# p-adic singular integrals and their commutators in generalized Morrey spaces

• Hui X. Mo
• Zhe Han
• Liu Yang
• Xiao J. Wang
Open Access
Research

## Abstract

For a prime number p, let $$\mathbb{Q}_{p}$$ be the field of p-adic numbers. In this paper, we establish the boundedness of a class of p-adic singular integral operators on the p-adic generalized Morrey spaces. We also consider 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.

42B20 42B25

## 1 Introduction

Let p be a prime number, and let $$x\in \mathbb{Q}$$. Then the non-Archimedean p-adic norm $$|x|_{p}$$ is defined as follows: if $$x=0$$, then $$|0|_{p}=0$$; if $$x\neq 0$$ is an arbitrary rational number with unique representation $$x=p^{\gamma }\frac{m}{n}$$, where m, n are not divisible by p, and $$\gamma =\gamma (x)\in \mathbb{Z}$$, then $$|x|_{p}=p^{-\gamma }$$. This norm has the following properties: $$|xy|_{p}=|x|_{p}|y|_{p}$$, $$|x+y|_{p}\leq \max \{|x|_{p}, |y|_{p}\}$$, and $$|x|_{p}=0$$ if and only if $$x=0$$. Moreover, when $$|x|_{p}\neq |y|_{p}$$, we have $$|x+y|_{p}=\max \{|x|_{p}, |y|_{p}\}$$. Let $$\mathbb{Q}_{p}$$ be the field of p-adic numbers defined as the completion of the field of rational numbers $$\mathbb{Q}$$ with respect to the non-Archimedean p-adic norm $$|\cdot |_{p}$$. For $$\gamma \in \mathbb{Z}$$, we denote the ball $$B_{\gamma }(a)$$ with center at $$a\in \mathbb{Q}_{p}$$ and radius $$p^{\gamma }$$ and its boundary $$S_{\gamma }(a)$$ by
\begin{aligned}& B_{\gamma }(a)=\bigl\{ x\in \mathbb{Q}_{p}: \vert x-a \vert _{p}\leq p^{\gamma }\bigr\} , \qquad S_{\gamma }(a)=\bigl\{ x\in \mathbb{Q}_{p}: \vert x-a \vert _{p}=p^{\gamma }\bigr\} , \end{aligned}
respectively. It is easy to see that
$$B_{\gamma }(a)=\bigcup_{k\leq \gamma }S_{k}(a).$$
For $$n\in \mathbb{N}$$, the space $$\mathbb{Q}_{p}^{n}=\mathbb{Q}_{p} \times \cdots \times \mathbb{Q}_{p}$$ consists of all points $${x}=(x_{1},\ldots , x_{n})$$ where $$x_{i}\in \mathbb{Q}_{p}$$, $$i=1,\dots ,n$$, $$n\geq 1$$. The p-adic norm of $$\mathbb{Q}_{p}^{n}$$ is defined by
$$\vert {x} \vert _{p}=\max_{1\leq i\leq n}{ \vert x_{i} \vert _{p}}, \quad {x}\in \mathbb{Q}_{p}^{n}.$$
Thus it is easy to see that $$|{x}|_{p}$$ is a non-Archimedean norm on $$\mathbb{Q}_{p}^{n}$$. The balls $$B_{\gamma }({a})$$ and the sphere $$S_{\gamma }({a})$$ in $$\mathbb{Q}_{p}^{n}$$ for $$\gamma \in \mathbb{Z}$$ are defined similarly to the case $$n=1$$.

Since $$\mathbb{Q}_{p}^{n}$$ is a locally compact commutative group under addition, by the standard analysis there exists the Haar measure dx on the additive group $$\mathbb{Q}_{p}^{n}$$ normalized by $$\int _{B_{0}}dx=|B _{0}|_{H}=1$$, where $$|E|_{H}$$ denotes the Haar measure of a measurable set $$E\subset \mathbb{Q}_{p}^{n}$$. 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 $$|B_{\gamma }({a})|_{H}=p^{n\gamma }$$ and $$|S_{\gamma }({a})|_{H}=p^{n\gamma }(1-p^{-n})$$ for any $${a}\in \mathbb{Q}_{p}^{n}$$. For a more complete introduction to the p-adic analysis, we refer to [1, 2, 3, 4, 5, 6, 7, 8] and the references therein.

The p-adic numbers have been applied in the 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 $$\varOmega \in L^{\infty }(\mathbb{Q}_{p}^{n})$$ be such that $$\varOmega (p^{j}x)=\varOmega (x)$$ for all $$j\in \mathbb{Z}$$ and $$\int _{|x|_{p}=1}\varOmega (x)\,dx=0$$. Then the p-adic singular integral operators defined by Taibleson [5] are as follows:
$$T_{k}(f) (x)= \int _{ \vert y \vert _{p}>p^{k}}f(x-y)\frac{\varOmega (y)}{ \vert y \vert _{p}^{n}}\,dz \quad \text{for } k\in \mathbb{Z}.$$
The p-adic singular integral operator T is defined as the limit of $$T_{k}$$ as k goes to −∞.
Moreover, let $$\overrightarrow{b}=(b_{1},b_{2},\ldots,b_{m})$$, where $$b_{i}\in L_{\mathrm{loc}}{(\mathbb{Q}_{p}^{n})}$$ for $$1\leq i\leq m$$. Then the higher commutator generated by b⃗ and $$T_{k}$$ can be defined as
$$T_{k}^{\vec{b}}f(x)= \int _{ \vert y \vert _{p}>p^{k}}\prod_{i=1}^{m} \bigl(b_{i}(x)-b _{i}(x-y)\bigr)f(x-y)\frac{\varOmega (y)}{ \vert y \vert _{p}^{n}}\,dz \quad \text{for } k\in \mathbb{Z},$$
and the commutator generated by $$\overrightarrow{b}=(b_{1},b_{2},\ldots,b _{m})$$ and the p-adic singular integral operator T is defined as the limit of $$T_{k}^{\vec{b}}$$ as k goes to −∞.

Under some conditions, the authors in [5, 11] showed that $$T_{k}$$ were of type $$(q, q)$$ for $$1 < q < \infty$$ and of weak type $$(1,1)$$ on local fields. Wu et al. [12] established the boundedness of $$T_{k}$$ on p-adic central Morrey spaces. Furthermore, the λ-central BMO estimates for commutators of these singular integral operators on p-adic central Morrey spaces were obtained in [12]. Moreover, in the p-adic linear space $$\mathbb{Q}_{p} ^{n}$$, Volosivets [13] gave sufficient conditions for the boundedness of the maximal function and Riesz potential in p-adic generalized Morrey spaces. Mo et al. [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 consider the boundedness of $$T_{k}$$ on the p-adic generalized Morrey type spaces, as well as the boundedness of the commutators generated by $$T_{k}$$ and p-adic generalized Campanato functions.

