Estimates for p-adic fractional integral operator and its commutators on p-adic Morrey–Herz spaces

Abstract

This research investigates the boundedness of a p-adic fractional integral operator on p-adic Morrey–Herz spaces. In particular, p-adic central bounded mean oscillations $(C\dot{M}O)$ and Lipschitz estimate for commutators of the p-adic fractional integral operator are provided as well.

1 Introduction

According to the well-known Ostrowski theorem [1], any nontrivial valuation on the field of rational numbers $\mathbb{Q}$ is equivalent either to the p-adic valuation $|\cdot |_{p}$ or to one of real valuations $|\cdot |$, where p is a prime number. The former norm is defined as follows: if any rational number $x\neq 0$ is denoted as $x = p^{\gamma}s/t$, where $\gamma =\gamma (x)\in \mathbb{Z}$ and the integers s, t are not divisible by p, then $|x|_{p}=p^{\gamma}$, $|0|_{p} = 0$. The norm $|\cdot |_{p}$ satisfies the strong triangle inequality $|x + y|_{p}\leq \max (|x|_{p}, |y|_{p})$. The extended form of any $x\neq 0\in \mathbb{Q}_{p}$ (field of p-adic numbers) is given in [2] as

$$ x=p^{\gamma}\sum_{i=0}^{\infty} \alpha _{i}p^{i},$$

(1)

where $\alpha _{i},\gamma \in \mathbb{Z}$, $\alpha _{i}\in \frac{\mathbb{Z}}{p\mathbb{Z}_{p}}$, $\alpha _{0}\neq 0$.

In what follows the n-dimensional vector space $\mathbb{Q}_{p}^{n}$ over $\mathbb{Q}_{p}$ is equipped with the following absolute value:

$$ \vert \mathbf{x} \vert _{p}=\max _{1\leq r \leq n} \vert x_{r} \vert _{p} \cdot $$

(2)

Let $B_{\gamma}(\mathbf{a})$ and $S_{\gamma}(\mathbf{a})$ represent respectively the ball and sphere of $\mathbb{Q}_{p}^{n}$ centered at $\mathbf{a}\in \mathbb{Q}_{p}^{n}$ and radius $p^{\gamma}>0$:

$$\begin{aligned}& B_{\gamma}(\mathbf{a})=\bigl\{ \mathbf{x} \in \mathbb{Q}_{p}^{n}: \vert \mathbf{x}-\mathbf{a} \vert _{p}\leq p^{\gamma}\bigr\} , \quad \text{and} \\& S_{\gamma}(\mathbf{a})=\bigl\{ \mathbf{x} \in \mathbb{Q}_{p}^{n}: \vert \mathbf{x}-\mathbf{a} \vert _{p}=p^{\gamma}\bigr\} \cdot \end{aligned}$$

Since $\mathbb{Q}_{p}^{n}$ is a locally compact commutative group with respect to addition, in $\mathbb{Q}_{p}^{n}$ there exists a positive Haar measure dx invariant under shift $d(\mathbf{x}+\mathbf{a})=d(\mathbf{x})\cdot$ It is worth noting that dx is normalized by the equality $\int _{{B_{0}(\mathbf{0})}}d\mathbf{x}=1$. It is easy to find $\int _{B_{\gamma}(\mathbf{a})}d\mathbf{x}=p^{n\gamma}$ and $\int _{S_{\gamma}(\mathbf{a})}d\mathbf{x}=p^{n\gamma}(1-p^{-n})$ for any $\mathbf{a}\in \mathbb{Q}_{p}^{n}$.

The field of p-adic numbers can be applied in many scientific fields. In physics, the groundbreaking application is the p-adic AdS/CFT [3]. Khrennikov et al. [4] enhanced p-adic wavelet for modeling reaction-diffusion dynamics. Its application in biology includes the models for hierarchical structures of genetic code [5] and protein [6]. Furthermore, p-adic numbers have found a novel application in harmonic analysis and mathematical physics (see, for example, [2, 7–11] and related references).

The topic of fractional calculus is undergoing fast development with more and more appealing applications in the real world (see, for instance, [12–16]). One of the principal options behind the popularity of the area is that fractional-order differentiations and integrations are more beneficial tools in expressing real-world matters than the integer-order ones. Moreover, it is applied in the solution of linear constant coefficient fractional differential equations of any commensurate order and the CRONE control-system design toolbox for the control engineering community (see [17, 18] and related references therein). Recently many researchers have made tremendous contributions to the topic of fractional calculus by developing multiple fractional expressions in diverse publications (see, for instance, [19–25]). Also, its applications have been found in various fields of science and engineering, such as rheology, fluid flow, probability, and electrical networks. In this context, fractional integral operators are an important part of the mathematical analysis as they construct and formulate inequalities which have multiple applications in scientific areas that can be seen in the existing literature [26–28]. The boundedness of fractional integral operator on several function spaces is a key area not only in harmonic analysis but also in differentiation theory, partial differential equation, and potential theory [29–31]. In this link, Taibleson [32] defined the p-adic fractional integral operator as

$$ I^{p}_{\beta}f(\mathbf{x})= \int _{\mathbb{Q}_{p}^{n}} \frac{f(\mathbf{t})}{ \vert \mathbf{x}-\mathbf{t} \vert _{p}^{n-\beta}}\,d\mathbf{t},\quad 0< \beta < n.$$

In the past, the above operator has received a considerable attention. The explicit formula of p-adic fractional integral $\mathbb{Q}_{p}$ along with the development of analytical potential theory on $\mathbb{Q}_{p}^{n}$ is acquired in [33] and [34] respectively. The fundamental properties of p-adic fractional integral operator on $\mathbb{Q}_{p}^{n}$ can be seen in [32]. Moreover, the boundedness of commutators of p-adic fractional integral operator respectively on central Morrey spaces and generalized Morrey spaces in the p-adic field is obtained in [11] and [35]. Recently, the boundedness of $I^{p}_{\beta}$ on Herz spaces has been reported in [36].

