The boundedness of commutators of rough p-adic fractional Hardy type operators on Herz-type spaces

Abstract

In this paper, we obtain some inequalities about commutators of a rough p-adic fractional Hardy-type operator on Herz-type spaces when the symbol functions belong to two different function spaces.

1 Introduction

During the last several decades, the p-adic analysis has cemented its role in the field of mathematical physics (see, for example, [1, 22, 32, 33]). That stimulates researchers to pay attention to harmonic analysis on p-adic fields [18–21, 24, 30, 31, 35], which has direct implications in the stochastic process [2, 3], theoretical biology [6], and p-adic pseudo-differential equations [23, 34]. In continuation of the ongoing research, the present paper considers an extension of the investigation of p-adic Hardy-type operators discussed in [19–21, 25, 36, 37].

For every non-zero rational number x there is a unique $\gamma =\gamma (x)\in \mathbb{Z}$ such that $x=p^{\gamma }m/n$, where $p\geq 2$ is a fixed prime number which is coprime to $m,n\in \mathbb{Z}$. We define a mapping $|\cdot |_{p}:\mathbb{Q}\rightarrow \mathbb{R_{+}}$ as follows:

$$ \vert x \vert _{p}= \textstyle\begin{cases} p^{-\gamma } & \text{if } x\neq 0, \\ 0 & \text{if } x=0. \end{cases} $$

(1.1)

The p-adic absolute value $|\cdot |_{p}$ has many properties of the usual real norm $|\cdot |$ with an additional non-Archimedean property,

$$ \vert x+y \vert _{p}\le \max \bigl\{ \vert x \vert _{p}, \vert y \vert _{p}\bigr\} . $$

The field of p-adic numbers, denoted by $\mathbb{Q}_{p}$, is the completion of rational numbers with respect to the p-adic absolute value $|\cdot |_{p}$. A p-adic number $x\in \mathbb{Q}_{p}$ can be written in the formal power series as [34]:

$$ x=p^{\gamma }\bigl(\beta _{0}+\beta _{1}p+\beta _{2}p^{2}+\cdots \bigr), $$

(1.2)

where $\gamma \in \mathbb{Z}$ and $\beta _{i}\in \{0,1,\ldots ,p-1\}$, $i=0,1,2,\ldots$ . The p-adic absolute value ensures the convergence of series (1.2) in $\mathbb{Q}_{p}$, because the inequality $|p^{\gamma }\beta _{i}p^{i}|_{p}\leq p^{-\gamma -i}$ holds for all $\gamma \in \mathbb{Z}$ and $i \in \mathbb{N}$.

The n-dimensional vector space $\mathbb{Q}_{p}^{n}$, $n \geq 1$, consists of the vectors $\mathbf{x} = (x_{1}, x_{2}, \ldots ,x_{n})$, where $x_{j}\in \mathbb{Q}_{p}$ and $j=1,2,\ldots ,n$, with the following absolute value:

$$ \vert \mathbf{x} \vert _{p}=\max _{1\leq k \leq n} \vert x_{k} \vert _{p}. $$

(1.3)

For $\gamma \in \mathbb{Z}$ and $\mathbf{a}=(a_{1}, a_{2}, \ldots , a_{n}) \in \mathbb{Q}_{p}^{n}$, we denote by

$$ B_{\gamma }(\mathbf{a})=\bigl\{ \mathbf{x} \in \mathbb{Q}_{p}^{n} \colon \vert \mathbf{x}-\mathbf{a} \vert _{p} \leq p^{\gamma }\bigr\} $$

the closed ball with the center a and radius $p^{\gamma }$ and by

$$ S_{\gamma }(\mathbf{a})=\bigl\{ \mathbf{x} \in \mathbb{Q}_{p}^{n} \colon \vert \mathbf{x}-\mathbf{a} \vert _{p} = p^{\gamma }\bigr\} $$

the corresponding sphere. For $\mathbf{a}=\mathbf{0}$, we write $B_{\gamma }(\mathbf{0})=B_{\gamma }$, and $S_{\gamma }(\mathbf{0})=S_{\gamma }$. It is easy to see that the equalities

$$ \mathbf{a}_{0} + B_{\gamma }=B_{\gamma }( \mathbf{a}_{0}) \quad \text{and} \quad \mathbf{a}_{0} + S_{\gamma }= S_{\gamma }(\mathbf{a}_{0}) = B_{\gamma }(\mathbf{a}_{0}) \setminus B_{\gamma -1}( \mathbf{a}_{0}) $$

hold for all $\mathbf{a}_{0}\in \mathbb{Q}_{p}^{n}$ and $\gamma \in \mathbb{Z}$.

Since $\mathbb{Q}_{p}^{n}$ is a locally compact commutative group under addition, there exists a unique Haar measure dx on $\mathbb{Q}_{p}^{n}$, such that

$$ \int _{B_{0}}d\mathbf{x}= \vert B_{0} \vert _{h} =1, $$

where $|B|_{h}$ denotes the Haar measure of measurable subset B of $\mathbb{Q}_{p}^{n}$. Furthermore, a simple calculation shows that

$$ \bigl\vert B_{\gamma }(\mathbf{a}) \bigr\vert _{h} = p^{n\gamma } \quad \text{and}\quad \bigl\vert S_{ \gamma }(\mathbf{a}) \bigr\vert _{h} = p^{n\gamma }\bigl(1-p^{-n}\bigr) $$

hold for all $\mathbf{a}\in \mathbb{Q}_{p}^{n}$ and $\gamma \in \mathbb{Z}$.

The one-dimensional Hardy operator

$$ \mathcal{H}f(x)=\frac{1}{x} \int _{0}^{x}f(y)\,dy,\quad x>0, $$

where $f \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ is a measurable functions, was introduced by Hardy in [13]. This operator satisfies the inequality:

$$ \Vert \mathcal{H} f \Vert _{L^{q}(\mathbb{R}^{+})}\leq \frac{q}{q-1} \Vert f \Vert _{L^{q}( \mathbb{R}^{+})},\quad 1< q< \infty , $$

(1.4)

where the constant $q/(q-1)$ is sharp. In [7], Faris proposed an extension of the operator $\mathcal{H}$ on higher dimensional Euclidean space $\mathbb{R}^{n}$ which is given by

$$ Hf(\mathbf{x}) = \frac{1}{ \vert \mathbf{x} \vert ^{n}} \int _{ \vert \mathbf{y} \vert \leq \vert \mathbf{x} \vert } f(\mathbf{y}) \,d\mathbf{y}, $$

(1.5)

for $\mathbf{x} = (x_{1}, \ldots , x_{n})$. In addition, Christ and Grafakos [4] obtained the exact value of the norm of operator H defined by (1.5). For boundedness results for these operators on function spaces we refer to some recent publications including [8, 10, 16, 17, 28, 29, 38].