Throughout this paper, the letter C will be used to denote constants varying from line to line. The relation $$A\lesssim B$$ means that $$A\leq CB$$ with some positive constant C independent of appropriate quantities.

## 2 Some notation and lemmas

### Definition 2.1

([13])

Let $$1\leq q<\infty$$, and let $$\omega (x)$$ be a nonnegative measurable function in $$\mathbb{Q}_{p} ^{n}$$. A function $$f\in L^{q}_{\mathrm{loc}}(\mathbb{Q}_{p}^{n} )$$ is said to belong to the generalized Morrey space $$GM_{q,\omega }(\mathbb{Q}_{p} ^{n} )$$ if
$$\Vert f \Vert _{GM_{q,\omega }}=\sup_{a\in \mathbb{Q}^{n}_{p},\gamma \in \mathbb{Z}}\frac{1}{ \omega (B_{\gamma }(a))} \biggl(\frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert f(y) \bigr\vert ^{q}\,dy \biggr)^{1/q}< \infty ,$$
where $$\omega (B_{\gamma }(a))=\int _{B_{\gamma }(a)}\omega (x)\,dx$$.

Let $$\lambda \in \mathbb{R}$$. If $$\omega (B_{\gamma }(a))=|B_{\gamma }(a)|^{\lambda }$$, then $$GM_{q,\omega }(\mathbb{Q}_{p}^{n} )$$ is the classical Morrey space $$M_{q,\lambda }(\mathbb{Q}_{p}^{n} )$$. About the generalized Morrey space, see [15], and for the classical Morrey spaces, see [16] and so on.

Moreover, let $$\lambda \in \mathbb{R}$$ and $$1\leq q<\infty$$. The p-adic central Morrey space $$CM_{q,\lambda }(\mathbb{Q}^{n}_{p})$$ (see [8]) is defined by
$$\Vert f \Vert _{CM_{q,\lambda }}=\sup_{\gamma \in \mathbb{Z}} \biggl( \frac{1}{ \vert B _{\gamma }(0) \vert _{H}^{1+\lambda q}} \int _{B_{\gamma }(0)} \bigl\vert f(y) \bigr\vert ^{q}\,dy \biggr)^{1/q}< \infty .$$

### Definition 2.2

([17])

For $$0<\beta <1$$, the the p-adic Lipschitz space $$\varLambda _{\beta }(\mathbb{Q}^{n}_{p})$$ is defined as the set of all functions $$f: \mathbb{Q}_{p}^{n}\mapsto \mathbb{C}$$ such that
$$\Vert f \Vert _{\varLambda _{\beta }(\mathbb{Q}_{p}^{n})}=\sup_{x,h\in \mathbb{Q}_{p}^{n}, h\neq 0} \frac{ \vert f(x+h)-f(x) \vert }{ \vert h \vert ^{ \beta}_{p}} < \infty .$$

### Definition 2.3

([13])

Let B be a ball in $$\mathbb{Q} _{p}^{n}$$, $$1\leq q<\infty$$, and let $$\omega (x)$$ be a nonnegative measurable function in $$\mathbb{Q}_{p}^{n}$$. A function $$f\in L^{q} _{\mathrm{loc}}(\mathbb{Q}_{p}^{n} )$$ is said to belong to the generalized Campanato space $$GC_{q,\omega }(\mathbb{Q}_{p}^{n} )$$ if
$$\Vert f \Vert _{GC_{q,\omega }}=\sup_{a\in \mathbb{Q}^{n}_{p},\gamma \in \mathbb{Z}}\frac{1}{ \omega (B_{\gamma }(a))} \biggl(\frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert f(y)-f_{B_{\gamma }(a)} \bigr\vert ^{q}\,dy \biggr)^{1/q}< \infty ,$$
where $$f_{B_{\gamma }(a)}=\frac{1}{|B_{\gamma }(a)|_{H}} \int _{B_{\gamma }(a)}f(x)\,dx$$ and $$\omega (B_{\gamma }(a))= \int _{B_{\gamma }(a)}\omega (x)\,dx$$.
The classical Campanato spaces can be found in [18, 19], and so on. The important particular case of $$GC_{q,\omega }( \mathbb{Q}_{p}^{n} )$$ is $$BMO_{q,\lambda }(\mathbb{Q}_{p}^{n} )$$, where $$1< q<\infty$$ and $$0<\lambda <1/n$$. The central BMO space $$CBMO_{q, \lambda }(\mathbb{Q}_{p}^{n} )$$ is defined by
$$\Vert f \Vert _{CBMO^{q,\lambda }(\mathbb{Q}_{p}^{n} )}=\sup_{\gamma \in \mathbb{Z}}\frac{1}{ \vert B_{\gamma }(0) \vert _{H}^{\lambda }} \biggl(\frac{1}{ \vert B_{\gamma }(0) \vert _{H}} \int _{B_{\gamma }(0)} \bigl\vert f(y)-f _{B_{\gamma }(0)} \bigr\vert ^{q}\,dy \biggr)^{1/q}< \infty .$$
(2.1)

### Lemma 2.1

([14])

Let$$1\leq q<\infty$$, and letωbe a nonnegative measurable function. Let$$b\in GC_{q, \omega }(\mathbb{Q}_{p}^{n} )$$. Then
$$\vert b_{B_{k}(a)}-b_{B_{j}(a)} \vert \leq \Vert b \Vert _{GC_{q,\omega }} \vert j-k \vert \max \bigl\{ \omega \bigl(B_{k}(a) \bigr),\omega \bigl(B_{j}(a)\bigr)\bigr\}$$
for$$j,k\in \mathbb{Z}$$and any fixed$$a\in \mathbb{Q}_{p}^{n}$$.
Thus, for $$j>k$$, from Lemma 2.1 we deduce that
$$\biggl( \int _{B_{j}(a)} \bigl\vert b(y)-b_{B_{k}(a)} \bigr\vert ^{q}\,dy \biggr)^{1/q} \leq (j+1-k) \bigl\vert B _{j}(a) \bigr\vert _{H}^{1/q}\omega \bigl(B_{j}(a)\bigr) \Vert b \Vert _{GC_{q,\omega }}.$$
(2.2)

### Lemma 2.2

([5])