Furthermore, suppose that $b \colon \mathbb{Q}_{p}^{n} \to \mathbb{R}$ and $f \colon \mathbb{Q}_{p}^{n} \to \mathbb{R}$ are measurable mappings, then the commutators of fractional integral operator can be defined as follows:

$$ I^{p,b}_{\beta}f(\mathbf{x})= \int _{\mathbb{Q}_{p}^{n}} \frac{(b(x)-b(t))f(\mathbf{t})}{ \vert \mathbf{x}-\mathbf{t} \vert _{p}^{n-\beta}}\,d\mathbf{t}.$$

Motivated by the above results, we obtain the boundedness of p-adic fractional integral operator on Morrey–Herz spaces. Furthermore, $C\dot{M}O$ and Lipschitz estimates for commutators of p-adic fractional integral operator on Morrey–Herz spaces are also studied. Before turning to our main results, let us specify that $\chi _{k}$ is the characteristic function of the sphere and a letter C denotes a constant with values that may differ at its occurrence. Also, we use the following generalized Hölder inequality in the article. Let $q>1$ be a real number such that $1/q+1/q'=1$, and let $f\in L^{q}(\mathbb{Q}_{p}^{n})$ and $g\in L^{q'}(\mathbb{Q}_{p}^{n})$, then fg is integrable on $\mathbb{Q}_{p}^{n}$ and

$$ \int _{\mathbb{Q}_{p}^{n}} \bigl\vert f(\mathbf{x})g(\mathbf{x}) \bigr\vert \,d\mathbf{x} \leq \Vert f \Vert _{L^{q}(\mathbb{Q}_{p}^{n})} \Vert g \Vert _{L^{q'}(\mathbb{Q}_{p}^{n})}.$$

Definition 1.1

([37])

Let $1< p<\infty$. The p-adic space $(C\dot{M}O)$ is defined as follows:

$$ \Vert f \Vert _{C\dot{M}O^{p}(\mathbb{Q}_{p}^{n})} = \sup_{\gamma \in \mathbb{Z}} \biggl(\frac{1}{ \vert B_{\gamma} \vert _{H}} \int _{B_{\gamma}} \bigl\vert f( \mathbf{t}) - f_{B_{\gamma}} \bigr\vert ^{p} \,d\mathbf{t} \biggr)^{1/p} < \infty , $$

(3)

where $f_{B_{\gamma}}=\frac{1}{|B_{\gamma}|_{H}} \int _{B_{\gamma}} f( \mathbf{t}) \,d\mathbf{t}$, $|B_{\gamma}|_{H}$ denotes the Haar measure of $B_{\gamma}$.

Definition 1.2

([38])

Let $\alpha \in \mathbb{R}$, $\lambda \geq 0$, $0< p<\infty$, and $0< q<\infty$. The homogeneous p-adic Morrey–Herz space is defined by

$$ M\dot{K}^{\alpha ,\lambda}_{q,p}\bigl(\mathbb{Q}_{p}^{n} \bigr)=\bigl\{ f\in L^{q}_{\mathrm{loc}}\bigl( \mathbb{Q}_{p}^{n} \setminus \{0\}\bigr): \Vert f \Vert _{M\dot{K}^{\alpha ,\lambda}_{q,p}( \mathbb{Q}_{p}^{n})}< \infty \bigr\} ,$$

where

$$ \Vert f \Vert _{M\dot{K}^{\alpha ,\lambda}_{q,p}(\mathbb{Q}_{p}^{n})}=\sup_{k_{0} \in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{k \alpha q} \Vert f\chi _{k} \Vert ^{q}_{L^{p}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/q}.$$

The above space is reduced to Herz space in p-adic field if $M\dot{K}^{\alpha ,0}_{q,p}(\mathbb{Q}_{p}^{n})=\dot{K}^{\alpha ,q}_{p}( \mathbb{Q}_{p}^{n})$, see [39] for details.

Definition 1.3

([38])

Let $\gamma \in \mathbb{R}^{+}$. The Lipschitz space $\Lambda _{\gamma}(\mathbb{Q}_{p}^{n})$ is the space of all measurable function f on $\mathbb{Q}_{p}^{n}$ such that

$$ \Vert f \Vert _{\Lambda _{\gamma}(\mathbb{Q}_{p}^{n})}=\sup_{\mathbf{x}, \mathbf{h}\in \mathbb{Q}_{p}^{n},\mathbf{h}\neq 0} \frac{ \vert f(\mathbf{x}+\mathbf{h})-f(\mathbf{x}) \vert }{ \vert \mathbf{h} \vert _{p}^{\gamma}}< \infty .$$

Now, we move towards the first key result of the article.

2 Estimates for $I^{p}_{\beta}$

In the following section we obtain the boundedness of p-adic fractional integral operator on Morrey–Herz spaces.

Theorem 2.1

Let $0<\beta <n$, $1< p_{1},p_{2}<\infty$, $p_{1}'< p_{2}<\infty$, $0< q_{1},q_{2}<\infty$, and $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{\beta}{n}$. Suppose that $-n/p_{2}<\alpha <n/p_{1}'$ and $0<\lambda <\alpha$, then

$$ \bigl\Vert I^{p}_{\beta}f \bigr\Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{2},p_{2}}( \mathbb{Q}_{p}^{n})} \leq C \Vert f \Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}( \mathbb{Q}_{p}^{n})} $$

for all $f\in M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})$.

Proof of Theorem 2.1

Since $f\in M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})$ and $i\in \mathbb{Z}$, we can write

$$ f(\mathbf{x})=\sum_{i=-\infty}^{\infty}f( \mathbf{x})\chi _{i}( \mathbf{x})=\sum_{i=-\infty}^{\infty}f_{i}( \mathbf{x}).$$

We proceed as

$$ \begin{aligned}[b] \bigl\Vert I^{p}_{\beta}(f) \bigr\Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{2},p_{2}}( \mathbb{Q}_{p}^{n})}&=\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{\alpha q_{2}k} \bigl\Vert I^{p}_{\beta}(f) \chi _{k} \bigr\Vert ^{q_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/q_{2}} \\ &\leq \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{\alpha q_{2}k}\sum _{i=-\infty}^{k} \bigl\Vert I^{p}_{\beta}(f_{i}) \chi _{k} \bigr\Vert ^{q_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/q_{2}} \\ &\quad {} +\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{\alpha q_{2}k}\sum _{i=k+1}^{\infty} \bigl\Vert I^{p}_{\beta}(f_{i}) \chi _{k} \bigr\Vert ^{q_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/q_{2}} \\ &=I+\mathit{II}. \end{aligned} $$

