$p$-adic singular integral and their commutator in generalized Morrey space

For a prime number $p,$ let $\mathbb{Q}_p$ be the field of $p$-adic numbers. In this paper, we established the boundedness of a class of $p$-adic singular integral operators on the $p$-adic generalized Morrey spaces. The corresponding boundedness for the commutators generalized by the $p$-adic singular integral operators and $p$-adic Lipschitz functions or $p$-adic generalized Campanato functions is also considered.


Introduction
Let p be a prime number and x ∈ Q. Then the non-Archimedean p-adic normal |x| p is defined as follows: if x = 0, |0| p = 0; if x = 0 is an arbitrary rational number with the unique representation x = p γ m n , where m, n are not divisible by p, γ = γ(x) ∈ Z, then |x| p = p −γ . This normal satisfies |xy| p = |x| p |y| p , |x + y| p ≤ max{|x| p , |y| p } and |x| p = 0 if and only if x = 0. Moreover, when |x| p = |y| p , we have |x + y| p = max{|x| p , |y| p }. Let Q p be the field of p-adic numbers, which is defined as the completion of the field of rational numbers Q with respect to the non-Archimedean p-adic normal | · | p . For γ ∈ Z, we denote the ball B γ (a) with center at a ∈ Q n p and radius p γ and its boundary S γ (a) by B γ (a) = {x ∈ Q n p : |x − a| p ≤ p γ }, S γ (a) = {x ∈ Q n p : |x − a| p = p γ }, respectively. It is easy to see that For n ∈ N, the space Q n p = Q p × · · · × Q p consists of all points x = (x 1 , · · · , x n ) where x i ∈ Q p , i = 1, . . . , n, n ≥ 1. The p-adic norm of Q n p is defined by Thus, it is easy to see that |x| p is a non-Archimedean norm on Q n p . The balls B γ (a) and the sphere S γ (a) in Q n p , γ ∈ Z are defined similar to the case n = 1.
Since Q n p is a locally compact commutative group under addition, thus from the standard analysis there exists the Haar measure dx on the additive group Q n p normalized by B 0 dx = |B 0 | H = 1, where |E| H denotes the Haar measure of a measurable set E ⊂ Q n p . Then by a simple calculation the Haar measures of any balls and spheres can be obtained. From the integral theory, it is easy to see that |B γ (a)| H = p nγ and |S γ (a)| H = p nγ (1 − p −n ) for any a ∈ Q n p . For a more complete introduction to the p-adic analysis, one can refer to [1,2,3,4,5,6,7,8] and the references therein.
The p-adic numbers have been applied in string theory, turbulence theory, statistical mechanics, quantum mechanics, and so forth(see [1,9,10] for detail). In the past few years, there is an increasing interest in the study of harmonic analysis on p-adic field (see [5,6,7,8] for detail).
Let Ω ∈ L ∞ (Q n p ), Ω(p j x) = Ω(x) for all j ∈ Z and |x|p=1 Ω(x)dx = 0. Then the p-adic singular integral operator defined by Taibleson [5] is as follows And the p-adic singular integral operator T is defined as the limit of T k when k goes to −∞.
Then the higher commutator generated by b and T k can be defined by And the commutator generated by − → b = (b 1 , b 2 , ..., b m ) and p-adic singular integral operator T is defined as the limit of T b k , when k goes to −∞.
Under some conditions, the authors in [5,11], obtained that T k were of type (q, q), 1 < q < ∞, and of weak type (1, 1) on local fields. In [12], Wu etal. established the boundedness of T k on p-adic central Morrey spaces. Furtherly, the λ-central BMO estimates for commutators of these singular integral operators on p-adic central Morrey spaces were obtained in [12]. Moreover, in p-adic linear space Q n p , Volosivets [13] gave the sufficient conditions for the maximal function and Riesz potential in p-adic generalized Morrey spaces. Mo etal. [14] established the boundedness of the commutators generated by the p-adic Riesz potential and p-adic generalized Campanato functions in p-adic generalized Morrey spaces.
Motivated by the works of [12,13,14], we are going to consider the boundedness of T k on the p-adic generalized Morrey type spaces, as well as the boundedness of the commutators generated by L k and p-adic generalized Campanato functions.
Throughout this paper, the letter C will be used to denote various constants and the various uses of the letter do not, however, denote the same constant. And, A B means that A ≤ CB, with some positive constant C independent of appropriate quantities.