Let$$\varOmega \in L^{\infty }( \mathbb{Q}_{p}^{n})$$be such that$$\varOmega (p^{j}x)=\varOmega (x)$$for all$$j\in \mathbb{Z}$$and$$\int _{|x|_{p}=1}\varOmega (x)\,dx=0$$. If
$$\sup_{ \vert y \vert _{p}=1}\sum_{j=1}^{\infty } \int _{ \vert x \vert _{p}=1} \bigl\vert \varOmega \bigl(x+p^{j}y \bigr)-\varOmega (x) \bigr\vert \,dx< \infty ,$$
then for$$1 < p <\infty$$, there is a constant$$C > 0$$such that
$$\bigl\Vert T_{k}(f) \bigr\Vert _{L^{p}(\mathbb{Q}_{p}^{n})}\leq C \Vert f \Vert _{L^{p}(\mathbb{Q} _{p}^{n})}$$
for$$k\in \mathbb{Z}$$, whereCis independent offand$$k \in {\mathbb{Z}}$$.
Furthermore, $$T(f) = \lim_{k\rightarrow -\infty }T_{k}(f)$$ exists in the $$L^{p}$$ norm, and
$$\bigl\Vert T(f) \bigr\Vert _{L^{p}(\mathbb{Q}_{p}^{n})}\leq C \Vert f \Vert _{L^{p}(\mathbb{Q}_{p} ^{n})}.$$
Moreover, on the p-adic field, the Riesz potential $$I_{\alpha }^{p}$$ is defined by
$$I^{\alpha }_{p}f(x)=\frac{1}{\varGamma _{n}(\alpha )} \int _{\mathbb{Q}_{p}^{n}}\frac{f(y)}{ \vert x-y \vert _{p}^{n-\alpha }}\,dy,$$
where $$\varGamma _{n}(\alpha )=(1-p^{\alpha -n})/(1-p^{-\alpha })$$ for $$\alpha \in \mathbb{C}$$, $$\alpha \neq 0$$.

### Lemma 2.3

([14])

Letαbe a complex number with$$0< \operatorname{Re}\alpha <n$$, and let$$1< r<\infty$$, $$1< q< n/ \operatorname{Re}\alpha$$, and$$0<1/r=1/q- \operatorname{Re}\alpha /n$$. Suppose that bothωandνare nonnegative measurable functions such that
$$\sum_{j={\gamma }}^{\infty }p^{j \operatorname{Re}\alpha } \frac{\nu (B_{j}(a))}{ \omega (B_{\gamma }(a))}=C< \infty$$
for any$$a\in \mathbb{Q}^{n}_{p}$$and$$\gamma \in \mathbb{Z}$$. Then the Riesz potential$$I^{\alpha }_{p}$$is bounded from$$GM_{q,\nu }$$to$$GM_{r,\omega }$$.

## 3 Main results

In this section, we state the main results of the paper.

### Theorem 3.1

Let$$1< q<\infty$$, and let$$\varOmega (p ^{j}x)=\varOmega (x)$$for all$$j\in \mathbb{Z}$$, $$\int _{|x|_{p}=1}\varOmega (x)\,dx=0$$, and
$$\sup_{ \vert y \vert _{p}=1}\sum_{j=1}^{\infty } \int _{ \vert x \vert _{p}=1} \bigl\vert \varOmega \bigl(x+p^{j}y \bigr)-\varOmega (x) \bigr\vert \,dx< \infty .$$
Suppose that bothωandνare nonnegative measurable functions such that
$$\sum_{j={\gamma }}^{\infty }\nu \bigl(B_{j}(a) \bigr)/\omega \bigl(B_{\gamma }(a)\bigr)=C< \infty$$
(3.1)
for any$$\gamma \in \mathbb{Z}$$and$$a\in \mathbb{Q}^{n}_{p}$$. Then the singular integral operators$$T_{k}$$are bounded from$$GM_{q,\nu }$$to$$GM_{q,\omega }$$for all$$k\in \mathbb{Z}$$. Moreover, $$T(f)=\lim_{k\rightarrow -\infty }T_{k}(f)$$exists in$$GM_{q,\omega }$$, and the operatorTis bounded from$$GM_{q,\nu }$$to$$GM_{q,\omega }$$.

### Corollary 3.1

Let$$1< q<\infty$$, $$\lambda <0$$, and let$$\varOmega \in L^{\infty }(\mathbb{Q}_{p}^{n})$$be such that$$\varOmega (p ^{j}x)=\varOmega (x)$$for all$$j\in \mathbb{Z}$$, $$\int _{|x|_{p}=1}\varOmega (x)\,dx=0$$, and
$$\sup_{ \vert y \vert _{p}=1}\sum_{j=1}^{\infty } \int _{ \vert x \vert _{p}=1} \bigl\vert \varOmega \bigl(x+p^{j}y \bigr)-\varOmega (x) \bigr\vert \,dx< \infty .$$
Then the operators$$T_{k}$$andTare bounded on the space$$M_{q,\lambda }$$for all$$k\in \mathbb{Z}$$.

In fact, for $$\lambda <0$$, taking $$\omega (B)=\nu (B)=|B|_{H}^{\lambda }$$ in Theorem 3.1, we obtain Corollary 3.1. If the Morrey space $$M_{q,\lambda }(\mathbb{Q}_{p}^{n} )$$ is replaced by the central Morrey space $$CM_{ q,\lambda }(\mathbb{Q}_{p}^{n} )$$ in Corollary 3.1, then the conclusion is that of Theorem 4.1 in [12].

### Theorem 3.2

Let$$\varOmega \in L^{\infty }(\mathbb{Q}_{p} ^{n})$$be such that$$\varOmega (p^{j}x)=\varOmega (x)$$for all$$j\in \mathbb{Z}$$, $$\int _{|x|_{p}=1}\varOmega (x)\,dx=0$$, and
$$\sup_{ \vert y \vert _{p}=1}\sum_{j=1}^{\infty } \int _{ \vert x \vert _{p}=1} \bigl\vert \varOmega \bigl(x+p^{j}y \bigr)-\varOmega (x) \bigr\vert \,dx< \infty .$$
Let$$0<\beta _{i}<1$$for$$i=1,2,\dots ,m$$be such that$$0<\beta =\sum_{i=1}^{m}\beta _{i}<n$$, and let$$1< r<\infty$$and$$1< q< n/\beta$$be such that$$1/r=1/q-\beta /n$$. Suppose that$$b_{i}\in {\varLambda _{\beta _{i}}}$$, $$i=1,2, \dots ,m$$, and bothωandνare nonnegative measurable functions such that
$$\sum_{j={\gamma }}^{\infty }p^{j\beta }\nu \bigl(B_{j}(a)\bigr)/\omega \bigl(B_{ \gamma }(a)\bigr)=C< \infty$$
(3.2)
for any$$\gamma \in \mathbb{Z}$$and$$a\in \mathbb{Q}^{n}_{p}$$. Then the commutators$$T_{k}^{\vec{b}}$$are bounded from$$GM_{q,\nu }$$to$$GM_{r,\omega }$$for all$$k\in \mathbb{Z}$$. Moreover, the commutator$$T^{\vec{b}}(f)=\lim_{k\rightarrow -\infty }T_{k}^{\vec{b}}(f)$$exists in the space of$$GM_{q,\omega }$$, and$$T^{\vec{b}}$$is bounded from$$GM_{q,\nu }$$to$$GM_{q,\omega }$$.

