positive constant which may vary from line to line.
Proof of Theorem 2.3 For the sake of brevity, we write Note that 1 q 1 + 1 q 2 = α n and 1 Applying Hölder's inequality we have We use Hölder's inequality to estimate I 1 . We have In a similar fashion we can estimate II 2 . Using Hölder's inequality we have (2.6) From (2.3), (2.5) and (2.6) together with the Jensen inequality, we have For brevity, we may choose b CMO max{q 2 ,t} (Q n p ) = 1. Consequently, Case 1: When 0 < r 1 ≤ 1, we have . Case 2: When r 1 > 1, applying Hölder's inequality we get .
The proof of Theorem 2.3 is thus completed. holds for all ∈ L s (S 0 (0)), b ∈ CMO max{q 2 ,t} (Q n p ) and f ∈ L q 1 loc (Q n p ).
Proof of Theorem 3.3 The proof follows from standard analysis performed in our previous theorems. So, we omit the details.