Some notations and lemmas
Definition 2.1 [13] Let 1 ≤ q < ∞, and let ω(x) be a non-negative measurable function in Q n p . A function f ∈ L q loc (Q n p ) is said to belong to the generalized Morrey space GM q,ω (Q n p ), if where ω(B γ (a)) = Bγ (a) ω(x)dx.
Moreover, let λ ∈ R and 1 ≤ q < ∞. The p-adic central Morrey space CM q,λ (Q n p ) (see [8]) is defined by Definition 2.2 [17] Let 0 < β < 1 , then the p-adic Lipschitz space Λ β (Q n p ) is defined the set of all functions f : Q n p → C such that The classical Campanato spaces can be seen in [18], [19] and etc. The important particular case of GC q,ω (Q n p ) is CBMO q,λ (Q n p ), where for 1 < q < ∞ and 0 < λ < 1/n. And the central BMO space CBMO q,λ (Q n p ) is defined by for j, k ∈ Z and any fixed a ∈ Q n p .
Thus, for j > k, from Lemma 2.1, it deduce that for k ∈ Z, where C is independent of f and k ∈ Z. Furthermore, Moreover, on the p−adic field, the Riesz potential I p α is defined by [14] Let α be a complex number with 0 < Reα < n, and let 1 < r < ∞, 1 < q < n/Reα, 0 < 1/r = 1/q − Reα/n. Suppose that both ω and ν are non-negative measurable functions, such that for any a ∈ Q n p and γ ∈ Z. Then the Riesz potential I α p is bounded from GM q,ν to GM r,ω .

Main results
In this section, let us state the main results of the paper.
Suppose that both ω and ν are non-negative measurable functions, such that for any γ ∈ Z and a ∈ Q n p . Then the singular integral operators T k are bounded from Then the operators T k and T are bounded on the space CM q,λ for all k ∈ Z.
In fact, for λ < 0. Taking ω(B) = ν(B) = |B| λ H in Theorem 3.1, we can obtain the Corollary 3.1. If the Morrey space M q,λ (Q n p ) is replaced by the central Morrey space CM q,λ (Q n p ) in Corollary 3.1, then the conclusion is that of Theorem 4.1 in [12].

Proof of Theorem 3.1-3.3
Let us give the proof of Theorem 3.1, firstly.
For any fixed γ ∈ Z and a ∈ Q n p , it is easy to see that where B c γ (a) is the complement to B γ (a) in Q n p .
Since x ∈ B γ (a) and Ω ∈ L ∞ (Q n p ), then we have Combining the estimates of (4.1), (4.2) and (4.4), we have which means that T k is bounded from GM q,ν to GM q,ω .
Moreover, from Lemma 2.2 and the definition of GM q,ω (Q n p ), it is obvious that T (f ) = lim k→−∞ T k (f ) exists in GM q,ω and the operator T is bounded from GM q,ν to GM q,ω .

Proof of Theorem 3.2.
For any x ∈ Q n p , since Ω ∈ L ∞ (Q n p ), and b i ∈ Λ β i , i = 1, 2, . . . , m, then it is easy to see Thus, from Lemma 2.2 it is obvious that the commutators T b k are bounded from GM q,ν to GM r,ω , for all k ∈ Z.
Moreover, from the definition of GM q,ω (Q n p ), it is obvious that exists in the space of GM q,ω , and the commutator T b is bounded from GM q,ν to GM q,ω .
Proof of Theorem 3.3 Without loss of generality, we need only to show that the conclusion holds for m = 2.
For any fixed γ ∈ Z and a ∈ Q n , we write f 0 = f χ Bγ(a) and f ∞ = f χ B c γ (a) , then In the following, we will estimate every part, respectively.
Moreover, from Lemma 2.2 and the definition of GM q,ω (Q n p ), it is obvious that T b (f ) = lim k→−∞ T b k (f ) exists in the space of GM q,ω , and the commutator T b is bounded from GM q,ν to GM q,ω .
Therefore, the proof of Theorem 3.3 is complete.