### Theorem 3.3

Let$$\varOmega \in L^{\infty }(\mathbb{Q} _{p}^{n})$$be such that$$\varOmega (p^{j}x)=\varOmega (x)$$for all$$j\in \mathbb{Z}$$, $$\int _{|x|_{p}=1}\varOmega (x)\,dx=0$$, and
$$\sup_{ \vert y \vert _{p}=1}\sum_{j=1}^{\infty } \int _{ \vert x \vert _{p}=1} \bigl\vert \varOmega \bigl(x+p^{j}y \bigr)-\varOmega (x) \bigr\vert \,dx< \infty .$$
Let$$1< q,r,q_{1},\dots, q_{m}<\infty$$be such that$$1/r=1/q+1/q_{1}+1/q _{2}+\cdots +1/q_{m}$$. Suppose thatω, ν, and$$\nu _{i}$$ ($$i=1,2,\dots ,m$$) are nonnegative measurable functions. Suppose that$$b_{i}\in GC_{q_{i},\nu _{i}}(\mathbb{Q}_{p}^{n} )$$, $$i=1,2,\dots ,m$$, and the functionsω, ν, and$$\nu _{i}$$ ($$i=1,2,\dots ,m$$) satisfy the following conditions:
1. (i)

$$\prod_{i=1}^{m}\nu _{i}(B_{\gamma }(a))\nu (B_{\gamma }(a))/ \omega (B_{\gamma }(a))=C<\infty$$,

2. (ii)

$$\sum_{j=\gamma +1}^{\infty }\prod_{i=1}^{m}\nu _{i}(B _{j}(a))(j+1-\gamma )^{m}\nu (B_{j}(a))/\omega (B_{\gamma }(a))=C< \infty$$

for any$$\gamma \in \mathbb{Z}$$and$$a\in \mathbb{Q}^{n}_{p}$$. Then the commutators$$T_{k}^{\vec{b}}$$are bounded from$$GM_{q,\nu }$$to$$GM_{r,\omega }$$for all$$k\in \mathbb{Z}$$. The commutator$$T^{ \vec{b}}=\lim_{k\rightarrow -\infty }T_{k}^{\vec{b}}$$exists in the space of$$GM_{q,\omega }$$, and$$T^{\vec{b}}$$is bounded from$$GM_{q,\nu }$$to$$GM_{q,\omega }$$.

### Corollary 3.2

Let$$\varOmega \in L^{\infty }( \mathbb{Q}_{p}^{n})$$be such that$$\varOmega (p^{j}x)=\varOmega (x)$$for all$$j\in \mathbb{Z}$$, $$\int _{|x|_{p}=1}\varOmega (x)\,dx=0$$, and
$$\sup_{ \vert y \vert _{p}=1}\sum_{j=1}^{\infty } \int _{ \vert x \vert _{p}=1} \bigl\vert \varOmega \bigl(x+p^{j}y \bigr)-\varOmega (x) \bigr\vert \,dx< \infty .$$
Let$$1< q,r,q_{1},\dots, q_{m}<\infty$$be such that$$1/r=1/q+1/q_{1}+1/q _{2}+\cdots +1/q_{m}$$. Let$$0\leq \lambda _{1},\dots ,\lambda _{m}<1/n$$, $$\lambda <-\sum_{i=1}^{m}\lambda _{i}$$, and$$\tilde{\lambda }= \sum_{i=1}^{m}\lambda _{i}+\lambda$$. If$$b_{i}\in BMO_{q_{i}, \lambda _{i}}(\mathbb{Q}_{p}^{n} )$$, then the commutators$$T_{k}^{ \vec{b}}$$and$$T^{\vec{b}}$$are bounded from$$M_{q,\lambda }$$to$$M_{r,\tilde{\lambda }}$$for all$$k\in \mathbb{Z}$$.

Moreover, let $$1< r,q,q_{1}<\infty$$ be such that $$1/r=1/q+1/q_{1}$$. Let $$0\leq \lambda _{1}<1/n$$, $$\lambda <-\lambda _{1}$$, and $$\tilde{\lambda }=\lambda _{1}+\lambda$$. If $$b\in CBMO_{q_{1},\lambda _{1}}(\mathbb{Q}_{p}^{n} )$$, then from Corollary 3.1 it follows that the commutators $$T_{k}^{b}=[T_{k}, b]$$ and $$T^{b}=[T, b]$$ are bounded from $$CM_{q,\lambda }$$ to $$CM_{r,\tilde{\lambda }}$$ for all $$k\in \mathbb{Z}$$. These results are those of Theorem 4.2 in [12].

## 4 Proof of Theorems 3.1–3.3

Let us first give the proof of Theorem 3.1.

For any fixed $$\gamma \in \mathbb{Z}$$ and $$a\in \mathbb{Q}^{n}_{p}$$, it is easy to see that
\begin{aligned}& \frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}(f) (x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \\& \quad \leq \frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}(f) (f \chi _{B_{\gamma }(a)}) (x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \\& \qquad {}+\frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}(f \chi _{B^{c}_{\gamma }(a)}) (x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \\& \quad :=I+II, \end{aligned}
(4.1)
where $$B^{c}_{\gamma }(a)$$ is the complement to $$B_{\gamma }(a)$$ in $$\mathbb{Q}^{n}_{p}$$.
Using Lemma 2.2 and (3.1), it follows that
\begin{aligned} I \lesssim &\frac{1}{\omega (B_{\gamma }(a))}\frac{1}{ \vert B_{\gamma }(a) \vert _{H} ^{1/q}} \biggl( \int _{B_{\gamma }(a)} \bigl\vert f(x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \\ =&\frac{\nu (B_{\gamma }(a))}{\omega (B_{\gamma }(a))}\frac{1}{ \nu (B_{\gamma }(a))} \biggl(\frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert f(x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \\ \lesssim & \Vert f \Vert _{GM_{q,\nu }}. \end{aligned}
(4.2)

For II, let us first estimate $$|T_{k}(f\chi _{B^{c}_{\gamma }(a)})(x)|$$.