(4)

Now, we estimate I, noticing that for each $\mathbf{x}\in S_{k}$ and $\mathbf{t}\in S_{i}$ with $i\leq k$. Now, since $\frac{\beta}{n}=\frac{1}{p_{1}}-\frac{1}{p_{2}}$ and applying Hölder’s inequality, we have

$$\begin{aligned} \bigl\Vert I^{p}_{\beta}(f_{i}) \chi _{k} \bigr\Vert ^{p_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} &= \int _{S_{k}} \biggl\vert \int _{S_{i}} \frac{f(\mathbf{t})}{ \vert \mathbf{x}-\mathbf{t} \vert _{p}^{n-\beta}}\,d\mathbf{t} \biggr\vert ^{p_{2}} \,d\mathbf{x} \\ &\leq \int _{S_{k}} \int _{S_{i}} \biggl\vert \frac{f(\mathbf{t})}{ \vert \mathbf{x}-\mathbf{t} \vert _{p}^{n-\beta}}\,d\mathbf{t} \biggr\vert ^{p_{2}} \,d\mathbf{x} \\ &\leq p^{kp_{2}( \beta -n)} \int _{S_{k}} \biggl( \int _{S_{i}} \bigl\vert f( \mathbf{t}) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\leq p^{kp_{2}( \beta -n)}p^{kn} \biggl( \biggl( \int _{S_{i}} \bigl\vert f( \mathbf{t}) \bigr\vert ^{p_{1}}\,d\mathbf{t} \biggr)^{1/p_{1}} \biggl( \int _{S_{i}}\,d\mathbf{t} \biggr)^{1/p_{1}'} \biggr)^{p_{2}} \\ &\leq Cp^{(i-k)n/p_{1}'} \Vert f_{i} \Vert ^{p_{2}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned}$$

(5)

By means of Jensen’s inequality and (5), we have

$$\begin{aligned} \begin{aligned} I&=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{k\alpha q_{2}}\sum _{i=-\infty}^{k} \bigl\Vert I^{p}_{\beta}(f_{i}) \chi _{k} \bigr\Vert ^{q_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/q_{2}} \\ &\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{k\alpha q_{1}}\sum _{i=-\infty}^{k} \bigl\Vert I^{p}_{\beta}(f_{i}) \chi _{k} \bigr\Vert ^{q_{1}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/q_{1}} \\ &\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{k\alpha q_{1}}\sum _{i=-\infty}^{k} \bigl(p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \bigr)^{q_{1}} \Biggr)^{1/q_{1}}. \end{aligned} \end{aligned}$$

From now on, we divide our proof in two cases. When $1< q_{1}<\infty$ and $\alpha < n/p_{1}'$. Applying Hölder’s inequality, we get

$$\begin{aligned} I^{q_{1}}&=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda}\sum _{k=- \infty}^{k_{0}} \Biggl(\sum _{i=-\infty}^{k}p^{i\alpha} \Vert f_{i} \Vert _{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(i-k)(n/p_{1}'-\alpha )} \Biggr)^{q_{1}} \\ &\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda}\sum _{k=-\infty}^{k_{0}} \Biggl(\sum _{i=-\infty}^{k}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(i-k)(n/p_{1}'-\alpha )q_{1}/2} \Biggr) \\ &\quad {} \times \Biggl(\sum_{i=-\infty}^{k}p^{(i-k)(n/p_{1}'-\alpha )q_{1}/2} \Biggr)^{q_{1}/q_{1}'} \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda}\sum _{i=-\infty}^{k_{0}}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})}\sum _{k=i}^{ \infty}p^{(i-k)(n/p_{1}'-\alpha )q_{1}/2} \\ &\leq C \Vert f \Vert ^{q_{1}}_{\dot{MK}^{\alpha ,\lambda}_{q_{1},p_{1}}( \mathbb{Q}_{p}^{n})}. \end{aligned}$$

When $0< q_{1}\leq 1$, we tackle the case as follows:

$$\begin{aligned} I^{q_{1}}&\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda}\sum _{k=- \infty}^{k_{0}}p^{k\alpha q_{1}}\sum _{i=-\infty}^{k} \bigl(p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \bigr)^{q_{1}} \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda}\sum _{k=-\infty}^{k_{0}} \Biggl(\sum _{i=-\infty}^{k}p^{i\alpha} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})}p^{(i-k)(n/p_{1}'- \alpha )} \Biggr)^{q_{1}} \\ &= C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda}\sum _{k=-\infty}^{ \infty}\sum_{i=-\infty}^{k}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(i-k)(n/p_{1}'-\alpha )q_{1}} \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda}\sum _{i=-\infty}^{k_{0}}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})}\sum _{k=i}^{ \infty}p^{(i-k)(n/p_{1}'-\alpha )q_{1}} \\ &\leq C \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}( \mathbb{Q}_{p}^{n})}. \end{aligned}$$

Now we evaluate II for each $\mathbf{x}\in S_{k}$ and $\mathbf{t}\in S_{i}$ with $k+1\leq i$. By means of Hölder’s inequality, we arrive at

$$\begin{aligned} \bigl\Vert I^{p}_{\beta}(f_{i}) \chi _{k} \bigr\Vert ^{p_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} &= \int _{S_{k}} \biggl\vert \int _{S_{i}} \frac{f(\mathbf{t})}{ \vert \mathbf{x}-\mathbf{t} \vert _{p}^{n-\beta}}\,d\mathbf{t} \biggr\vert ^{p_{2}} \,d\mathbf{x} \\ &\leq \int _{S_{k}} \biggl( \int _{S_{i}}p^{i(\beta -n)} \bigl\vert f(\mathbf{t}) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\leq Cp^{(k-i)n/p_{2}} \Vert f_{i} \Vert ^{p_{2}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned}$$

(6)

