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

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.

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 let ω be 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 $k\in \mathbb{Z}$, where C is independent of f and $k \in {\mathbb{Z}}$.

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])

Let α be 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 both ω and ν are 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 both ω and ν are 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 operator T is 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 $T_{k}$ and T are 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 both ω and ν are 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 that ω, 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 functions ω, ν, and $\nu _{i}$ ($i=1,2,\dots ,m$) satisfy the following conditions:

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

• (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.