Since $$x\in B_{\gamma }(a)$$ and $$\varOmega \in L^{\infty }(\mathbb{Q} _{p}^{n})$$, we have
\begin{aligned} \bigl\vert T_{k}(f\chi _{B^{c}_{\gamma }(a)}) (x) \bigr\vert =& \biggl\vert \int _{ \vert y \vert _{p}>p^{k}}(f \chi _{B^{c}_{\gamma }(a)}) (x-y)\frac{\varOmega (y)}{ \vert y \vert _{p}^{n}}\,dy \biggr\vert \\ =& \biggl\vert \int _{ \vert x-z \vert _{p}>p^{k}}(f\chi _{B^{c}_{\gamma }(a)}) (z)\frac{ \varOmega (x-z)}{ \vert x-z \vert _{p}^{n}}\,dz \biggr\vert \\ \lesssim& \int _{B^{c}_{\gamma }(a)}\frac{ \vert f(z) \vert }{ \vert x-z \vert _{p}^{n}}\,dz \\ \lesssim& \sum_{j=\gamma +1}^{\infty } \int _{S_{j}(a)}p^{-jn} \bigl\vert f(y) \bigr\vert \,dy \\ \leq& \sum_{j=\gamma +1}^{\infty }p^{-jn} \biggl( \int _{B_{j}(a)} \bigl\vert f(y) \bigr\vert ^{q}\,dy \biggr)^{1/q} \bigl\vert B_{j}(a) \bigr\vert _{H}^{1-1/q} \\ =& \Vert f \Vert _{GM_{q,\nu }}\sum_{j=\gamma +1}^{\infty } \nu \bigl(B_{j}(a)\bigr). \end{aligned}
(4.3)
Thus from (3.1) and (4.3) it follows that
\begin{aligned} II =&\frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}(f \chi _{B^{c}_{\gamma }(a)}) (x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \\ \lesssim & \Vert f \Vert _{GM_{q,\nu }}\sum_{j=\gamma +1}^{\infty } \nu \bigl(B_{j}(a)\bigr)/\omega \bigl(B_{\gamma }(a)\bigr) \\ \lesssim & \Vert f \Vert _{GM_{q,\nu }}. \end{aligned}
(4.4)
Combining the estimates of (4.1), (4.2), and (4.4), we have
$$\frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}(f) (x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \lesssim \Vert f \Vert _{GM_{q,\nu }},$$
which means that $$T_{k}$$ is bounded from $$GM_{q,\nu }$$ to $$GM_{q, \omega }$$ for all $$k\in \mathbb{Z}$$.

Moreover, from Lemma 2.2 and the definition of $$GM_{q,\omega }( \mathbb{Q}_{p}^{n} )$$ it is obvious that $$T(f)=\lim_{k\rightarrow -\infty }T_{k}(f)$$ exists in $$GM_{q,\omega }$$ and the operator T is bounded from $$GM_{q,\nu }$$ to $$GM_{q,\omega }$$.

### Proof of Theorem 3.2

For any $$x\in \mathbb{Q}^{n}_{p}$$, since $$\varOmega \in L^{\infty }( \mathbb{Q}_{p}^{n})$$ and $$b_{i}\in {\varLambda _{\beta _{i}}}$$, $$i=1,2,\dots ,m$$, it is easy to see that
\begin{aligned}& \bigl\vert T_{k}^{\vec{b}}f(x) \bigr\vert \\& \quad \leq \int _{ \vert y \vert _{p}>p^{k}}\prod_{i=1}^{m} \bigl\vert b_{i}(x)-b_{i}(x-y) \bigr\vert \bigl\vert f(x-y) \bigr\vert \frac{ \vert \varOmega (y) \vert }{ \vert y \vert _{p}^{n}}\,dy \\& \quad \lesssim \int _{\mathbb{Q}_{p}^{n}}\frac{ \vert f(z) \vert }{ \vert x-z \vert _{p}^{n-\beta }}\,dz \\& \quad \lesssim I^{\beta }_{p}\bigl( \vert f \vert \bigr) (x). \end{aligned}

Thus from Lemma 2.3 it is obvious that the commutators $$T_{k}^{ \vec{b}}$$ are bounded from $$GM_{q,\nu }$$ to $$GM_{r,\omega }$$ for all $$k\in \mathbb{Z}$$.

Moreover, from the definition of $$GM_{q,\omega }(\mathbb{Q}_{p}^{n} )$$ it is obvious that $$T^{\vec{b}}(f)=\lim_{k\rightarrow -\infty }T_{k}^{\vec{b}}(f)$$ exists in the space of $$GM_{q,\omega }$$, and the commutator $$T^{\vec{b}}$$ is bounded from $$GM_{q,\nu }$$ to $$GM_{q, \omega }$$. □

### Proof of Theorem 3.3

Without loss of generality, we need only to show that the conclusion holds for $$m=2$$.

For any fixed $$\gamma \in \mathbb{Z}$$ and $$a\in \mathbb{Q}^{n}_{p}$$, we write $$f^{0}=f\chi _{B_{\gamma }(a)}$$ and $$f^{\infty }=f \chi _{B^{c}_{\gamma }(a)}$$. Then
\begin{aligned}& \frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}^{(b_{1},b _{2})}(f) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\& \quad \leq \frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{1}(x)-(b _{1})_{B_{\gamma }(a)}\bigr) \bigl(b_{2}(x)-(b_{2})_{B_{\gamma }(a)} \bigr)T_{k}\bigl(f^{0}\bigr) (x) \bigr\vert ^{r} \,dx \biggr)^{1/r} \\& \qquad {}+\frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{1}(x)-(b _{1})_{B_{\gamma }(a)}\bigr)T_{k}\bigl(\bigl(b_{2}-(b_{2})_{B_{\gamma }(a)} \bigr)f^{0}\bigr) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\& \qquad {}+\frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{2}(x)-(b _{2})_{B_{\gamma }(a)}\bigr)T_{k}\bigl(\bigl(b_{1}-(b_{1})_{B_{\gamma }(a)} \bigr)f^{0}\bigr) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\& \qquad {}+\frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}\bigl( \bigl(b_{1}-(b _{1})_{B_{\gamma }(a)}\bigr) \bigl(b_{2}-(b_{2})_{B_{\gamma }(a)}\bigr)f^{0} \bigr) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\& \qquad {}+\frac{1}{\omega (B_{\gamma }(a))} \\& \qquad {}\times \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{1}(x)-(b _{1})_{B_{\gamma }(a)}\bigr) \bigl(b_{2}(x)-(b_{2})_{B_{\gamma }(a)} \bigr)T_{k}\bigl(f^{ \infty }\bigr) (x) \bigr\vert ^{r} \,dx \biggr)^{1/r} \\& \qquad {}+\frac{1}{\omega (B_{\gamma }(a))} \\& \qquad {}\times \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{1}(x)-(b _{1})_{B_{\gamma }(a)}\bigr)T_{k}\bigl(\bigl(b_{2}-(b_{2})_{B_{\gamma }(a)} \bigr)f^{ \infty }\bigr) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\& \qquad {}+\frac{1}{\omega (B_{\gamma }(a))} \\& \qquad {}\times \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{2}(x)-(b _{2})_{B_{\gamma }(a)}\bigr)T_{k}\bigl(\bigl(b_{1}-(b_{1})_{B_{\gamma }(a)} \bigr)f^{ \infty }\bigr) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\& \qquad {}+\frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}\bigl( \bigl(b_{1}-(b _{1})_{B_{\gamma }(a)}\bigr) \bigl(b_{2}-(b_{2})_{B_{\gamma }(a)}\bigr)f^{\infty } \bigr) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\& \quad =:E_{1}+E_{2}+E_{3}+E_{4}+E_{5}+E_{6}+E_{7}+E_{8}. \end{aligned}
(4.5)

