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

  • Hui X. MoEmail author
  • Zhe Han
  • Liu Yang
  • Xiao J. Wang
Open Access
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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.

Keywords

p-adic field p-adic singular integral operator Commutator p-adic generalized Morrey function p-adic generalized Campanato function p-adic Lipschitz function 

MSC

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 3.13.3

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.

Authors’ contributions

All authors have made equal contributions in this article. All authors read and approved the final manuscript.

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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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina

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