Now, by Jensen’s inequality and (6), we have

$$ \mathit{II}\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{k\alpha q_{1}}\sum _{i=k+1}^{\infty} \bigl(p^{(k-i)n/p_{2}} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \bigr)^{q_{1}} \Biggr)^{1/q_{1}}. $$

In the rest of the proof we consider a couple of cases. When $1< q_{1}<\infty$, since $\alpha >-n/p_{2}$ and by the application of Hölder’s inequality, we have

$$ \begin{aligned} (\mathit{II})^{q_{1}}&\leq C\sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \Biggl(\sum_{i=k+1}^{\infty}p^{i\alpha }p^{(k-i)(n/p_{2}+ \alpha )} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \\ &\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=- \infty}^{k_{0}} \Biggl(\sum _{i=k+1}^{\infty}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \Biggr) \\ &\quad {} \times \Biggl(\sum_{i=k+1}^{\infty}p^{(k-i)(n/p_{2}+\alpha )q_{1}'/2} \Biggr)^{q_{1}/q_{1}'} \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=-\infty}^{k_{0}} \Biggl(\sum _{i=k+1}^{\infty}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \Biggr) \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=-\infty}^{k_{0}} \sum_{i=k+1}^{k_{0}+1}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \\ &\quad {} +C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=- \infty}^{k_{0}}\sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \\ &=\mathit{II}_{1}+\mathit{II}_{2}. \end{aligned} $$

Because of $\alpha >-n/p_{2}$, the estimation of $\mathit{II}_{1}$ is given by

$$ \begin{aligned} \mathit{II}_{1}&=C\sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{i=- \infty}^{k_{0}+1}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}\sum _{k=-\infty}^{i-1}p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{i=-\infty}^{k_{0}+1}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &\leq C \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}( \mathbb{Q}_{p}^{n})}. \end{aligned} $$

Next, since $\lambda <\alpha /2$, we have

$$ \begin{aligned} \mathit{II}_{2}&=C\sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=- \infty}^{k_{0}} \sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}}p^{(k-i)(n/p_{2}+ \alpha )q_{1}/2} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=- \infty}^{k_{0}}\sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}}p^{(k-i)(n/p_{2}+ \alpha )q_{1}/2}p^{i\lambda q_{1}} \\ &\quad {} \times p^{-i\lambda q_{1}} \Biggl(\sum_{l=-\infty}^{i}p^{ \alpha lq_{1}} \Vert f_{l} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr) \\ &\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=- \infty}^{k_{0}}\sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}}p^{(k-i)(n/p_{2}+ \alpha )q_{1}/2}p^{i\lambda q_{1}} \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha , \lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})} \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{k\alpha q_{1}/2} \Biggr) \Biggl(\sum _{i=k_{0}+2}^{ \infty}p^{i\lambda q_{1}(\lambda -\alpha /2)} \Biggr) \Vert f \Vert ^{q_{1}}_{M \dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})} \\ &=C \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned} $$

If $0< q_{1}\leq 1$, we have

$$ \begin{aligned} (\mathit{II})^{q_{1}}&\leq C\sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{\infty} \sum_{i=k+1}^{\infty}p^{i\alpha q_{1}}p^{(k-i)(n/p_{2}+ \alpha )q_{1}} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})}^{q_{1}} \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=-\infty}^{k_{0}} \sum_{i=k+1}^{k_{0}+1}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(k-i)(n/p_{2}+\alpha )q_{1}} \\ &\quad {} +C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=- \infty}^{k_{0}}\sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(k-i)(n/p_{2}+\alpha )q_{1}} \\ &=\mathit{II}_{1}'+\mathit{II}_{2}'. \end{aligned} $$

Since $\alpha >-n/p_{2}$, the estimate of $\mathit{II}_{1}'$ is done as follows:

$$ \begin{aligned} \mathit{II}_{1}'&=C\sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{i=- \infty}^{k_{0}+1}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}\sum _{k=-\infty}^{i-1}p^{(k-i)(n/p_{2}+\alpha )q_{1}} \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{i=-\infty}^{k_{0}+1}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &\leq C \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}( \mathbb{Q}_{p}^{n})}. \end{aligned} $$

Now $\alpha >0$ and $\lambda <\alpha$, the estimation of $\mathit{II}_{2}'$ is carried out as follows:

$$\begin{aligned} \mathit{II}_{2}'&=C\sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=- \infty}^{k_{0}} \sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}}p^{(k-i)(n/p_{2}+ \alpha )q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=- \infty}^{k_{0}}\sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}}p^{(k-i)(n/p_{2}+ \alpha )q_{1}}p^{i\lambda q_{1}} \\ &\quad {} \times p^{-i\lambda q_{1}} \Biggl(\sum_{l=-\infty}^{i}p^{ \alpha lq_{1}} \Vert f_{l} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr) \\ &\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=- \infty}^{k_{0}}\sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}}p^{(k-i)(n/p_{2}+ \alpha )q_{1}}p^{i\lambda q_{1}} \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha , \lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})} \\ &=C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{k\alpha q_{1}} \Biggr) \Biggl(\sum _{i=k_{0}+2}^{\infty}p^{i \lambda q_{1}(\lambda -\alpha )} \Biggr) \Vert f \Vert ^{q_{1}}_{M\dot{K}^{ \alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})} \\ &=C \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned}$$

By combining the estimates of I and II, the proof of Theorem 2.1 is completed. □

3 $C\dot{M}O$ and Lipschitz estimates for $I^{p,b}_{\beta}$

In the present section we establish the $C\dot{M}O(\mathbb{Q}_{p}^{n})$ and Lipschitz estimates for commutators of p-adic fractional integral operator on Morrey–Herz spaces. However, before doing that, we need an important result which is as follows.