We further estimate every part.

Since $$1/r=1/q+1/q_{1}+1/q_{2}$$, from Hölder’s inequality, Lemma 2.2, and (i) it follows that
\begin{aligned} E_{1} =&\frac{1}{\omega (B_{\gamma }(a))} \\ & {}\times \biggl(\frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{1}(x)-(b_{1})_{B_{\gamma }(a)} \bigr) \bigl(b _{2}(x)-(b_{2})_{B_{\gamma }(a)} \bigr)T_{k}\bigl(f^{0}\bigr) (x) \bigr\vert ^{r} \,dx \biggr)^{1/r} \\ \leq &\frac{1}{\omega (B_{\gamma }(a)) \vert B_{\gamma }(a) \vert _{H}^{1/r}} \prod_{i=1}^{2} \biggl( \int _{B_{\gamma }(a)} \bigl\vert b_{i}(x)-(b_{i})_{B _{\gamma }(a)} \bigr\vert ^{q_{i}}\,dx \biggr)^{1/q_{i}} \\ &{}\times \biggl( \int _{B_{\gamma }(a)} \bigl\vert T _{k}\bigl(f^{0} \bigr) (x) \bigr\vert ^{q}\,dx \biggr)^{1/{q}} \\ \lesssim &\frac{\nu _{1}(B_{\gamma }(a))\nu _{2}(B_{\gamma }(a))}{ \omega (B_{\gamma }(a)) \vert B_{\gamma }(a) \vert _{H}^{1/q}}\prod_{i=1} ^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \biggl( \int _{B_{\gamma }(a)} \bigl\vert f(x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \\ \leq &\frac{\nu (B_{\gamma }(a))\nu _{1}(B_{\gamma }(a))\nu _{2}(B_{ \gamma }(a))}{\omega (B_{\gamma }(a))}\prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }} \\ \lesssim &\prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }}. \end{aligned}
Let $$1/\bar{q}=1/q+1/q_{2}$$. Then $$1/r=1/q_{1}+1/\bar{q}$$. Thus, from Hölder’s inequality, Lemma 2.2, and (i) we obtain
\begin{aligned} E_{2} =& \frac{1}{\omega (B_{\gamma }(a))} \biggl(\frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{1}(x)-(b_{1})_{B_{\gamma }(a)} \bigr)T _{k}\bigl(\bigl(b_{2}-(b_{2})_{B_{\gamma }(a)} \bigr)f^{0}\bigr) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\ \leq &\frac{1}{\omega (B_{\gamma }(a)) \vert B_{\gamma }(a) \vert _{H}^{1/r}} \biggl( \int _{B_{\gamma }(a)} \bigl\vert b_{1}(x)-(b_{1})_{B_{\gamma }(a)} \bigr\vert ^{q _{1}}\,dx \biggr)^{1/q_{1}} \\ &{}\times \biggl( \int _{B_{\gamma }(a)} \bigl\vert T_{k}\bigl( \bigl(b_{2}-(b _{2})_{B_{\gamma }(a)}\bigr)f^{0} \bigr) (x) \bigr\vert ^{\bar{q}}\,dx \biggr)^{1/{\bar{q}}} \\ \lesssim & \frac{1}{\omega (B_{\gamma }(a)) \vert B_{\gamma }(a) \vert _{H}^{1/r}} \biggl( \int _{B_{\gamma }(a)} \bigl\vert b_{1}(x)-(b_{1})_{B_{\gamma }(a)} \bigr\vert ^{q_{1}}\,dx \biggr)^{1/q_{1}} \\ &{}\times \biggl( \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{2}(x)-(b_{2})_{B _{\gamma }(a)} \bigr)f(x) \bigr\vert ^{\bar{q}}\,dx \biggr)^{1/{\bar{q}}} \\ \leq &\frac{1}{\omega (B_{\gamma }(a)) \vert B_{\gamma }(a) \vert _{H}^{1/r}}\prod_{i=1}^{2} \biggl( \int _{B_{\gamma }(a)} \bigl\vert b_{i}(x)-(b_{i})_{B_{\gamma }(a)} \bigr\vert ^{q_{i}}\,dx \biggr)^{1/q_{i}} \biggl( \int _{B_{\gamma }(a)} \bigl\vert f(x) \bigr\vert ^{q}\,dx \biggr)^{1/ {q}} \\ \leq &\frac{\nu (B_{\gamma }(a))\nu _{1}(B_{\gamma }(a))\nu _{2}(B_{ \gamma }(a))}{\omega (B_{\gamma }(a))} \prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }} \\ \lesssim &\prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }}. \end{aligned}
Similarly,
$$E_{3} \lesssim \prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }}.$$
For $$E_{4}$$, from Lemma 2.2, Hölder’s inequality, and (i) we obtain
\begin{aligned} E_{4} =& \frac{1}{\omega (B_{\gamma }(a))} \biggl(\frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}\bigl(b_{1}-(b_{1})_{B_{\gamma }(a)} \bigr) \bigl(b _{2}-(b_{2})_{B_{\gamma }(a)} \bigr)f^{0}) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\ \lesssim & \frac{1}{\omega (B_{\gamma }(a)) \vert B_{\gamma }(a) \vert _{H}^{1/r}} \biggl( \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{1}(x)-(b_{1})_{B_{\gamma }(a)} \bigr) \bigl(b_{2}(x)-(b _{2})_{B_{\gamma }(a)}\bigr)f(x) \bigr\vert ^{r}\,dx \biggr)^{1/{r}} \\ \leq &\frac{1}{\omega (B_{\gamma }(a)) \vert B_{\gamma }(a) \vert _{H}^{1/r}} \prod_{i=1}^{2} \biggl( \int _{B_{\gamma }(a)} \bigl\vert b_{i}(x)-(b_{i})_{B _{\gamma }(a)} \bigr\vert ^{q_{i}}\,dx \biggr)^{1/q_{i}} \biggl( \int _{B_{\gamma }(a)} \bigl\vert f(x) \bigr\vert ^{q}\,dx \biggr)^{1/{q}} \\ \leq &\frac{\nu (B_{\gamma }(a))\nu _{1}(B_{\gamma }(a))\nu _{2}(B_{ \gamma }(a))}{\omega (B_{\gamma }(a))}\prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }} \\ \lesssim &\prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }}. \end{aligned}