On the other hand, the n-dimensional fractional p-adic Hardy operator

$$ H^{p}_{\alpha } f(\mathbf{x}) = \frac{1}{ \vert \mathbf{x} \vert _{p}^{n-\alpha }} \int _{ \vert \mathbf{y} \vert _{p}\leq \vert \mathbf{x} \vert _{p}}f(\mathbf{y})\,d\mathbf{y} $$

was defined and studied for $f\in L_{1}^{\mathrm{loc}}(\mathbb{Q}_{p}^{n})$ and $0\le \alpha < n$ in [36]. When $\alpha =0$, the operator $H^{p}_{\alpha }$ transfers to the p-adic Hardy-type operator (see [10] for more details). Fu et al. in [9], fixed the optimal bounds of p-adic Hardy operator on $L^{q}(\mathbb{Q}_{p}^{n})$. On the central Morrey space the p-adic Hardy-type operators and their commutators were discussed in [37]. In this connection see also [19, 21, 25].

There is still zero attention towards the rough Hardy operators on the p-adic linear spaces. Motivated by papers cited above and results of Fu et al. in [8], we define the special kind of p-adic rough fractional Hardy operator $H^{p}_{\Omega ,\alpha }$ and its commutators as follows.

Definition 1.1

Let $f \colon \mathbb{Q}_{p}^{n} \to \mathbb{R}$, $b \colon \mathbb{Q}_{p}^{n} \to \mathbb{R}$ be measurable mappings and let $0<\alpha <n$. Then, for $\mathbf{x} \in \mathbb{Q}_{p}^{n} \setminus \{\mathbf{0}\}$, we define the rough p-adic fractional Hardy operator $H^{p}_{\Omega , \alpha }$ by

$$ H^{p}_{\Omega , \alpha } f(\mathbf{x}) = \frac{1}{ \vert \mathbf{x} \vert _{p}^{n-\alpha }} \int _{ \vert \mathbf{y} \vert _{p} \leq \vert \mathbf{x} \vert _{p}} \Omega \bigl( \vert \mathbf{y} \vert _{p} \mathbf{y} \bigr) f( \mathbf{y}) \,d\mathbf{y}, $$

(1.6)

and its commutator $H^{p,b}_{\Omega ,\alpha }$ by

$$ H^{p,b}_{\Omega ,\alpha } f(\mathbf{x}) = \frac{1}{ \vert \mathbf{x} \vert _{p}^{n-\alpha }} \int _{ \vert \mathbf{y} \vert _{p} \leq \vert \mathbf{x} \vert _{p}} \bigl(b (\mathbf{x}) -b(\mathbf{y} )\bigr) \Omega \bigl( \vert \mathbf{y} \vert _{p} \mathbf{y} \bigr) f( \mathbf{y})\,d\mathbf{y}, $$

(1.7)

whenever

$$ \int _{ \vert \mathbf{y} \vert _{p} \leq \vert \mathbf{x} \vert _{p}} \bigl\vert \Omega \bigl( \vert \mathbf{y} \vert _{p} \mathbf{y} \bigr) f(\mathbf{y}) \bigr\vert \,d\mathbf{y} < \infty $$

(1.8)

and

$$ \int _{ \vert \mathbf{y} \vert _{p} \leq \vert \mathbf{x} \vert _{p}} \bigl\vert b(\mathbf{y}) \Omega \bigl( \vert \mathbf{y} \vert _{p} \mathbf{y} \bigr) f(\mathbf{y}) \bigr\vert \,d\mathbf{y} < \infty , $$

(1.9)

where $\Omega \in L^{s}(S_{0}(\mathbf{0}))$, $1\leq s<\infty$.

Remark 1.2

Obviously

$$ \bigl\{ \vert \mathbf{y} \vert _{p} \colon \mathbf{y} \in \mathbb{Q}_{p}^{n}\bigr\} = \bigl\{ p^{ \gamma } \colon \gamma \in \mathbb{Z}\bigr\} \cup \{0\} $$

holds for every integer $n \geq 1$ and prime $p \geq 2$. Since the inclusion

$$ \{0\} \cup \bigl\{ p^{\gamma } \colon \gamma \in \mathbb{Z}\bigr\} \subseteq \mathbb{Q}_{p} $$

holds and $\mathbb{Q}_{p}^{n}$ is a linear space over field $\mathbb{Q}_{p}$, the product $|\mathbf{y}|_{p} \mathbf{y}$ is well defined. Moreover, if a non-zero $\mathbf{y} \in \mathbb{Q}_{p}^{n}$ has the form $\mathbf{y} = (y_{1}, \ldots , y_{n})$ and

$$ y_{i} = p^{\gamma _{i}} \bigl(\beta _{0, i} + \beta _{1, i} p + \beta _{2, i} p^{2} + \cdots \bigr),\quad i = 1, \ldots , n $$

(1.10)

(see (1.2)), then there is $i_{0} \in \{1, \ldots , n\}$ such that

$$ \vert y_{i_{0}} \vert _{p} = p^{-\gamma _{i_{0}}} \geq p^{-\gamma _{i}} = \vert y_{i} \vert _{p} $$

(1.11)

whenever $y_{i} \neq 0$. Using (1.3) we obtain $|\mathbf{y}|_{p} = p^{-\gamma _{i_{0}}}$. Now from (1.10) and (1.11) it follows that

$$ \bigl\vert \vert \mathbf{y}|_{p} \mathbf{y}\bigr|_{p} = \max _{ \substack{1 \leq i \leq n \\ y_{i} \neq 0}} \bigl\vert p^{\gamma _{i} - \gamma _{i_{0}}} \bigr\vert _{p} = \max_{\substack{1 \leq i \leq n \\ y_{i} \neq 0}} p^{\gamma _{i_{0}} - \gamma _{i}} = p^{\gamma _{i_{0}} - \gamma _{i_{0}}} = 1. $$

Thus, for every non-zero $\mathbf{y} \in \mathbb{Q}_{p}^{n}$, the vector $|\mathbf{y}|_{p} \mathbf{y}$ belongs to the sphere

$$ S_{0}(\mathbf{0}) = \bigl\{ \mathbf{y} \in \mathbb{Q}_{p}^{n} \colon \vert \mathbf{y} \vert _{p} = 1\bigr\} . $$

From (1.8) it directly follows that $H^{p}_{\Omega , \alpha } \in \mathbb{R}$ for every non-zero $\mathbf{x} \in \mathbb{Q}_{p}^{n}$ and using (1.8), (1.9) we have

$$\begin{aligned} \bigl\vert H^{p, b}_{\Omega , \alpha } f(\mathbf{x}) \bigr\vert & \leq \frac{ \vert b(\mathbf{x}) \vert }{ \vert \mathbf{x} \vert _{p}^{n-\alpha }} \int _{ \vert \mathbf{y} \vert _{p} \leq \vert \mathbf{x} \vert _{p}} \bigl\vert \Omega \bigl( \vert \mathbf{y} \vert _{p} \mathbf{y} \bigr) f(\mathbf{y}) \bigr\vert \,d\mathbf{y} \\ & \quad {}+ \frac{1}{ \vert \mathbf{x} \vert _{p}^{n-\alpha }} \int _{ \vert \mathbf{y} \vert _{p} \leq \vert \mathbf{x} \vert _{p}} \bigl\vert b(\mathbf{y}) \Omega \bigl( \vert \mathbf{y} \vert _{p} \mathbf{y} \bigr) f(\mathbf{y}) \bigr\vert \,d\mathbf{y} < \infty \end{aligned}$$

for every $\mathbf{x} \in \mathbb{Q}_{p}^{n} \setminus \{\mathbf{0}\}$. Consequently, the operators $H^{p}_{\Omega , \alpha }$ and $H^{p,b}_{\Omega ,\alpha }$ are well defined.

The aim of the current paper is to study the boundedness of $H^{p,b}_{\Omega ,\alpha }$ on p-adic Herz-type spaces by considering the symbol function b belonging to the p-adic CMO and Lipschitz spaces. In Euclidean space $\mathbb{R}^{n}$, Herz spaces and Morrey–Herz spaces were firstly introduced in [14] and [26], respectively. For more recent developments in the said spaces we mention the articles [15, 27, 39] and the references therein. Also, some operators with rough kernels defined on Euclidian space were recently studied on function spaces; see for example [11, 12]. Before turning to our main results, let us recall the definitions of p-adic function spaces first.

Definition 1.3

([9])

Suppose $1< q<\infty$. The p-adic central bounded mean oscillation (CBMO) space $C\dot{M}O^{q}(\mathbb{Q}_{p}^{n})$ is the set of all measurable functions $f \colon \mathbb{Q}_{p}^{n} \to \mathbb{R}$ which satisfy

$$ \Vert f \Vert _{\mathit{CMO}^{q}(\mathbb{Q}_{p}^{n})} = \sup _{\gamma \in \mathbb{Z}} \biggl(\frac{1}{ \vert B_{\gamma } \vert _{h}} \int _{B_{\gamma }} \bigl\vert f(\mathbf{x}) - f_{B_{ \gamma }} \bigr\vert ^{q} \,d\mathbf{x} \biggr)^{1/q} < \infty , $$

(1.12)

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

Definition 1.4

([9])

Suppose $0< r<\infty$, $0< q<\infty$ and $\beta \in \mathbb{R}$. The homogeneous p-adic Herz space $\dot{K}^{\beta ,r}_{q}(\mathbb{Q}_{p}^{n})$ is defined by

$$ \dot{K}^{\beta ,r}_{q}\bigl(\mathbb{Q}_{p}^{n} \bigr)=\bigl\{ f\in L^{q}\bigl(\mathbb{Q}_{p}^{n} \bigr): \Vert f \Vert _{\dot{K}^{\beta ,r}_{q}(\mathbb{Q}_{p}^{n})}< \infty \bigr\} , $$

where

$$ \Vert f \Vert _{\dot{K}^{\beta ,r}_{q}(\mathbb{Q}_{p}^{n})}= \Biggl(\sum_{k=- \infty }^{\infty }p^{k\beta r} \Vert f\chi _{k} \Vert ^{r}_{L^{q}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/r}, $$

and $\chi _{k}$ is the characteristic function of $S_{k}$.

Obviously, the equalities $\dot{K}^{0,q}_{q}(\mathbb{Q}_{p}^{n})=L^{q}(\mathbb{Q}_{p}^{n})$ and $\dot{K}^{\beta /q,q}_{q}(\mathbb{Q}_{p}^{n})=L^{q}(|\mathbf{x}|_{p}^{ \beta })$ hold.

Definition 1.5

([5])

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

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

where

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

It is evident that $M\dot{K}^{\beta ,0}_{r,q}(\mathbb{Q}_{p}^{n})=\dot{K}^{\beta ,r}_{q}( \mathbb{Q}_{p}^{n})$ and $M\dot{K}^{\beta /q,0}_{q,q}(\mathbb{Q}_{p}^{n})=L^{q}(|\mathbf{x}|_{p}^{ \alpha })$.

Definition 1.6

([5])

Suppose δ is a positive real number. The Lipschitz space $\Lambda _{\delta }(\mathbb{Q}_{p}^{n})$ is defined to be the space of all measurable function f on $\mathbb{Q}_{p}^{n}$ such that

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

2 CBMO estimates for commutators of p-adic rough fractional Hardy operator

The present section discusses the boundedness of p-adic rough fractional Hardy operator on p-adic Herz-type spaces. We begin this section with the following useful lemma.

Lemma 2.1

([36])

Suppose b is a $\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})$ function and suppose $i, j\in \mathbb{Z}$. Then the inequality

$$ \bigl\vert b(\mathbf{y})-b_{B_{j}} \bigr\vert \leq \bigl\vert b( \mathbf{y})-b_{B_{i}} \bigr\vert +p^{n} \vert i-j \vert \Vert b \Vert _{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})}, $$

holds.

Remark 2.2

From now on the letter C indicates a positive constant which may vary from line to line.

Theorem 2.3

Let $0< r_{1}\leq r_{2}<\infty$, $1\leq q_{1}$, $q_{2}<\infty$. Also, let $\frac{1}{q_{1}}-\frac{1}{q_{2}}=\frac{\alpha }{n}$, $q_{1}'< s< \infty$, $\frac{1}{q_{1}'}-\frac{1}{t}=\frac{1}{s}$. If $\beta <\frac{n}{t}$, then the inequality

$$ \bigl\Vert H^{p,b}_{\Omega ,\alpha }f \bigr\Vert _{\dot{K}^{\beta ,r_{2}}_{q_{2}}( \mathbb{Q}_{p}^{n})}\leq C \Vert f \Vert _{\dot{K}^{\beta ,r_{1}}_{q_{1}}( \mathbb{Q}_{p}^{n})}, $$

holds for all $\Omega \in L^{s}(S_{\mathbf{0}}(\mathbf{0}))$, $b\in \mathit{CMO}^{\max \{q_{2},t\}}(\mathbb{Q}_{p}^{n})$, and $f\in L_{\mathrm{loc}}^{q_{1}}(\mathbb{Q}_{p}^{n})$.

Proof of Theorem 2.3

For the sake of brevity, we write

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

Since

$$\begin{aligned} \bigl\Vert \bigl(H^{p,b}_{\Omega ,\alpha }f\bigr) \chi _{k} \bigr\Vert ^{q_{2}}_{L^{q_{2}}( \mathbb{Q}_{p}^{n})} &= \int _{S_{k}} \vert \mathbf{x} \vert _{p}^{-q_{2}(n-\alpha )} \biggl\vert \int _{ \vert \mathbf{y} \vert _{p} \leq \vert \mathbf{x} \vert _{p}} \Omega \bigl( \vert \mathbf{y} \vert _{p}\mathbf{y}\bigr)f(\mathbf{y}) \bigl(b(\mathbf{x}) -b(\mathbf{y}) \bigr) \,d\mathbf{y} \biggr\vert ^{q_{2}} \,d\mathbf{x} \\ &\leq Cp^{-kq_{2}(n-\alpha )} \int _{S_{k}} \biggl( \int _{ \vert \mathbf{y} \vert _{p} \leq p^{k}} \bigl\vert \Omega \bigl( \vert \mathbf{y} \vert _{p}\mathbf{y}\bigr) f(\mathbf{y}) \bigl(b( \mathbf{x}) -b( \mathbf{y})\bigr) \bigr\vert \,d\mathbf{y} \biggr)^{q_{2}} \,d\mathbf{x} \\ &=Cp^{-kq_{2}(n-\alpha )} \int _{S_{k}} \Biggl(\sum_{j=-\infty }^{k} \int _{S_{j}} \bigl\vert f(\mathbf{y}) \Omega \bigl(p^{j}\mathbf{y}\bigr) \bigl(b(\mathbf{x})-b( \mathbf{y})\bigr)\bigr| \,d\mathbf{y} \Biggr)^{q_{2}}\,d\mathbf{x} \\ &\leq Cp^{-kq_{2}(n-\alpha )} \int _{S_{k}} \Biggl(\sum_{j=-\infty }^{k} \int _{S_{j}} \bigl\vert f(\mathbf{y}) \Omega \bigl(p^{j} \mathbf{y}\bigr) \bigl(b(\mathbf{x})-b_{B_{k}} \bigr) \bigr\vert \,d\mathbf{y} \Biggr)^{q_{2}}\,d\mathbf{x} \\ &\quad {} +Cp^{-kq_{2}(n-\alpha )} \int _{S_{k}} \Biggl(\sum_{j=-\infty }^{k} \int _{S_{j}} \bigl\vert f(\mathbf{y}) \Omega \bigl(p^{j}\mathbf{y}\bigr) \bigl(b(\mathbf{y})-b_{B_{k}} \bigr) \bigr\vert \,d\mathbf{y} \Biggr)^{q_{2}}\,d\mathbf{x} \\ &=I+\mathit{II}. \end{aligned}$$

(2.1)

For $j,k\in \mathbb{Z}$ with $j\leq k$, we get

$$ \int _{S_{j}} \bigl\vert \Omega \bigl(p^{j} \mathbf{y}\bigr) \bigr\vert ^{s}\,d\mathbf{y}= \int _{ \vert \mathbf{z} \vert _{p}=1} \bigl\vert \Omega (\mathbf{z}) \bigr\vert ^{s}p^{jn}\,d\mathbf{z}\leq Cp^{kn}. $$

(2.2)

Note that $\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{\alpha }{n}$ and $\frac{1}{q_{1}}+\frac{1}{s}+\frac{1}{t}=1$, where $\frac{1}{t}=\frac{1}{q_{1}'}-\frac{1}{s}$. Applying Hölder’s inequality we have

$$\begin{aligned} I&\leq Cp^{-kq_{2}(n-\alpha )} \int _{B_{k}} \bigl\vert b(\mathbf{x})-b_{B_{k}} \bigr\vert ^{q_{2}} \\ & \quad {}\times \Biggl\{ \sum_{j=-\infty }^{k} \biggl( \int _{S_{j}} \bigl\vert f( \mathbf{y}) \bigr\vert ^{q_{1}}\,d\mathbf{y} \biggr)^{1/q_{1}} \biggl( \int _{S_{j}} \bigl\vert \Omega \bigl(p^{j} \mathbf{y}\bigr) \bigr\vert ^{s}\,d\mathbf{y} \biggr)^{1/s}p^{jn(1/q_{1}'-1/s)} \Biggr\} ^{q_{2}}\,d\mathbf{x} \\ &\leq C \Vert b \Vert ^{q_{2}}_{\mathit{CMO}^{q_{2}}(\mathbb{Q}_{p}^{n})}p^{kn-kq_{2}(n- \alpha )} \Biggl\{ \sum_{j=-\infty }^{k}p^{jn(1/q_{1}'-1/s)}p^{kn/s} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr\} ^{q_{2}} \\ &=C \Vert b \Vert ^{q_{2}}_{\mathit{CMO}^{q_{2}}(\mathbb{Q}_{p}^{n})} \Biggl\{ \sum _{j=- \infty }^{k}p^{(j-k)n(1/q_{1}'-1/s)} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr\} ^{q_{2}}. \end{aligned}$$

(2.3)

Lemma 2.1 will be helpful for estimating II. Thus

$$\begin{aligned} \mathit{II}&\leq Cp^{-kq_{2}(n-\alpha )} \int _{S_{k}} \Biggl(\sum_{j=-\infty }^{k} \int _{S_{j}} \bigl\vert f(\mathbf{y}) \Omega \bigl(p^{j}\mathbf{y}\bigr) \bigl(b(\mathbf{y})-b_{B_{j}} \bigr) \bigr\vert \,d\mathbf{y} \Biggr)^{q_{2}}\,d\mathbf{x} \\ &\quad {} +C \Vert b \Vert ^{q_{2}}_{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})}p^{-kq_{2}(n- \alpha )} \int _{S_{k}} \Biggl(\sum_{j=-\infty }^{k}(k-j) \int _{S_{j}} \bigl\vert f( \mathbf{y})\Omega \bigl(p^{j}\mathbf{y}\bigr) \bigr\vert \,d\mathbf{y} \Biggr)^{q_{2}}\,d\mathbf{x} \\ &=I_{1}+\mathit{II}_{2}. \end{aligned}$$

(2.4)

We use Hölder’s inequality to estimate $I_{1}$. We have

$$\begin{aligned} I_{1}&\leq Cp^{-kq_{2}(n-\alpha )} \int _{S_{k}} \Biggl\{ \sum_{j=- \infty }^{k} \biggl( \int _{S_{j}} \bigl\vert b(\mathbf{y})-b_{B_{j}} \bigr\vert ^{t}\,d\mathbf{y} \biggr)^{1/t} \\ &\quad {} \times \biggl( \int _{S_{j}} \bigl\vert \Omega \bigl(p^{j} \mathbf{y}\bigr) \bigr\vert ^{s}\,d\mathbf{y} \biggr)^{1/s} \biggl( \int _{S_{j}} \bigl\vert f(\mathbf{y}) \bigr\vert ^{q_{1}} \,d\mathbf{y} \biggr)^{1/q_{1}} \Biggr\} ^{q_{2}} \,d\mathbf{x} \\ &\leq \Vert b \Vert ^{q_{2}}_{\mathit{CMO}^{t}(\mathbb{Q}_{p}^{n})}\sum _{j=-\infty }^{k} \biggl\{ p^{-kn/q_{1}'}p^{kn/s}p^{jn/t} \biggl(\frac{1}{ \vert B_{j} \vert _{H}} \int _{B_{j}} \bigl\vert b(\mathbf{y})-b_{B_{j}} \bigr\vert ^{t} \biggr)^{1/t} \Vert f_{j} \Vert _{L^{q_{1}}( \mathbb{Q}_{p}^{n})} \biggr\} ^{q_{2}} \\ &=C \Vert b \Vert ^{q_{2}}_{\mathit{CMO}^{t}(\mathbb{Q}_{p}^{n})} \Biggl\{ \sum _{j=-\infty }^{k}p^{(j-k)n(1/q_{1}'-1/s)} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr\} ^{q_{2}}. \end{aligned}$$

(2.5)

In a similar fashion we can estimate $\mathit{II}_{2}$. Using Hölder’s inequality we have

$$\begin{aligned} \mathit{II}_{2}&\leq C \Vert b \Vert ^{q_{2}}_{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})}p^{-kq_{2}(n- \alpha )} \\ &\quad {} \times \int _{S_{k}} \Biggl\{ \sum_{j=-\infty }^{k}(k-j) \biggl( \int _{S_{j}} \bigl\vert f(\mathbf{y}) \bigr\vert ^{q_{1}}\,d\mathbf{y} \biggr)^{1/q_{1}} \biggl( \int _{S_{j}} \bigl\vert \Omega \bigl(p^{j} \mathbf{y}\bigr) \bigr\vert ^{s}\,d\mathbf{y} \biggr)^{1/s}p^{jn/t} \Biggr\} ^{q_{2}}\,d\mathbf{x} \\ &=C \Vert b \Vert ^{q_{2}}_{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})} \Biggl(\sum _{j=-\infty }^{k}(k-j)p^{(j-k)n(1/q_{1}'-1/s)} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{q_{2}}. \end{aligned}$$

(2.6)

From (2.3), (2.5) and (2.6) together with the Jensen inequality, we have

$$\begin{aligned} & \bigl\Vert H^{p,b}_{\Omega ,\alpha }f \bigr\Vert _{\dot{K}^{\beta ,r_{2}}_{q_{2}}( \mathbb{Q}_{p}^{n})} \\ &\quad = \Biggl(\sum_{k=-\infty }^{\infty }p^{k\beta r_{2}} \bigl\Vert \bigl(H^{p,b}_{\Omega , \alpha }f\bigr)\chi _{k} \bigr\Vert ^{r_{2}}_{L^{q_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/r_{2}} \\ &\quad \leq \Biggl(\sum_{k=-\infty }^{\infty }p^{k\beta r_{1}} \bigl\Vert \bigl(H^{p,b}_{ \Omega ,\alpha }f\bigr)\chi _{k} \bigr\Vert ^{r_{1}}_{L^{q_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/r_{1}} \\ &\quad \leq C \Vert b \Vert _{\mathit{CMO}^{q_{2}}(\mathbb{Q}_{p}^{n})} \Biggl(\sum _{k=-\infty }^{ \infty }p^{k\beta r_{1}} \Biggl(\sum _{j=-\infty }^{k}p^{(j-k)n/t} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}} \Biggr)^{1/r_{1}} \\ &\qquad {} +C \Vert b \Vert _{\mathit{CMO}^{t}(\mathbb{Q}_{p}^{n})} \Biggl(\sum _{k=-\infty }^{ \infty }p^{k\beta r_{1}} \Biggl(\sum _{j=-\infty }^{k}p^{(j-k)n/t} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}} \Biggr)^{1/r_{1}} \\ &\qquad {} +C \Vert b \Vert _{\mathit{CMO}^{1}(\mathbb{Q}_{p}^{n})} \Biggl(\sum _{k=-\infty }^{ \infty }p^{k\beta r_{1}} \Biggl(\sum _{j=-\infty }^{k}(k-j)p^{(j-k)n/t} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}} \Biggr)^{1/r_{1}} \\ &\quad =J. \end{aligned}$$

For brevity, we may choose $\|b\|_{\mathit{CMO}^{\max \{q_{2},t\}}(\mathbb{Q}_{p}^{n})}=1$. Consequently,

$$ J\leq C \Biggl(\sum_{k=-\infty }^{\infty }p^{k\beta r_{1}} \Biggl(\sum_{j=- \infty }^{k} (k-j)p^{(j-k)n/t} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}} \Biggr)^{1/r_{1}}. $$

Case 1: When $0< r_{1}\leq 1$, we have

$$\begin{aligned} J^{r_{1}}&=C\sum_{k=-\infty }^{\infty }p^{k\beta r_{1}} \Biggl(\sum_{j=- \infty }^{k}(k-j)p^{(j-k)n/t} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}} \\ &=C\sum_{k=-\infty }^{\infty } \Biggl(\sum _{j=-\infty }^{k}p^{j\beta } \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})}(k-j)p^{(j-k)(n/t-\beta )} \Biggr)^{r_{1}} \\ &\leq C\sum_{k=-\infty }^{\infty }\sum _{j=-\infty }^{k}p^{j\beta r_{1}} \Vert f_{j} \Vert ^{r_{1}}_{L^{q_{1}}(\mathbb{Q}_{p}^{n})}(k-j)^{r_{1}}p^{(j-k)(n/t -\beta )r_{1}} \\ &=C\sum_{k=-\infty }^{\infty }p^{j\beta r_{1}} \Vert f_{j} \Vert ^{r_{1}}_{L^{q_{1}}( \mathbb{Q}_{p}^{n})}\sum _{k=j}^{\infty }(k-j)^{r_{1}}p^{(j-k)(n/t - \beta )r_{1}} \\ &=C \Vert f \Vert ^{r_{1}}_{\dot{K}^{\beta ,r_{1}}_{q_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned}$$

Case 2: When $r_{1}>1$, applying Hölder’s inequality we get

$$\begin{aligned} J^{r_{1}}&=C\sum_{k=-\infty }^{\infty } \Biggl(\sum_{j=-\infty }^{k}p^{j \beta } \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})}(k-j)p^{(j-k)(n/t- \beta )} \Biggr)^{r_{1}} \\ &\leq C\sum_{k=-\infty }^{\infty }\sum _{j=-\infty }^{k}p^{j\beta r_{1}} \Vert f_{j} \Vert ^{r_{1}}_{L^{q_{1}}(\mathbb{Q}_{p}^{n})}p^{(j-k)(n/t -\beta )r_{1}/2} \\ &\quad {} \times \Biggl(\sum_{j=-\infty }^{k}(k-j)^{r_{1}'}p^{(j-k)(n/t - \beta )r_{1}'/2} \Biggr)^{r_{1}/r_{1}'} \\ &=C\sum_{k=-\infty }^{\infty }p^{j\beta r_{1}} \Vert f_{j} \Vert ^{r_{1}}_{L^{q_{1}}( \mathbb{Q}_{p}^{n})}\sum _{k=j}^{\infty }p^{(j-k)(n/t -\beta )r_{1}/2} \\ &=C \Vert f \Vert ^{r_{1}}_{\dot{K}^{\beta ,r_{1}}_{q_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned}$$

The proof of Theorem 2.3 is thus completed. □

Theorem 2.4

Let $0< r_{1}\leq r_{2}<\infty$, $1\leq q_{1}$, $q_{2}<\infty$. Also, let $\frac{1}{q_{1}}-\frac{1}{q_{2}}=\frac{\alpha }{n}$, $q_{1}'< s<\infty$, $\frac{1}{q_{1}'}-\frac{1}{t}=\frac{1}{s}$, and $\lambda >0$. If $\beta <\frac{n}{t}+\lambda$, then the inequality

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

holds for all $\Omega \in L^{s}(S_{\mathbf{0}}(\mathbf{0}))$, $b\in \mathit{CMO}^{\max \{q_{2},t\}}(\mathbb{Q}_{p}^{n})$ and $f\in L_{\mathrm{loc}}^{q_{1}}(\mathbb{Q}_{p}^{n})$.

Proof of Theorem 2.4

From the proof of Theorem 2.3 and

$$ \bigl\Vert \bigl(H^{p,b}_{\Omega ,\alpha }f\bigr)\chi _{k} \bigr\Vert _{L^{q_{2}}(\mathbb{Q}_{p}^{n})} \leq C\sum _{j=-\infty }^{k}(k-j)p^{\frac{(j-k)n}{t}} \Vert f_{j} \Vert _{L^{q_{1}}( \mathbb{Q}_{p}^{n})}, $$

together with the definition of a Morrey–Herz space, the Jensen inequality, $\beta < n/t+\lambda$, $\lambda >0$ and $1< r_{1}<\infty$, it follows that

$$ \begin{aligned} & \bigl\Vert H^{p,b}_{\Omega ,\alpha }f \bigr\Vert _{M\dot{K}^{\beta ,\lambda }_{r_{2},q_{2}}( \mathbb{Q}_{p}^{n})} \\ &\quad =\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda } \Biggl(\sum _{k=-\infty }^{k_{0}}p^{k \beta r_{2}} \bigl\Vert \bigl(H^{p,b}_{\Omega ,\alpha }f\bigr)\chi _{k} \bigr\Vert ^{r_{2}}_{L^{q_{2}}( \mathbb{Q}_{p}^{n})} \Biggr)^{1/r_{2}} \\ &\quad \leq \sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda } \Biggl(\sum _{k=- \infty }^{k_{0}}p^{k\beta r_{1}} \bigl\Vert \bigl(H^{p,b}_{\Omega ,\alpha }f\bigr)\chi _{k} \bigr\Vert ^{r_{1}}_{L^{q_{2}}(\mathbb{Q}_{p}^{n})} \Biggr)^{1/r_{1}} \\ &\quad \leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda } \Biggl(\sum _{k=- \infty }^{k_{0}}p^{k\beta r_{1}} \Biggl(\sum _{j=-\infty }^{k}(k-j)p^{ \frac{(j-k)n}{t}} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}} \Biggr)^{1/r_{1}} \\ &\quad \leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda } \Biggl(\sum _{k=- \infty }^{k_{0}} \Biggl(\sum _{j=-\infty }^{k}p^{k\beta }(k-j)p^{ \frac{(j-k)n}{t}}p^{-j\beta }p^{j\lambda }p^{j\lambda } \\ &\qquad {} \times \Biggl(\sum_{l=-\infty }^{j}p^{l\beta r_{1}} \Vert f_{j} \Vert ^{r_{1}}_{L^{q_{1}}( \mathbb{Q}_{p}^{n})} \Biggr)^{1/r_{1}} \Biggr)^{r_{1}} \Biggr)^{1/r_{1}} \\ &\quad \leq C\sup_{k_{0}\in \mathbb{Z}}p^{-k_{0}\lambda } \Biggl(\sum _{k=- \infty }^{k_{0}}p^{k\lambda r_{1}} \Biggl(\sum _{j=-\infty }^{k}(k-j)p^{(j-k)(n/t- \beta +\lambda )} \Vert f \Vert _{M\dot{K}^{\beta ,\lambda }_{r_{1},q_{1}}( \mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}} \Biggr)^{1/r_{1}} \\ &\quad \leq C \Vert f \Vert _{M\dot{K}^{\beta ,\lambda }_{r_{1},q_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned} $$

 □

3 Lipschitz estimates for commutators of p-adic rough fractional Hardy operator

The current section deals with the boundedness for the commutators of p-adic rough fractional Hardy operator on homogeneous p-adic Herz-type spaces by considering the symbol function from Lipschitz space. We open the discussion for this section from the following lemma.

Lemma 3.1

Suppose $f\in \Lambda _{\delta }(\mathbb{Q}_{p}^{n})$ and $0<\delta <1$, then

$$ \bigl\vert f(\mathbf{x})-f(\mathbf{y}) \bigr\vert \leq \vert \mathbf{x}- \mathbf{y} \vert _{p}^{\delta } \Vert f \Vert _{\Lambda _{\delta }(\mathbb{Q}_{p}^{n})}. $$

Proof

Proof immediately follows from Definition 1.6. □

Theorem 3.2

Let $1\leq q_{1}$, $q_{2}<\infty$, $0< r_{1}\leq r_{2}<\infty$. Also, let $\frac{1}{q_{1}}-\frac{1}{q_{2}}=\frac{\delta +\alpha }{n}$, $q_{1}'< s<\infty$, $\frac{1}{q_{1}'}-\frac{1}{t}=\frac{1}{s}$, and $0<\delta <1$. If $\beta < n(\frac{1}{q_{1}'}-\frac{1}{s})$, then the inequality

$$ \bigl\Vert H^{p,b}_{\Omega ,\alpha }f \bigr\Vert _{\dot{K}^{\beta ,r_{2}}_{q_{2}}( \mathbb{Q}_{p}^{n})}\leq C \Vert f \Vert _{\dot{K}^{\beta ,r_{1}}_{q_{1}}( \mathbb{Q}_{p}^{n})} $$

holds for all $\Omega \in L^{s}(S_{\mathbf{0}}(\mathbf{0}))$, $b\in \Lambda _{\delta }(\mathbb{Q}_{p}^{n})$, and $f\in L_{\mathrm{loc}}^{q_{1}}(\mathbb{Q}_{p}^{n})$.

Proof of Theorem 3.2

By Hölder’s inequality along with Lemma 3.1, we have

$$ \begin{aligned}[b] &\bigl\Vert \bigl(H^{p,b}_{\Omega ,\alpha }f\bigr)\chi _{k} \bigr\Vert ^{q_{2}}_{L^{q_{2}}( \mathbb{Q}_{p}^{n})} \\ &\quad = \int _{S_{k}} \vert \mathbf{x} \vert _{p}^{-q_{2}(n-\alpha )} \biggl\vert \int _{ \vert \mathbf{y} \vert _{p}\leq \vert \mathbf{x} \vert _{p}}\Omega \bigl( \vert \mathbf{y} \vert _{p} \mathbf{y}\bigr)f(\mathbf{y}) \bigl(b(\mathbf{x})-b(\mathbf{y}) \bigr)\,d\mathbf{y} \biggr\vert ^{q_{2}}\,d\mathbf{x} \\ &\quad \leq Cp^{-kq_{2}(n-\alpha )} \int _{S_{k}} \biggl( \int _{ \vert \mathbf{y} \vert _{p} \leq p^{k}} \bigl\vert \Omega \bigl( \vert \mathbf{y} \vert _{p}\mathbf{y}\bigr)f(\mathbf{y}) \bigl(b( \mathbf{x})-b( \mathbf{y})\bigr) \bigr\vert \,d\mathbf{y} \biggr)^{q_{2}}\,d\mathbf{x} \\ &\quad \leq Cp^{-kq_{2}(n-\alpha )} \Vert b \Vert ^{q_{2}}_{\Lambda _{\delta }( \mathbb{Q}_{p}^{n})} \int _{S_{k}} \Biggl(\sum_{j=-\infty }^{k} \int _{S_{j}} \bigl\vert \Omega \bigl(p^{j} \mathbf{y}\bigr)f(\mathbf{y}) \bigr\vert \vert \mathbf{x}-\mathbf{y} \vert _{p}^{ \delta }\,d\mathbf{y} \Biggr)^{q_{2}}\,d\mathbf{x} \\ &\quad \leq Cp^{-kq_{2}(n-\alpha -\delta )} \Vert b \Vert ^{q_{2}}_{\Lambda _{\delta }( \mathbb{Q}_{p}^{n})} \int _{S_{k}} \Biggl(\sum_{j=-\infty }^{k} \int _{S_{j}} \bigl\vert \Omega \bigl(p^{j} \mathbf{y}\bigr)f(\mathbf{y}) \bigr\vert \,d\mathbf{y} \Biggr)^{q_{2}}\,d\mathbf{x} \\ &\quad \leq C \Vert b \Vert ^{q_{2}}_{\Lambda _{\delta }(\mathbb{Q}_{p}^{n})} p^{-kq_{2}(n- \alpha -\delta )+kn} \Biggl(\sum_{j=-\infty }^{k} \biggl( \int _{S_{j}} \bigl\vert \Omega \bigl(p^{j} \mathbf{y}\bigr) \bigr\vert ^{s}\,d\mathbf{y} \biggr)^{1/s} \\ &\qquad {} \times \biggl( \int _{S_{j}} \bigl\vert f(\mathbf{y}) \bigr\vert ^{q_{1}}\,d\mathbf{y} \biggr)^{1/q_{1}} \biggl( \int _{S_{j}}\,d\mathbf{y} \biggr)^{1-1/q-1/s} \Biggr)^{q_{2}} \\ &\quad =I. \end{aligned} $$

(3.1)

By virtue of (2.2), inequality (3.1) takes the following form:

$$ \begin{aligned} I&\leq C \Vert b \Vert ^{q_{2}}_{\Lambda _{\delta }(\mathbb{Q}_{p}^{n})}p^{-kq_{2}(n- \alpha -\delta )+kn} \Biggl(\sum _{j=-\infty }^{k}p^{kn/s+jn(1/q_{1}'-1/s)} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{q_{2}} \\ &\leq C \Vert b \Vert ^{q_{2}}_{\Lambda _{\delta }(\mathbb{Q}_{p}^{n})} \Biggl( \sum _{j=-\infty }^{k}p^{(j-k)n(1/q'-1/s)} \Vert f_{j} \Vert _{L^{q_{1}}( \mathbb{Q}_{p}^{n})} \Biggr)^{q_{2}}. \end{aligned} $$

For the sake of brevity, we take $\|b\|^{q_{2}}_{\Lambda _{\delta }(\mathbb{Q}_{p}^{n})}=1$. Now, by definition of Herz spaces and the Jensen inequality, it follows that

$$ \begin{aligned} \bigl\Vert H^{p,b}_{\Omega ,\alpha }f \bigr\Vert ^{r_{1}}_{\dot{K}^{\beta ,r_{2}}_{q_{2}}( \mathbb{Q}_{p}^{n})}&= \Biggl(\sum _{k=-\infty }^{\infty }p^{k\beta r_{2}} \bigl\Vert \bigl(H^{p,b}_{\Omega ,\alpha }f\bigr)\chi _{k} \bigr\Vert ^{r_{2}}_{L^{q_{2}}( \mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}/r_{2}} \\ &\leq \sum_{k=-\infty }^{\infty }p^{k\beta r_{1}} \bigl\Vert \bigl(H^{p,b}_{\Omega , \alpha }f\bigr)\chi _{k} \bigr\Vert ^{r_{1}}_{L^{q_{2}}(\mathbb{Q}_{p}^{n})} \\ &\leq C\sum_{k=-\infty }^{\infty }p^{k\beta r_{1}} \Biggl(\sum_{j=- \infty }^{k}p^{(j-k)n(1/q_{1}'-1/s)} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}} \\ &=C\sum_{k=-\infty }^{\infty } \Biggl(\sum _{j=-\infty }^{k}p^{j\beta }p^{(j-k)(n/q_{1}'-n/s- \beta )} \Vert f_{j} \Vert _{L^{q_{1}}(\mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}}. \end{aligned} $$

Case 1: If $0< r_{1}\leq 1$, then

$$ \begin{aligned} \bigl\Vert H^{p,b}_{\Omega ,\alpha }f \bigr\Vert ^{r_{1}}_{\dot{K}^{\beta ,r_{2}}_{q_{2}}( \mathbb{Q}_{p}^{n})}&\leq C\sum _{k=-\infty }^{\infty }\sum_{j=-\infty }^{k}p^{j \beta r_{1}}p^{(j-k)(n/q_{1}'-n/s-\beta )r_{1}} \Vert f_{j} \Vert _{L^{q_{1}}( \mathbb{Q}_{p}^{n})}^{r_{1}} \\ &=C\sum_{j=-\infty }^{\infty }p^{j\beta r_{1}} \Vert f_{j} \Vert _{L^{q_{1}}( \mathbb{Q}_{p}^{n})}^{r_{1}}\sum _{k=j}^{\infty }p^{(j-k)(n/q_{1}'-n/s- \beta )r_{1}} \\ &\leq C \Vert f \Vert ^{r_{1}}_{\dot{K}^{\beta ,r_{1}}_{q_{2}}(\mathbb{Q}_{p}^{n})}. \end{aligned} $$

Case 2: When $r_{1}>1$, applying Hölder’s inequality, we have

$$ \begin{aligned} \bigl\Vert H^{p,b}_{\Omega ,\alpha }f \bigr\Vert ^{r_{1}}_{\dot{K}^{\beta ,r_{2}}_{q_{2}}( \mathbb{Q}_{p}^{n})}&\leq C\sum _{k=-\infty }^{\infty } \Biggl(\sum _{j=- \infty }^{k}p^{j\beta }p^{(j-k)(n/q_{1}'-n/s-\beta )} \Vert f_{j} \Vert ^{r_{1}}_{L^{q_{1}}( \mathbb{Q}_{p}^{n})} \Biggr)^{r_{1}} \\ &\leq C\sum_{k=-\infty }^{\infty }\sum _{j=-\infty }^{k}p^{j\beta r_{1}} \Vert f_{j} \Vert ^{r_{1}}_{L^{q_{1}}(\mathbb{Q}_{p}^{n})}p^{(j-k)(n/q_{1}'-n/s- \beta )r_{1}/2} \\ &\quad {} \times \Biggl(\sum_{j=-\infty }^{k}p^{(j-k)(n/q_{1}'-n/s-\beta )r_{1}'/2} \Biggr)^{r_{1}/r_{1}'} \\ &\leq C\sum_{j=-\infty }^{\infty }p^{j\beta r_{1}} \Vert f_{j} \Vert ^{r_{1}}_{L^{q_{1}}( \mathbb{Q}_{p}^{n})}\sum _{k=j}^{\infty }p^{(j-k)(n/q_{1}'-n/s-\beta )r_{1}/2} \\ &\leq C \Vert f \Vert ^{r_{1}}_{\dot{K}^{\beta ,r_{1}}_{q_{1}}(\mathbb{Q}_{p}^{n})}. \end{aligned} $$

 □

Theorem 3.3

Let $1\leq q_{1}$, $q_{2}<\infty$, $0< r_{1}\leq r_{2}<\infty$. Also, let $\frac{1}{q_{1}}-\frac{1}{q_{2}}=\frac{\delta +\alpha }{n}$, $s>q_{1}'$, $\frac{1}{q_{1}'}-\frac{1}{t}=\frac{1}{s}$, $\lambda \geq 0$ and $0<\delta <1$. If $n(\frac{1}{q_{1}'}-\frac{1}{s})+\lambda >\beta$, then the inequality

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

holds for all $\Omega \in L^{s}(S_{\mathbf{0}}(\mathbf{0}))$, $b\in \Lambda _{\delta }(\mathbb{Q}_{p}^{n})$, and $f\in L_{\mathrm{loc}}^{q_{1}}(\mathbb{Q}_{p}^{n})$.

Proof of Theorem 3.3

The proof follows from standard analysis performed in our previous theorems. So, we omit the details. □