Lemma 3.1

([40])

Let $b\in C\dot{M}O^{1}(\mathbb{Q}_{p}^{n})$ and let k, $i\in \mathbb{Z}$. Then

$$ \bigl\vert b(\mathbf{t})-b_{B_{i}} \bigr\vert \leq \bigl\vert b( \mathbf{t})-b_{B_{k}} \bigr\vert +p^{n} \vert k-i \vert \Vert b \Vert _{C\dot{M}O^{1}(\mathbb{Q}_{p}^{n})}. $$

Theorem 3.2

Let $<\beta <n$, $b\in C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})$, $1< p_{1},p_{2}<\infty$, $p_{1}'< p_{2}<\infty$, $0< q_{1},q_{2}<\infty$, and $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{\beta}{n}$. Suppose $-n/p_{2}<\alpha <n/p_{1}'$ and $0<\lambda <\alpha$, then

$$ \bigl\Vert I^{p,b}_{\beta}f \bigr\Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{2},p_{2}}( \mathbb{Q}_{p}^{n})}\leq C \Vert b \Vert _{C\dot{M}O^{\max \{p_{1}',p_{2}\}}( \mathbb{Q}_{p}^{n})} \Vert f \Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}( \mathbb{Q}_{p}^{n})} $$

for all $f\in M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})$.

Theorem 3.3

Let $0<\beta <n$, $b\in \Lambda _{\gamma}(\mathbb{Q}_{p}^{n})$ $(0<\gamma <1)$, $1< p_{1}$, $p_{2}<\infty$, $p_{1}'< p_{2}<\infty$, $0< q_{1}$, $q_{2}<\infty$, and $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{\beta +\gamma}{n}$. Suppose $-n/p_{2}<\alpha <n/p_{1}'$ and $0<\lambda <\alpha$, then

$$ \bigl\Vert I^{p,b}_{\beta}f \bigr\Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{2},p_{2}}( \mathbb{Q}_{p}^{n})}\leq C \Vert b \Vert _{\Lambda _{\gamma}(\mathbb{Q}_{p}^{n})} \Vert f \Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})} $$

for all $f\in M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})$.

Proof of Theorem 3.2

For $i\in \mathbb{Z}$, we can write

$$ f(\mathbf{x})=\sum_{i=-\infty}^{\infty}f( \mathbf{x})\chi _{i}( \mathbf{x})=\sum_{i=-\infty}^{\infty}f_{i}( \mathbf{x}).$$

Thus, we begin

$$ \begin{aligned}[b] \bigl\Vert I^{p,b}_{\beta}(f) \bigr\Vert _{M\dot{K}^{\alpha ,\lambda}_{q_{2},p_{2}}( \mathbb{Q}_{p}^{n})}&=\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{\alpha q_{2}k} \bigl\Vert I^{p,b}_{\beta}(f) \chi _{k} \bigr\Vert ^{q_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/q_{2}} \\ &\leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{\alpha q_{2}k}\sum _{i=-\infty}^{k} \bigl\Vert I^{p,b}_{\beta}(f_{i}) \chi _{k} \bigr\Vert ^{q_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/q_{2}} \\ &\quad {} + C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{\alpha q_{2}k}\sum _{i=k+1}^{\infty} \bigl\Vert I^{p,b}_{ \beta}(f_{i}) \chi _{k} \bigr\Vert ^{q_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/q_{2}} \\ &=L+\mathit{LL}. \end{aligned} $$

(7)

The estimate of L is carried out as follows. For each $\mathbf{x}\in S_{k}$ and $\mathbf{t}\in S_{i}$ with $i\leq k$, we have

$$\begin{aligned} \bigl\Vert I^{p,b}_{\beta}(f_{i}) \chi _{k} \bigr\Vert ^{p_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} &= \int _{S_{k}} \biggl\vert \int _{S_{i}} \frac{f(\mathbf{t})}{ \vert \mathbf{x}-\mathbf{t} \vert _{p}^{n-\beta}}\bigl(b( \mathbf{x}) -b( \mathbf{t})\bigr) \,d\mathbf{t} \biggr\vert ^{p_{2}} \,d\mathbf{x} \\ &\leq Cp^{kp_{2}(\beta -n)} \int _{S_{k}} \biggl( \int _{S_{i}} \bigl\vert f( \mathbf{t}) \bigl(b( \mathbf{x})-b(\mathbf{t})\bigr) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\leq Cp^{kp_{2}(\beta -n)} \int _{S_{k}} \biggl( \int _{S_{i}} \bigl\vert f( \mathbf{t}) \bigl(b( \mathbf{x})-b_{B_{k}}\bigr) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\quad {} +Cp^{kp_{2}(\beta -n)} \int _{S_{k}} \biggl( \int _{S_{i}} \bigl\vert f( \mathbf{t}) \bigl(b( \mathbf{t})-b_{B_{k}}\bigr) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &=L_{1}+L_{2}. \end{aligned}$$

(8)

By noticing that $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{\beta}{n}$ and applying Hölder’s inequality, we get

$$\begin{aligned} L_{1}&\leq Cp^{kp_{2}(\beta -n)} \int _{B_{k}} \bigl\vert b(\mathbf{x})-b_{B_{k}} \bigr\vert ^{p_{2}} \\ &\quad {} \times \biggl( \biggl( \int _{S_{i}} \bigl\vert f(\mathbf{t}) \bigr\vert ^{p_{1}}\,d\mathbf{t} \biggr)^{1/p_{1}} \biggl( \int _{S_{i}}\,d\mathbf{t} \biggr)^{1/p_{1}'} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\leq C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{p_{2}}(\mathbb{Q}_{p}^{n})} \bigl\{ p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \bigr\} ^{p_{2}}. \end{aligned}$$

(9)

Next we move towards the estimation of $L_{2}$, for this we use Lemma 3.1.

$$\begin{aligned} L_{2}&\leq Cp^{kp_{2}(\beta -n)} \int _{S_{k}} \biggl( \int _{S_{i}} \bigl\vert f( \mathbf{t}) \bigl(b( \mathbf{t})-b_{B_{i}}\bigr) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\quad {} +C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})}p^{kp_{2}(\beta -n)} \int _{S_{k}} \biggl((k-i) \int _{S_{i}} \bigl\vert f(\mathbf{t}) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &=L_{21}+L_{22}. \end{aligned}$$

(10)

We make use of Hölder’s inequality along with $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{\beta}{n}$ to evaluate $L_{21}$.

$$\begin{aligned} L_{21}&\leq Cp^{kp_{2}(\beta -n)} \int _{S_{k}} \biggl( \biggl( \int _{S_{i}} \bigl\vert b( \mathbf{t})-b_{B_{i}} \bigr\vert ^{p_{1}'}\,d\mathbf{t} \biggr)^{1/p_{1}'} \\ &\quad {} \times \biggl( \int _{S_{i}} \bigl\vert f(\mathbf{t}) \bigr\vert ^{p_{1}} \,d\mathbf{t} \biggr)^{1/p_{1}} \biggr)^{p_{2}} \,d\mathbf{x} \\ &\leq C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{p_{1}'}(\mathbb{Q}_{p}^{n})}p^{(i-k)np_{2}/p_{1}'} \Vert f_{i} \Vert ^{p_{2}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &=C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{p_{1}'}(\mathbb{Q}_{p}^{n})} \bigl(p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \bigr)^{p_{2}}. \end{aligned}$$

In a similar way we can estimate $L_{22}$. Since $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{\beta}{n}$, using Hölder’s inequality, we obtain

$$\begin{aligned} L_{22}&\leq C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})}p^{kp_{2}( \beta -n)} \\ &\quad {} \times \int _{S_{k}} \biggl((k-i) \biggl( \int _{S_{i}} \bigl\vert f( \mathbf{t}) \bigr\vert ^{p_{1}}\,d\mathbf{t} \biggr)^{1/p_{1}} \biggl( \int _{S_{i}}\,d\mathbf{t} \biggr)^{1/p_{1}'} \biggr)^{p_{2}}\,d\mathbf{x} \\ &=C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})} \bigl((k-i)p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \bigr)^{p_{2}}. \end{aligned} $$

Combining the above values and by means of Jensen’s inequality, we have

$$\begin{aligned} L &\leq C \Vert b \Vert _{\mathit{CMO}^{p_{2}}(\mathbb{Q}_{p}^{n})}\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{k \alpha q_{1}} \Biggl(\sum_{i=-\infty}^{k}p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \Biggr)^{1/q_{1}} \\ &\quad {} +C \Vert b \Vert _{\mathit{CMO}^{p_{1}'}(\mathbb{Q}_{p}^{n})} \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{k \alpha q_{1}} \Biggl(\sum_{i=-\infty}^{k}p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \Biggr)^{1/q_{1}} \\ &\quad {} +C \Vert b \Vert _{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})}\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{k \alpha q_{1}} \Biggl(\sum_{i=-\infty}^{k}(k-i)p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \Biggr)^{1/q_{1}} \\ &\leq C \Vert b \Vert _{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})}\sup_{k_{0} \in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{k \alpha q_{1}} \Biggl(\sum_{i=-\infty}^{k}(k-i)p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \Biggr)^{1/q_{1}}. \end{aligned}$$

From here on, we consider a couple of cases. When $1< q_{1}<\infty$, applying Hölder’s inequality from the outset, we have

$$\begin{aligned} L^{q_{1}}&=C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \Biggl(\sum_{i=-\infty}^{k}p^{i\alpha} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &\quad {} \times (k-i)p^{(i-k)(n/p_{1}'-\alpha )} \Biggr)^{q_{1}} \\ &\leq C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{i=-\infty}^{k}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(i-k)(n/p_{1}' -\alpha )q_{1}/2} \\ &\quad {} \times \Biggl(\sum_{i=-\infty}^{k}(k-i)^{q_{1}'}p^{(i-k)(n/p_{1}' -\alpha )q_{1}'/2} \Biggr)^{q_{1}/q_{1}'} \\ &=C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{i=-\infty}^{k_{0}}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &\quad {} \times \sum_{k=i}^{k_{0}}p^{(i-k)(n/p_{1}' -\alpha )q_{1}/2} \\ &=C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned}$$

We consider the other case, when $0< q_{1}\leq 1$, since $\alpha < n/p_{1}'$, we handle the case as

$$\begin{aligned} L^{q_{1}}&=C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=-\infty}^{k_{0}}p^{k \alpha q_{1}} \Biggl(\sum_{i=-\infty}^{k}(k-i) \\ &\quad {} \times p^{(i-k)n/p_{1}'} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \\ &\leq C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{i=-\infty}^{k}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \\ &\quad {} \times (k-i)^{q_{1}}p^{(i-k)(n/p_{1}' -\alpha )q_{1}} \\ &=C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{i=-\infty}^{k_{0}}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ & \quad {}\times \sum_{k=i}^{k_{0}}(k-i)^{q_{1}}p^{(i-k)(n/p_{1}' - \alpha )q_{1}} \\ &=C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned}$$

Next we turn our attention towards estimating LL. For each $\mathbf{x}\in S_{k}$ and $\mathbf{t}\in S_{i}$ with $k+1\leq i$, we proceed as follows:

$$\begin{aligned} \bigl\Vert I^{p,b}_{\beta}(f_{i}) \chi _{k} \bigr\Vert ^{p_{2}}_{L^{p_{2}}(\mathbb{Q}_{p}^{n})} &= \int _{S_{k}} \biggl\vert \int _{S_{i}} \frac{f(\mathbf{t})}{ \vert \mathbf{x}-\mathbf{t} \vert _{p}^{n-\beta}}\bigl(b( \mathbf{x}) -b( \mathbf{t})\bigr) \,d\mathbf{t} \biggr\vert ^{p_{2}} \,d\mathbf{x} \\ &\leq \int _{S_{k}} \biggl( \int _{S_{i}}p^{i(\beta -n)} \bigl\vert f(\mathbf{t}) \bigl(b( \mathbf{x})-b(\mathbf{t})\bigr) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\leq C \int _{S_{k}} \biggl( \int _{S_{i}}p^{i(\beta -n)} \bigl\vert f(\mathbf{t}) \bigl(b( \mathbf{x})-b_{B_{k}}\bigr) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\quad {} +C \int _{S_{k}} \biggl( \int _{S_{i}}p^{i(\beta -n)} \bigl\vert f( \mathbf{t}) \bigl(b(\mathbf{t})-b_{B_{k}}\bigr) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &=\mathit{LL}_{1}+\mathit{LL}_{2}. \end{aligned}$$

(11)

We apply Hölder’s inequality for evaluating $\mathit{LL}_{1}$. Since $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{\beta}{n}$, we obtain

$$\begin{aligned} \mathit{LL}_{1}&\leq C \int _{B_{k}} \bigl\vert b(\mathbf{x})-b_{B_{k}} \bigr\vert ^{p_{2}} \biggl(p^{i(\beta -n)} \biggl( \int _{S_{i}} \bigl\vert f(\mathbf{t}) \bigr\vert ^{p_{1}}\,d\mathbf{t} \biggr)^{1/p_{1}} \biggl( \int _{S_{i}}\,d\mathbf{t} \biggr)^{1/p_{1}'} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\leq C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{p_{2}}(\mathbb{Q}_{p}^{n})} \bigl(p^{(k-i)n/p_{2}} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \bigr)^{p_{2}}. \end{aligned}$$

(12)

The estimation of $\mathit{LL}_{2}$ is done by using Lemma 3.1.

$$\begin{aligned} \mathit{LL}_{2}&\leq C \int _{S_{k}} \biggl( \int _{S_{i}}p^{i(\beta -n)} \bigl\vert f( \mathbf{t}) \bigl(b(\mathbf{t})-b_{B_{i}}\bigr) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &\quad {} +C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})} \int _{S_{k}} \biggl((i-k) \int _{S_{i}}p^{i(\beta -n)} \bigl\vert f(\mathbf{t}) \bigr\vert \,d\mathbf{t} \biggr)^{p_{2}}\,d\mathbf{x} \\ &=\mathit{LL}_{21}+\mathit{LL}_{22}. \end{aligned}$$

(13)

Implying Hölder’s inequality to evaluate $\mathit{LL}_{21}$, we have

$$\begin{aligned}[b] \mathit{LL}_{21}&\leq C \int _{S_{k}} \biggl(p^{i(\beta -n)} \biggl( \int _{S_{i}} \bigl\vert b( \mathbf{t})-b_{B_{i}} \bigr\vert ^{p_{1}'}\,d\mathbf{t} \biggr)^{1/p_{1}'} \\ &\quad {} \times \biggl( \int _{S_{i}} \bigl\vert f(\mathbf{t}) \bigr\vert ^{p_{1}} \,d\mathbf{t} \biggr)^{1/p_{1}} \biggr)^{p_{2}} \,d\mathbf{x} \\ &\leq C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{p_{1}'}(\mathbb{Q}_{p}^{n})} \bigl(p^{(k-i)n/p_{2}} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \bigr)^{p_{2}}. \end{aligned} $$

Now, we estimate $\mathit{LL}_{22}$, for this Hölder’s inequality is helpful, therefore we arrive at

$$\begin{aligned} \mathit{LL} _{22}&\leq C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})} \\ &\quad {} \times \int _{S_{k}} \biggl((i-k)p^{i(\beta -n)} \biggl( \int _{S_{i}} \bigl\vert f( \mathbf{t}) \bigr\vert ^{p_{1}}\,d\mathbf{t} \biggr)^{1/p_{1}} \biggl( \int _{S_{i}}\,d\mathbf{t} \biggr)^{1/p_{1}'} \biggr)^{p_{2}}\,d\mathbf{x} \\ &=C \Vert b \Vert ^{p_{2}}_{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})} \bigl((i-k)p^{(k-i)n/p_{2}} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \bigr)^{p_{2}}. \end{aligned}$$

(14)

Combining the above values and by means of Jensen’s inequality, we have

$$\begin{aligned} \mathit{LL} &\leq C \Vert b \Vert _{C\dot{M}O^{p_{2}}(\mathbb{Q}_{p}^{n})}\sup_{k_{0} \in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{k \alpha q_{1}} \Biggl(\sum_{i=k+1}^{\infty}p^{(k-i)n/p_{2}} \Vert f_{i} \Vert _{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \Biggr)^{1/q_{1}} \\ &\quad {} +C \Vert b \Vert _{C\dot{M}O^{p_{1}'}(\mathbb{Q}_{p}^{n})} \sup_{k_{0} \in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{k \alpha q_{1}} \Biggl(\sum_{i=k+1}^{\infty}p^{(k-i)n/p_{2}} \Vert f_{i} \Vert _{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \Biggr)^{1/q_{1}} \\ &\quad {}+ C \Vert b \Vert _{C\dot{M}O^{1}(\mathbb{Q}_{p}^{n})}\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{k \alpha q_{1}} \Biggl(\sum_{i=k+1}^{\infty}(i-k) \\ &\quad {} \times p^{(k-i)n/p_{2}} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \Biggr)^{1/q_{1}} \\ &\leq C \Vert b \Vert _{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda} \Biggl(\sum_{k=-\infty}^{k_{0}}p^{k \alpha q_{1}} \Biggl(\sum_{i=k+1}^{\infty}(i-k) \\ &\quad {} \times p^{(k-i)n/p_{2}} \Vert f_{i} \Vert _{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \Biggr)^{1/q_{1}}. \end{aligned}$$

From this point on, we consider two cases. When $1< q_{1}<\infty$ and by Hölder’s inequality, we get

$$ \begin{aligned} (\mathit{LL})^{q_{1}}&\leq C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}( \mathbb{Q}_{p}^{n})}\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \Biggl(\sum _{i=k+1}^{\infty}p^{i\alpha }p^{(k-i)(n/p_{2}+ \alpha )} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{q_{1}} \\ &\leq C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \Biggl(\sum_{i=k+1}^{\infty}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \\ &\quad {} \times p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \Biggr) \Biggl(\sum_{i=k+1}^{ \infty}p^{(k-i)(n/p_{2}+\alpha )q_{1}'/2} \Biggr)^{q_{1}/q_{1}'} \\ &=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{i=k+1}^{\infty}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})} p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \\ &=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=-\infty}^{k_{0}} \sum_{i=k+1}^{k_{0}+1}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})}p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \\ &\quad {} +C \Vert b \Vert ^{q_{1}}_{C \dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=-\infty}^{k_{0}}\sum_{i=k_{0}+2}^{ \infty}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})}p^{(k-i)(n/p_{2}+ \alpha )q_{1}/2} \\ &=\mathit{LL}'_{1}+\mathit{LL}'_{2}. \end{aligned} $$

Since $\alpha >-n/p_{2}$, the evaluation of $\mathit{LL}'_{1}$ is obtained by

$$ \begin{aligned} \mathit{LL}'_{1}&=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \sum_{i=-\infty}^{k_{0}+1}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &\quad {} \times \sum_{k=-\infty}^{i-1}p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \\ &=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{i=-\infty}^{k_{0}+1}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &\leq C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned} $$

Next, by noticing that $\lambda <\alpha /2$, we have

$$\begin{aligned} \mathit{LL}'_{2}&=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}} \sum _{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}} \\ &\quad {} \times p^{(k-i)(n/p_{2}+\alpha )q_{1}/2} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \\ &\leq C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}}p^{(k-i)(n/p_{2}+\alpha )q_{1}/2}p^{i \lambda q_{1}} \\ &\quad {} \times p^{-i\lambda q_{1}} \Biggl(\sum_{l=-\infty}^{i}p^{ \alpha lq_{1}} \Vert f_{l} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr) \\ &\leq C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum _{k=-\infty}^{k_{0}} \sum_{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}} \\ &\quad {} \times p^{(k-i)(n/p_{2}+\alpha )q_{1}/2}p^{i\lambda q_{1}} \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})} \\ &=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{k\alpha q_{1}/2} \Biggr) \Biggl(\sum _{i=k_{0}+2}^{ \infty}p^{i\lambda q_{1}(\lambda -\alpha /2)} \Biggr) \\ &\quad {} \times \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}( \mathbb{Q}_{p}^{n})} \\ &=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned}$$

For the second case $0< q_{1}\leq 1$, we are down to

$$ \begin{aligned} (\mathit{LL})^{q_{1}}&\leq C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}( \mathbb{Q}_{p}^{n})}\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{\infty}\sum _{i=k+1}^{\infty}p^{i\alpha q_{1}} \\ &\quad {} \times p^{(k-i)(n/p_{2}+\alpha )q_{1}} \Vert f_{i} \Vert _{L^{p_{1}}( \mathbb{Q}_{p}^{n})}^{q_{1}} \\ &=C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{i=k+1}^{k_{0}+1}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \\ &\quad {} \times p^{(k-i)(n/p_{2}+\alpha )q_{1}}+C \Vert b \Vert ^{q_{1}}_{\mathit{CMO}^{ \max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0} \lambda q_{1}} \\ &\quad {} \times \sum_{k=-\infty}^{k_{0}}\sum _{i=k_{0}+2}^{\infty}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} p^{(k-i)(n/p_{2}+ \alpha )q_{1}} \\ &=\mathit{LL}_{1}''+\mathit{LL}_{2}''. \end{aligned} $$

Because $\alpha >-n/p_{2}$, we estimate $\mathit{LL}_{1}''$ as

$$ \begin{aligned} \mathit{LL}_{1}''&=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}( \mathbb{Q}_{p}^{n})}\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \sum_{i=-\infty}^{k_{0}+1}p^{i\alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \\ &\quad {} \times \sum_{k=-\infty}^{i-1}p^{(k-i)(n/p_{2}+\alpha )q_{1}} \\ &=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{i=-\infty}^{k_{0}+1}p^{i \alpha q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \\ &\leq C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned} $$

Since $\alpha >0$ and $\lambda <\alpha$, the estimation of $\mathit{LL}_{2}''$ is given by

$$\begin{aligned} \mathit{LL}_{2}''&=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}( \mathbb{Q}_{p}^{n})}\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \sum_{k=-\infty}^{k_{0}}\sum _{i=k_{0}+2}^{\infty}p^{i\alpha q_{1}} \\ &\quad {} \times p^{(k-i)(n/p_{2}+\alpha )q_{1}} \Vert f_{i} \Vert ^{q_{1}}_{L^{p_{1}}( \mathbb{Q}_{p}^{n})} \\ &\leq C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}}\sum_{k=-\infty}^{k_{0}} \sum_{j=k_{0}+2}^{\infty}p^{i\alpha q_{1}}p^{(k-i)(n/p_{2}+\alpha )q_{1}}p^{i \lambda q_{1}} \\ &\quad {} \times p^{-i\lambda q_{1}} \Biggl(\sum_{l=-\infty}^{i}p^{ \alpha lq_{1}} \Vert f_{l} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{Q}_{p}^{n})} \Biggr) \\ &=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \sup _{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda q_{1}} \Biggl(\sum _{k=- \infty}^{k_{0}}p^{k\alpha q_{1}} \Biggr) \\ &\quad {} \times \Biggl(\sum_{i=k_{0}+2}^{\infty}p^{i\lambda q_{1}( \lambda -\alpha )} \Biggr) \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}( \mathbb{Q}_{p}^{n})} \\ &=C \Vert b \Vert ^{q_{1}}_{C\dot{M}O^{\max \{p_{1}',p_{2}\}}(\mathbb{Q}_{p}^{n})} \Vert f \Vert ^{q_{1}}_{M\dot{K}^{\alpha ,\lambda}_{q_{1},p_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned}$$

Combining the estimates of L and LL, we are done with the proof of Theorem 3.2. □

Proof of Theorem 3.3

When $b\in \Lambda _{\gamma}(\mathbb{Q}_{p}^{n})$, the inequality

$$ \bigl\vert I^{p,b}_{\beta}f(\mathbf{x}) \bigr\vert \leq C \Vert b \Vert _{\Lambda _{\gamma}( \mathbb{Q}_{p}^{n})} \bigl\vert I^{p,b}_{\beta +\gamma}f(x) \bigr\vert $$

can be easily obtained under the assumptions of Theorem 3.3, and the proof of Theorem 3.3 is a consequence of Theorem 2.1. □

4 Application

As an application, we can characterize p-adic Morrey–Herz space in terms of p-adic wavelets as they are used to construct solutions of p-adic Navier–Stokes equation and semi-linear evolutionary pseudo-differential equations, see for instance [41, 42].

5 Conclusion

Herein, the boundedness of a p-adic fractional integral operator on p-adic Morrey–Herz spaces is studied. In addition, the property of boundedness for commutators of the p-adic fractional integral operator on Morrey–Herz spaces in the p-adic field is also obtained when the symbol function is from $C\dot{M}O$ and Lipschitz spaces.