To estimate $$E_{5}$$, we first need to consider $$|T_{k}(f^{\infty })(x)|$$. In fact, by (4.3) it is easy to see that
$$\bigl\vert T_{k}\bigl(f^{\infty }\bigr) (x) \bigr\vert \lesssim \Vert f \Vert _{GM_{q,\nu }}\sum_{j=\gamma +1}^{\infty } \nu \bigl(B_{j}(a)\bigr).$$
(4.6)
Therefore from Hölder’s inequality, (4.6), and (ii) we get
\begin{aligned} E_{5} =&\frac{1}{\omega (B_{\gamma }(a))} \\ &{}\times \biggl(\frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{1}(x)-(b_{1})_{B_{\gamma }(a)} \bigr) \bigl(b _{2}(x)-(b_{2})_{B_{\gamma }(a)} \bigr)T_{k}\bigl(f^{\infty }\bigr) (x) \bigr\vert ^{r} \,dx \biggr)^{1/r} \\ \leq &\frac{1}{\omega (B_{\gamma }(a)) \vert B_{\gamma }(a) \vert _{H}^{1/r}} \prod_{i=1}^{2} \biggl( \int _{B_{\gamma }(a)} \bigl\vert b_{i}(x)-(b_{i})_{B _{\gamma }(a)} \bigr\vert ^{q_{i}}\,dx \biggr)^{1/q_{i}} \\ &{}\times \biggl( \int _{B_{\gamma }(a)} \bigl\vert T _{k}\bigl(f^{\infty } \bigr) (x)f(x) \bigr\vert ^{q}\,dx \biggr)^{1/{q}} \\ \lesssim &\sum_{j=\gamma +1}^{\infty }\frac{\nu (B_{j}(a))\nu _{1}(B _{\gamma }(a))\nu _{2}(B_{\gamma }(a))}{\omega (B_{\gamma }(a))} \prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q, \nu }} \\ \lesssim &\prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }}. \end{aligned}
It is similar to estimate (4.3) for $$x\in B_{\gamma }(a)$$. By $$\varOmega \in L^{\infty }(\mathbb{Q}_{p}^{n})$$ and (2.2) we can deduce that
\begin{aligned}& \big|T_{k}\bigl(b_{2}-(b_{2})_{B_{\gamma }(a)} \bigr)f^{\infty }) (x)\big| \\& \quad = \biggl\vert \int _{ \vert y \vert _{p}>p^{k}}\bigl(b_{2}(x-y)-(b_{2})_{B_{\gamma }(a)} \bigr)f \chi _{B^{c}_{\gamma }(a)}(x-y)\frac{\varOmega (y)}{ \vert y \vert _{p}^{n}}\,dy \biggr\vert \\& \quad \leq \int _{B^{c}_{\gamma }} \bigl\vert b_{2}(z)-(b_{2})_{B_{\gamma }(a)} \bigr\vert \bigl\vert f(z) \bigr\vert \frac{ \vert \varOmega (x-z) \vert }{ \vert x-z \vert _{p}^{n}}\,dz \\& \quad \lesssim \int _{B^{c}_{\gamma }}\frac{ \vert b_{2}(z)-(b_{2})_{B_{\gamma }(a)} \vert \vert f(z) \vert }{ \vert x-z \vert _{p} ^{n}}\,dz \\& \quad \lesssim \sum_{j=\gamma +1}^{\infty } \int _{S_{j}(a)}p^{-jn} \bigl\vert b_{2}(z)-(b _{2})_{B_{\gamma }(a)} \bigr\vert \bigl\vert f(y) \bigr\vert \,dy \\& \quad =\sum_{j=\gamma +1}^{\infty }p^{-jn} \bigl\vert B_{j}(a) \bigr\vert _{H}^{1-1/q-1/q_{2}} \biggl( \int _{S_{j}(a)} \bigl\vert f(y) \bigr\vert ^{q}\,dy \biggr)^{1/{q}} \biggl( \int _{S_{j}(a)} \bigl\vert b _{2}(y)-(b_{2})_{B_{\gamma }(a)} \bigr\vert ^{q_{2}}\,dy \biggr)^{1/q_{2}} \\& \quad \leq \Vert f \Vert _{GM_{q,\nu }}\sum_{j=\gamma +1}^{\infty }p^{-jn} \bigl\vert B_{j}(a) \bigr\vert _{H} ^{1-1/q_{2}}\nu \bigl(B_{j}(a)\bigr) \biggl( \int _{B_{j}(a)} \bigl\vert b_{2}(y)-(b_{2})_{B _{\gamma }(a)} \bigr\vert ^{q_{2}}\,dy \biggr)^{1/q_{2}} \\& \quad \lesssim \Vert b_{2} \Vert _{GC_{q_{2},\nu _{2}}} \Vert f \Vert _{GM_{q,\nu }} \sum_{j=\gamma +1}^{\infty }(j+1- \gamma )\nu \bigl(B_{j}(a)\bigr)\nu _{2}\bigl(B_{j}(a) \bigr). \end{aligned}
(4.7)
Let $$1/\bar{q}=1/q+1/q_{2}$$. Then $$1/r=1/q_{1}+1/\bar{q}$$. Thus from Hölder’s inequality, (4.7), and (ii) it follows that
\begin{aligned} E_{6} =&\frac{1}{\omega (B_{\gamma }(a))} \biggl(\frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert \bigl(b_{1}(x)-(b_{1})_{B_{\gamma }(a)} \bigr)T _{k}\bigl(\bigl(b_{2}-(b_{2})_{B_{\gamma }(a)} \bigr)f^{\infty }\bigr) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\ \leq &\frac{1}{\omega (B_{\gamma }(a)) \vert B_{\gamma }(a) \vert _{H}^{1/r}} \biggl( \int _{B_{\gamma }(a)} \bigl\vert b_{1}(x)-(b_{1})_{B_{\gamma }(a)} \bigr\vert ^{q _{1}}\,dx \biggr)^{1/q_{1}} \\ &{}\times \biggl( \int _{B_{\gamma }(a)} \bigl\vert T_{k}\bigl( \bigl(b_{2}-(b _{2})_{B_{\gamma }(a)}\bigr)f^{\infty } \bigr) (x) \bigr\vert ^{\bar{q}}\,dx \biggr)^{1/{\bar{q}}} \\ \leq &\prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM _{q,\nu }} \frac{1}{\omega (B_{\gamma }(a))}\sum_{j=\gamma +1}^{ \infty }(j+1-\gamma )\nu \bigl(B_{j}(a)\bigr)\nu _{2}\bigl(B_{j}(a) \bigr)\nu _{1}\bigl(B_{\gamma }(a)\bigr) \\ \lesssim &\prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }}. \end{aligned}
Similarly estimating $$E_{6}$$, we obtain
$$E_{7}\lesssim \prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }}.$$
Moreover, since $$\varOmega \in L^{\infty }(\mathbb{Q}_{p}^{n})$$, by (2.2) it is easy to see that
\begin{aligned}& \bigl\vert T_{k}\bigl(\bigl(b_{1}-(b_{1})_{B_{\gamma }(a)} \bigr) \bigl(b_{2}-(b_{2})_{B_{\gamma }(a)}\bigr)f ^{\infty }\bigr) (x) \bigr\vert \\& \quad = \biggl\vert \int _{ \vert x-z \vert _{p}>p^{k}}\bigl(b_{1}(z)-(b_{1})_{B_{\gamma }(a)} \bigr) \bigl(b _{2}(z)-(b_{2})_{B_{\gamma }(a)}\bigr)f\chi _{B^{c}_{\gamma }(a)}(z)\frac{ \varOmega (x-z)}{ \vert x-z \vert _{p}^{n}}\,dz \biggr\vert \\& \quad \leq \int _{B^{c}_{\gamma }} \bigl\vert b_{1}(z)-(b_{1})_{B_{\gamma }(a)} \bigr\vert \bigl\vert b_{2}(z)-(b _{2})_{B_{\gamma }(a)} \bigr\vert \bigl\vert f(z) \bigr\vert \frac{ \vert \varOmega (x-z) \vert }{ \vert x-z \vert _{p}^{n}}\,dz \\& \quad \lesssim \sum_{j=\gamma +1}^{\infty } \int _{S_{j}(a)}p^{-jn} \bigl\vert b_{1}(z)-(b _{1})_{B_{\gamma }(a)} \bigr\vert \bigl\vert b_{2}(z)-(b_{2})_{B_{\gamma }(a)} \bigr\vert \bigl\vert f(y) \bigr\vert \,dy \\& \quad =\sum_{j=\gamma +1}^{\infty }p^{-jn} \bigl\vert B_{j}(a) \bigr\vert _{H}^{1-1/q-1/q_{1}-1/q _{2}} \biggl( \int _{S_{j}(a)} \bigl\vert f(y) \bigr\vert ^{q}\,dy \biggr)^{1/{q}} \\& \qquad {}\times \biggl( \int _{S_{j}(a)} \bigl\vert b_{1}(y)-(b_{1})_{B_{\gamma }(a)} \bigr\vert ^{q_{1}}\,dy \biggr)^{1/q _{1}} \\& \qquad {} \times \biggl( \int _{S_{j}(a)} \bigl\vert b_{2}(y)-(b_{2})_{B_{\gamma }(a)} \bigr\vert ^{q _{2}}\,dy \biggr)^{1/q_{2}} \\& \quad \lesssim \prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }}\sum _{j=\gamma +1}^{\infty }(j+1-\gamma )^{2}\nu \bigl(B_{j}(a)\bigr) \nu _{1}\bigl(B_{j}(a)\bigr) \nu _{2}\bigl(B_{j}(a)\bigr). \end{aligned}
(4.8)
Therefore from (4.8) and (ii) we get that
\begin{aligned} E_{8} =&\frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{B_{\gamma }(a)|_{H}} \int _{B} \bigl\vert T_{k}\bigl( \bigl(b_{1}-(b_{1})_{B_{ \gamma }(a)}\bigr) \bigl(b_{2}-(b_{2})_{B_{\gamma }(a)}\bigr)f^{\infty } \bigr) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \\ \leq& \prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM _{q,\nu }} \frac{1}{\omega (B_{\gamma }(a))}\sum_{j=\gamma +1}^{ \infty }(j+1-\gamma )^{2} \nu \bigl(B_{j}(a)\bigr)\nu _{1} \bigl(B_{j}(a)\bigr)\nu _{2}\bigl(B _{j}(a)\bigr) \\ \lesssim& \prod_{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }}. \end{aligned}
Combining (4.5) and the estimates of $$E_{1},E_{2},\dots , E_{8}$$, we have
$$\frac{1}{\omega (B_{\gamma }(a))} \biggl( \frac{1}{ \vert B_{\gamma }(a) \vert _{H}} \int _{B_{\gamma }(a)} \bigl\vert T_{k}^{(b_{1},b _{2})}(f) (x) \bigr\vert ^{r}\,dx \biggr)^{1/r} \leq \prod _{i=1}^{2} \Vert b_{i} \Vert _{GC_{q_{i},\nu _{i}}} \Vert f \Vert _{GM_{q,\nu }},$$
which means that the commutator $$T_{k}^{(b_{1},b_{2})}$$ is bounded from $$GM_{q,\nu }$$ to $$GM_{r,\omega }$$.

Moreover, by Lemma 2.2 and the definition of $$GM_{q,\omega }( \mathbb{Q}_{p}^{n} )$$ it is obvious that the commutator $$T^{\vec{b}}(f)= \lim_{k\rightarrow -\infty }T_{k}^{\vec{b}}(f)$$ exists in the space of $$GM_{q,\omega }$$, and $$T^{\vec{b}}$$ is bounded from $$GM_{q,\nu }$$ to $$GM_{q,\omega }$$.

Therefore the proof of Theorem 3.3 is complete. □

## 5 Conclusion

In this paper, we established the boundedness of a class of p-adic singular integral operators on the p-adic generalized Morrey spaces. We also considered 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.

## Notes

### Acknowledgements

The authors are grateful to the editor and referees for carefully reading the manuscript.

### Availability of data and materials

No data were used to support this study.

### Funding

The work is supported by National Natural Science Foundation of China (No. 11601035).

### Competing interests

The authors declare that they have no competing interests.

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## Authors and Affiliations

• Hui X. Mo
• 1
• Zhe Han
• 1
• Liu Yang
• 1
• Xiao J. Wang
• 1
1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina