Some estimates for p-adic rough multilinear Hausdorff operators and commutators on weighted Morrey-Herz type spaces

TL;DR

This paper investigates the boundedness of a new class of p-adic rough multilinear Hausdorff operators and their commutators on various weighted Morrey-Herz type spaces, expanding understanding in p-adic harmonic analysis.

Contribution

It introduces and analyzes the boundedness of p-adic rough multilinear Hausdorff operators and their commutators on weighted Morrey-Herz spaces, a novel extension in p-adic analysis.

Findings

Boundedness established for p-adic rough multilinear Hausdorff operators.

Boundedness results for commutators with symbols in central BMO space.

Extension of operator theory to weighted Morrey-Herz spaces in p-adic setting.

Abstract

The aim of this paper is to introduce and study the boundedness of a new class of p-adic rough multilinear Hausdorff operators on the product of Herz, central Morrey and Morrey-Herz spaces with power weights and Muckenhoupt weights. We also establish the boundedness for the commutators of p-adic rough multilinear Hausdorff operators on the weighted spaces with symbols in central BMO space.

Equations

$${\mathcal{H}}_{\phi ,A}(f)(x)={\int }_{{\mathbb{Q}}_p^n}\phi (t)f(A(t)x)dt,x\in {\mathbb{Q}}_p^n,$$

$${\mathcal{H}}_{\Phi ,\Omega }(f)(x)={\int }_{{\mathbb{R}}^n}\frac{\Phi \left(x|y{|}^{-1}\right)}{|y{|}^n}\Omega \left(y|y{|}^{-1}\right)f(y)dy,x\in {\mathbb{R}}^n.$$

$${\mathcal{H}}_{\Phi ,\Omega }(f)(x)={\int }_0^{\infty }{\int }_{S^{n-1}}\frac{\Phi (t)}{t}\Omega (y^{\prime })f(t^{-1}|x|y^{\prime })d\sigma (y^{\prime })dt.$$

$${{\mathcal{H}}^p}_{\Phi ,\Omega }(f)(x)={\int }_{{\mathbb{Q}}_p^n}\frac{\Phi (x|y{|}_p)}{|y{|}_p^n}\Omega (y|y{|}_p)f(y)dy.$$

$${\mathcal{H}}_{\Phi ,\Omega }^p(\vec{f})(x)=\sum _{\gamma \in \mathbb{Z}}{\int }_{S_0}\frac{\Phi (p^{\gamma })}{p^{\gamma }}\Omega (y)\prod _{i=1}^m{f}_i(p^{\gamma }|x{|}_p^{-1}y)dy,x\in {\mathbb{Q}}_p^n,$$

$${\mathcal{H}}^p(f)(x)=\frac{1}{|x{|}_p^n}{\int }_{B(0,|x{|}_p)}f(t)dt,x\in {\mathbb{Q}}_p^n\setminus \{0\},$$

$${\mathcal{H}}^{p}_{\Phi,\Omega,\vec{b}}(\vec{f})(x)=\sum_{\gamma\in\mathbb{Z}}\int_{S_{0}}\dfrac{\Phi(p^{\gamma})}{p^{\gamma}}\Omega(y)\prod\limits_{i=1}^{m}\Big{(}b_{i}(x)-b_{i}(p^{\gamma}|x|_{p}^{-1}y)\Big{)}\prod\limits_{i=1}^{m}f_{i}(p^{\gamma}|x|_{p}^{-1}y)dy,$$

$$x=p^{\alpha }(x_0+x_{1}p+x_2{p}^2+\cdot \cdot \cdot ),$$

$$|x{|}_p=\underset{1\le j\le n}{\max }|x_j{|}_p.$$

$$B_{\alpha }(a)=\left\{x\in {\mathbb{Q}}_p^n:|x-a{|}_p\le p^{\alpha }\right\}$$

$$S_{\alpha }(a)=\left\{x\in {\mathbb{Q}}_p^n:|x-a{|}_p=p^{\alpha }\right\}$$

$$\underset{B_0}{\int }dx=|B_0|=1,$$

$${\int }_{{\mathbb{Q}}_p^n}f(x)dx=\underset{\alpha \to +\infty }{\lim }{\int }_{B_{\alpha }}f(x)dx=\underset{\alpha \to +\infty }{\lim }\sum _{-\infty <\gamma \le \alpha }{\int }_{S_{\gamma }}f(x)dx.$$

$${\int }_{{\mathbb{Q}}_p^n}f(x)dx=\sum _{\alpha =-\infty }^{+\infty }{\int }_{S_{\alpha }}f(x)dx,$$

$${\int }_{{\mathbb{Q}}_p^n}f(tx)dx=\frac{1}{|t{|}_p^n}{\int }_{{\mathbb{Q}}_p^n}f(x)dx,$$

$$\|f\|_{L^{q}_{\omega}(\mathbb{Q}^{n}_{p})}=\Big{(}\int_{\mathbb{Q}^{n}_{p}}|f(x)|^{q}\omega(x)dx\Big{)}^{1/q}<\infty.$$

$$\|f\|_{{\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})}=\mathop{\rm sup}\limits_{\gamma\in\mathbb{Z}}\Big{(}\dfrac{1}{\omega(B_{\gamma})^{1+\lambda q}}\int_{B_{\gamma}}|f(x)|^{q}\omega(x)dx\Big{)}^{1/q}.$$

$$\|f\|_{K_{q,\omega}^{\beta,\ell}(\mathbb{Q}^{n}_{p})}=\Big{(}\sum\limits_{k=-\infty}^{\infty}p^{k\beta\ell}\|f\chi_{k}\|_{L^{q}_{\omega}(\mathbb{Q}^{n}_{p})}^{\ell}\Big{)}^{1/\ell}.$$

$$\|f\|_{{\mathop{K}\limits^{.}}_{q,\omega}^{\beta,\ell}(\mathbb{Q}^{n}_{p})}=\Big{(}\sum\limits_{k=-\infty}^{\infty}\omega^{\beta\ell/n}(B_{k})\|f\chi_{k}\|_{L^{q}_{\omega}(\mathbb{Q}^{n}_{p})}^{\ell}\Big{)}^{1/\ell}.$$

$$M{K}_{\ell ,q,\omega }^{\beta ,\lambda }({\mathbb{Q}}_p^n)=\left\{f\in L_{\omega ,\mathrm{loc}}^q({\mathbb{Q}}_p^n\setminus \{0\}):\Vert f{\Vert }_{M{K}_{\ell ,q,\omega }^{\beta ,\lambda }({\mathbb{Q}}_p^n)}<\infty \right\},$$

$$\|f\|_{MK_{\ell,q,\omega}^{\beta,\lambda}(\mathbb{Q}^{n}_{p})}=\mathop{\rm sup}\limits_{k_{0}\in\mathbb{Z}}p^{-k_{0}\lambda}\Big{(}\sum\limits_{k=-\infty}^{k_{0}}p^{k\beta\ell}\|f\chi_{k}\|_{L^{q}_{\omega}(\mathbb{Q}^{n}_{p})}^{\ell}\Big{)}^{1/\ell}.$$

$$\big{\|}f\big{\|}_{{\mathop{CMO}\limits^{.}}^{q}_{\omega}(\mathbb{Q}^{n}_{p})}=\mathop{\rm sup}\limits_{\gamma\in\mathbb{Z}}\Big{(}\frac{1}{\omega(B_{\gamma})}\int\limits_{B_{\gamma}}{|f(x)-f_{\omega,B_{\gamma}}|^{q}\omega(x)dx}\Big{)}^{\frac{1}{q}},$$

$$f_{\omega ,B_{\gamma }}=\frac{1}{\omega (B_{\gamma })}\underset{B_{\gamma }}{\int }f(x)\omega (x)dx.$$

$$\Big{(}\dfrac{1}{|B|}\int_{B}\omega(x)dx\Big{)}\Big{(}\dfrac{1}{|B|}\int_{B}\omega(x)^{-1/(\ell-1)}dx\Big{)}^{\ell-1}\leq C.$$

$$\frac{1}{|B|}{\int }_B\omega (x)dx\le C\underset{x\in B}{\mathrm{essinf}}\omega (x).$$

$$\Big{(}\dfrac{1}{|B|}\int_{B}\omega(x)^{r}dx\Big{)}^{1/r}\leq\dfrac{C}{|B|}\int_{B}\omega(x)dx,$$

$$C_1{\left(\frac{|E|}{|B|}\right)}^{\ell }\le \frac{\omega (E)}{\omega (B)}\le C_2{\left(\frac{|E|}{|B|}\right)}^{(r-1)/r}$$

$$\frac{1}{|B|}{\int }_B|f(x)|dx\le C{\left(\frac{1}{\omega (B)}{\int }_B|f(x){|}^{\ell }\omega (x)dx\right)}^{1/\ell }.$$

$$\frac{1}{q_1}+\frac{1}{q_2}+\cdots +\frac{1}{q_m}=\frac{1}{q},$$

$$\frac{{\alpha }_1}{q_1}+\frac{{\alpha }_2}{q_2}+\cdots +\frac{{\alpha }_m}{q_m}=\frac{\alpha }{q},$$

Full text

This paper is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED)

1. Introduction

The $p$-adic analysis in the past decades has received a lot of attention due to its important applications in mathematical physics as well as its necessity in sciences and technologies (see e.g. [2, 3, 5, 9, 10, 15, 22, 24, 35, 36, 37, 38] and references therein). All these developments have been motivated for two physical ideas. The first is the conjecture in particle physics that at very small, so-called Planck distances physical space-time has a complicated non-Archimedean structure and that $p$-adic numbers correctly reflect this structure. As a consequence of this idea have emerged the $p$-adic quantum mechanics and $p$-adic quantum field theory. The second idea comes from statistical physics, in particular in connection with models describing relaxation in glasses, macromolecules and proteins. It is also known that the theory of functions from $\mathbb{Q}_{p}$ into $\mathbb{C}$ play an important role in $p$-adic quantum mechanics, the theory of $p$-adic probability in which real-valued random variables have to be considered to solve covariance problems. In recent years, there is an increasing interest in the study of harmonic analysis and wavelet analysis over the $p$-adic fields (see e.g. [1, 9, 10, 13, 20, 21, 23, 24]). One of the important operators in harmonic analysis is the Hausdorff operator, and it is used to solve certain classical problems in analysis. In 2010, Volosivets [39] introduced the Hausdorff operator on the $p$-adic field as follows

[TABLE]

where $\varphi(t)$ is a locally intefrable function on $\mathbb{Q}_{p}^{n}$ and $A(t)$ is an $n\times n$ invertible matrix for almost everywhere $t$ in the support of $\varphi$. It should be pointed out that if we take $\varphi(t)=\psi(t_{1})\chi_{{\mathbb{Z}^{*}_{p}}^{n}}(t)$ and $A(t)=t_{1}.I_{n}$ ($I_{n}$ is an identity matrix), for $t=(t_{1},t_{2},...,t_{n})$, where $\psi:\mathbb{Q}_{p}\to\mathbb{C}$ is a measurable function, ${\mathcal{H}}_{\varphi,A}$ then reduces to the $p$-adic weighted Hardy-Littlewood average operator due to Rim and Lee [32]. For all we know, the theory of the Hardy operators, the Hausdorff operators over the $p$-adic numbers field has been significantly developed into different contexts, and they are actually useful for $p$-adic analysis (see e.g. [11, 12, 17, 40, 41, 43]). In 2012, Chen, Fan and Li [7] introduced another version of Hausdorff operators, so-called the rough Hausdorff operators, as follow

[TABLE]

Note that if $\Phi$ is a radial function, then by using the change of variable in polar coordinates, the operator ${{\mathcal{H}}}_{\Phi,\Omega}$ is rewritten under of the form

[TABLE]

Very recently, Volosivets [42] introduced and investigated the rough Hausdorff operators on the Lorentz space on $p$-adic fields which are defined by

[TABLE]

Motivated by above results, we shall introduce and investigate in this paper a new class of rough multilinear Hausdorff operators defined as follows.

Definition 1.1**.**

Let $\Phi:\mathbb{Q}_{p}\longrightarrow[0,\infty)$ and $\Omega:S_{0}\longrightarrow\mathbb{C}$ be measurable functions such that $\Omega(y)\neq 0$ for almost everywhere $y$ in $S_{0}$. Let $f_{1},f_{2},...,f_{m}$ be measurable complex-valued functions on $\mathbb{Q}^{n}_{p}$. The $p$-adic rough multilinear Hausdorff operator is defined by

[TABLE]

for \vec{f}=\big{(}f_{1},...,f_{m}\big{)}.

It is clear that if $m=1$, $\Omega\equiv 1$ and $\Phi(t)=|t|_{p}^{n-1}\chi_{B(0,1)}$, then ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is reduced to the $p$-adic Hardy operator $\mathcal{H}^{p}$ (see, for example, [26], [43]) defined by

[TABLE]

where $B(0,|x|_{p})$ is a ball in $\mathbb{Q}^{n}_{p}$ with center at [math] and radius $|x|_{p}$. Let $b$ be a measurable function. We denote by $\mathcal{M}_{b}$ the multiplication operator defined by $\mathcal{M}_{b}f(x)=b(x)f(x)$ for any measurable function $f$. If $\mathcal{H}$ is a linear operator on some measurable function space, the commutator of Coifman-Rochberg-Weiss type formed by $\mathcal{M}_{b}$ and $\mathcal{H}$ is defined by $[\mathcal{M}_{b},\mathcal{H}]f(x)=(\mathcal{M}_{b}\mathcal{H}-\mathcal{H}\mathcal{M}_{b})f(x)$. Next, let us give the definition for the commutators of Coifman-Rochberg-Weiss type of $p$-adic rough multilinear Hausdorff operator.

Definition 1.2**.**

Let $\Phi,\Omega$ be as above. The Coifman-Rochberg-Weiss type commutator of $p$-adic rough multilinear Hausdorff operator is defined by

[TABLE]

where $x\in\mathbb{Q}^{n}_{p}$, \vec{b}=\big{(}b_{1},...,b_{m}\big{)} and $b_{i}$ are locally integrable functions on $\mathbb{Q}_{p}^{n}$ for all $i=1,...,m$.

The main purpose of this paper is to extend and study the new $p$-adic rough multilinear Hausdorff operators on the $p$-adic numbers field. We then obtain the necessary and sufficient conditions for the boundedness of ${\mathcal{H}}^{p}_{\Phi,\Omega}(\vec{f})$ on the product of Herz, central Morrey and Morrey-Herz spaces on $p$-adic field. In each case, we estimate the corresponding operator norms. Moreover, the boundedness of ${\mathcal{H}}^{p}_{\Phi,\Omega,\vec{b}}(\vec{f})$ on the such spaces with symbols in central BMO space is also established. It should be pointed out that all our results are new even in the case of linear $p$-adic rough Hausdorff operators. Our paper is organized as follows. In Section 2, we present some notations and preliminaries about $p$-adic analysis as well as give some definitions of the Herz, central Morrey and Morrey-Herz spaces associated with power weights and Muckenhoupt weights. Our main theorems are given and proved in Section 3 and Section 4.

2. Some notations and definitions

For a prime number $p$, let $\mathbb{Q}_{p}$ be the field of $p$-adic numbers. This field is the completion of the field of rational numbers $\mathbb{Q}$ with respect to the non-Archimedean $p$-adic norm $|\cdot|_{p}$. This norm is defined as follows: if $x=0$, $|0|_{p}=0$; if $x\not=0$ is an arbitrary rational number with the unique representation $x=p^{\alpha}\frac{m}{n}$, where $m,n$ are not divisible by $p$, $\alpha=\alpha(x)\in\mathbb{Z}$, then $|x|_{p}=p^{-\alpha}$. This norm satisfies the following properties:

(i) $|x|_{p}\geq 0,\;\;\forall x\in\mathbb{Q}_{p}$, $|x|_{p}=0\Leftrightarrow x=0;$

(ii) $|xy|_{p}=|x|_{p}|y|_{p},\;\;\forall x,y\in\mathbb{Q}_{p};$

(iii) $|x+y|_{p}\leq\max(|x|_{p},|y|_{p}),\;\forall x,y\in\mathbb{Q}_{p}$, and when $|x|_{p}\not=|y|_{p}$, we have $|x+y|_{p}=\max(|x|_{p},|y|_{p}).$

It is also well-known that any non-zero $p$-adic number $x\in\mathbb{Q}_{p}$ can be uniquely represented in the canonical series

[TABLE]

where $\alpha=\alpha(x)\in\mathbb{Z}$,  $x_{k}=0,1,...,p-1,\;x_{0}\not=0,\;k=0,1,...$. This series converges in the $p$-adic norm since $|x_{k}p^{k}|_{p}\leq p^{-k}$.

The space $\mathbb{Q}^{n}_{p}=\mathbb{Q}_{p}\times\cdot\cdot\cdot\times\mathbb{Q}_{p}$ consists of all points $x=(x_{1},...,x_{n})$, where $x_{i}\in\mathbb{Q}_{p},\;i=1,...,n,\;\;n\geq 1$. The $p$-adic norm of $\mathbb{Q}^{n}_{p}$ is defined by

[TABLE]

Let

[TABLE]

be a ball of radius $p^{\alpha}$ with center at $a\in\mathbb{Q}^{n}_{p}$. Similarly, denote by

[TABLE]

the sphere with center at $a\in\mathbb{Q}^{n}_{p}$ and radius $p^{\alpha}$. If $B_{\alpha}=B_{\alpha}(0),S_{\alpha}=S_{\alpha}(0)$, then for any $x_{0}\in\mathbb{Q}^{n}_{p}$ we have $x_{0}+B_{\alpha}=B_{\alpha}(x_{0})$ and $x_{0}+S_{\alpha}=S_{\alpha}(x_{0})$. Especially, we denote $\mathbb{Z}_{p}$ instead of $B_{0}$, $\mathbb{Z}^{*}_{p}=B_{0}\setminus\{0\}$ in $\mathbb{Q}_{p}$, $\mathbb{Q}^{*}_{p}=\mathbb{Q}_{p}\setminus\{0\}$ and $\chi_{k}$ be the characteristic function of the sphere $S_{k}$.

Since $\mathbb{Q}^{n}_{p}$ is a locally compact commutative group under addition, it follows from the standard theory that there exists a Haar measure $dx$ on $\mathbb{Q}^{n}_{p}$, which is unique up to positive constant multiple and is translation invariant. This measure is unique by normalizing $dx$ such that

[TABLE]

where $|B|$ denotes the Haar measure of a measurable subset $B$ of $\mathbb{Q}^{n}_{p}$. By simple calculation, it is easy to obtain that $|B_{\alpha}(a)|=p^{n\alpha},|S_{\alpha}(a)|=p^{n\alpha}(1-p^{-n})\simeq p^{n\alpha}$, for any $a\in\mathbb{Q}^{n}_{p}$. For $f\in L^{1}_{\text{loc}}(\mathbb{Q}^{n}_{p})$, we have

[TABLE]

In particular, if $f\in L^{1}(\mathbb{Q}^{n}_{p})$, we can write

[TABLE]

and

[TABLE]

where $t\in\mathbb{Q}_{p}\setminus\{0\}$. For a more complete introduction to the $p$-adic analysis, we refer the readers to [22, 38] and the references therein.

Let $\omega(x)$ be a non-negative measurable function in $\mathbb{Q}^{n}_{p}$. The weighted Lebesgue space $L^{q}_{\omega}(\mathbb{Q}^{n}_{p})\left(0<q<\infty\right)$ is defined to be the space of all measurable functions $f$ on $\mathbb{Q}^{n}_{p}$ such that

[TABLE]

The space $L^{q}_{\omega,\,\rm loc}(\mathbb{Q}^{n}_{p})$ is defined as the set of all measurable functions $f$ on $\mathbb{Q}^{n}_{p}$ satisfying $\int_{K}|f(x)|^{q}\omega(x)dx<\infty$ for any compact subset $K$ of $\mathbb{Q}^{n}_{p}$. The space $L^{q}_{\omega,\rm loc}(\mathbb{Q}^{n}_{p}\setminus\{0\})$ is also defined in a similar way as the space $L^{q}_{\omega,\rm loc}(\mathbb{Q}^{n}_{p})$. Throught the whole paper, we denote by $C$ a positive geometric constant that is independent of the main parameters, but can change from line to line. Denote $\omega^{\lambda}(B)=\left(\int_{B}{\omega(x)}dx\right)^{\lambda}$, for $\lambda\in\mathbb{R}$ and $\omega(B):=\omega^{1}(B)$. We also write $a\lesssim b$ to mean that there is a positive constant $C$, independent of the main parameters, such that $a\leq Cb$. The symbol $f\simeq g$ means that $f$ is equivalent to $g$ (i.e. $C^{-1}f\leq g\leq Cf$). For any real number $\ell>1$, denote by $\ell^{\prime}$ conjugate real number of $\ell$, i.e. $\frac{1}{\ell}+\frac{1}{\ell^{\prime}}=1$. Next, let us give the definition of weighted $\lambda$-central Morrey spaces on $p$-adic field.

Definition 2.1**.**

Let $\lambda\in\mathbb{R}$ and $1<q<\infty$. The weighted $\lambda$-central Morrey $p$-adic spaces ${\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})$ consists of all Haar measurable functions $f\in L^{q}_{\omega,\rm loc}(\mathbb{Q}_{p}^{n})$ satisfying $\|f\|_{{\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})}<\infty$, where

[TABLE]

Remark that ${\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})$ is a Banach space and reduces to $\{0\}$ when $\lambda<-\frac{1}{q}$.

We also present some definitions of the weighted Herz $p$-adic space and Morrey-Herz $p$-adic space.

Definition 2.2**.**

Let $\beta\in\mathbb{R},0<q<\infty$ and $0<\ell<\infty$. The weighted Herz $p$-adic space $K_{q,\omega}^{\beta,\ell}(\mathbb{Q}^{n}_{p})$ is defined as the set of all Haar measurable functions $f\in L^{q}_{\omega,\rm loc}(\mathbb{Q}_{p}^{n}\setminus\{0\})$ such that $\|f\|_{K_{q,\omega}^{\beta,\ell}(\mathbb{Q}^{n}_{p})}<\infty$, where

[TABLE]

Definition 2.3**.**

Let $\beta\in\mathbb{R},0<q<\infty$ and $0<\ell<\infty$. The weighted Herz $p$-adic space ${\mathop{K}\limits^{.}}_{q,\omega}^{\beta,\ell}(\mathbb{Q}^{n}_{p})$ is defined as the set of all Haar measurable functions $f\in L^{q}_{\omega,\rm loc}(\mathbb{Q}_{p}^{n}\setminus\{0\})$ such that $\|f\|_{{\mathop{K}\limits^{.}}_{q,\omega}^{\beta,\ell}(\mathbb{Q}^{n}_{p})}<\infty$, where

[TABLE]

Definition 2.4**.**

Let $\beta\in\mathbb{R},0<q<\infty$, $0<\ell<\infty$ and $\lambda$ be a non-negative real number. The weighted Morrey-Herz $p$-adic space is defined by

[TABLE]

where

[TABLE]

Let us recall to define the central BMO $p$-adic space.

Definition 2.5**.**

Let $1\leq q<\infty$ and $\omega$ be a weight function. The central bounded mean oscillation space ${\mathop{CMO}\limits^{.}}^{q}_{\omega}(\mathbb{Q}^{n}_{p})$ is defined as the set of all functions $f\in L^{q}_{\omega,\rm loc}(\mathbb{Q}^{n}_{p})$ such that

[TABLE]

where

[TABLE]

The theory of $A_{p}$ weight was first introduced by Benjamin Muckenhoupt on the Euclidean spaces in order to characterise the boundedness of Hardy-Littlewood maximal functions on the weighted $L^{p}$ spaces (see [29]). For $A_{p}$ weights on the $p$-adic fields, more generally, on the local fields or homogeneous type spaces, one can refer to [14, 18] for more details.

Definition 2.6**.**

Let $1<\ell<\infty$. We say that a weight $\omega\in A_{\ell}(\mathbb{Q}^{n}_{p})$ if there exists a constant $C$ such that for all balls $B$,

[TABLE]

We say that a weight $\omega\in A_{1}(\mathbb{Q}^{n}_{p})$ if there is a constant $C$ such that for all balls $B$,

[TABLE]

We denote by $A_{\infty}(\mathbb{Q}^{n}_{p})=\bigcup\limits_{1\leq\ell<\infty}A_{\ell}(\mathbb{Q}^{n}_{p})$.

We give the following standard result related to the Muckenhoupt weights.

Proposition 2.7**.**

1. (i)

$A_{\ell}(\mathbb{Q}^{n}_{p})\subsetneq A_{q}(\mathbb{Q}^{n}_{p})$, for $1\leq\ell<q<\infty$. 2. (ii)

If $\omega\in A_{\ell}(\mathbb{Q}^{n}_{p})$ for $1<\ell<\infty$, then there is an $\varepsilon>0$ such that $\ell-\varepsilon>1$ and $\omega\in A_{\ell-\varepsilon}(\mathbb{Q}^{n}_{p})$.

A closing relation to $A_{\infty}(\mathbb{Q}^{n}_{p})$ is the reverse Hölder condition. If there exist $r>1$ and a fixed constant $C$ such that

[TABLE]

for all balls $B\subset\mathbb{Q}^{n}_{p}$, we then say that $\omega$ satisfies the reverse Hölder condition of order $r$ and write $\omega\in RH_{r}(\mathbb{Q}^{n}_{p})$. According to Theorem 19 and Corollary 21 in [19], $\omega\in A_{\infty}(\mathbb{Q}^{n}_{p})$ if and only if there exists some $r>1$ such that $\omega\in RH_{r}(\mathbb{Q}^{n}_{p})$. Moreover, if $\omega\in RH_{r}(\mathbb{Q}^{n}_{p})$, $r>1$, then $\omega\in RH_{r+\varepsilon}(\mathbb{Q}^{n}_{p})$ for some $\varepsilon>0$. We thus write $r_{\omega}={\rm sup}\{r>1:\omega\in RH_{r}(\mathbb{Q}^{n}_{p})\}$ to denote the critical index of $\omega$ for the reverse Hölder condition.

An important example of $A_{p}(\mathbb{Q}^{n}_{p})$ weight is the power function $|x|_{p}^{\alpha}$. By the similar arguments as Propositions 1.4.3 and 1.4.4 in [27], we obtain the following properties of power weights.

Proposition 2.8**.**

Let $x\in\mathbb{Q}^{n}_{p}$. Then, we have

• (i)

$|x|^{\alpha}_{p}\in A_{1}(\mathbb{Q}^{n}_{p})$* if and only if $-n<\alpha\leq 0$;*

• (ii)

$|x|^{\alpha}_{p}\in A_{\ell}(\mathbb{Q}^{n}_{p})$* for $1<\ell<\infty$, if and only if $-n<\alpha<n(\ell-1)$.*

Let us give the following standard characterization of $A_{\ell}$ weights which it is proved in the similar way as the real setting (see [16, 33] for more details).

Proposition 2.9**.**

Let $\omega\in A_{\ell}(\mathbb{Q}^{n}_{p})\cap RH_{r}(\mathbb{Q}^{n}_{p})$, $\ell\geq 1$ and $r>1$. Then, there exist constants $C_{1},C_{2}>0$ such that

[TABLE]

for any measurable subset $E$ of a ball $B$.

Proposition 2.10**.**

If $\omega\in A_{\ell}(\mathbb{Q}^{n}_{p})$, $1\leq\ell<\infty$, then for any $f\in L^{1}_{\rm loc}(\mathbb{Q}^{n}_{p})$ and any ball $B\subset\mathbb{Q}^{n}_{p}$,

[TABLE]

3. The main results about the boundness of ${\mathcal{H}}^{p}_{\Phi,\Omega}$

Let us now assume that $q$ and $q_{i}\in[1,\infty)$, $\alpha,\alpha_{i}$, $\beta_{i},\beta,\ell_{i},\ell$ are real numbers such that $\alpha_{i}\in(-n,\infty)$, $\ell,\ell_{i}\in[1,\infty)$, $i=1,2,...,m$ and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Theorem 3.1**.**

Let $\omega_{i}(x)=|x|^{\alpha_{i}}_{p}$, \lambda_{i}\in\big{(}\frac{-1}{q_{i}},0\big{)}, for all $i=1,...,m$, $\omega(x)=|x|^{\alpha}_{p}$, $\Omega\in L^{q^{\prime}}(S_{0})$ and the following condition is true:

[TABLE]

Then, we have that ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is bounded from ${\mathop{B}\limits^{.}}^{q_{1},\lambda_{1}}_{\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{B}\limits^{.}}^{{q_{m}},{\lambda_{m}}}_{\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})$ if and only if

[TABLE]

Furthermore, we obtain {\big{\|}{\mathcal{H}}^{p}_{\Phi,\Omega}\big{\|}_{{\mathop{B}\limits^{.}}^{{q_{1}},{\lambda_{1}}}_{{\omega_{1}}}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{B}\limits^{.}}^{{q_{m}},{\lambda_{m}}}_{{\omega_{m}}}(\mathbb{Q}^{n}_{p})\to{\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})}}\simeq\mathcal{C}_{1}.

Proof.

We prove the sufficient condition of the theorem. For $R\in\mathbb{Z}$, denote

[TABLE]

By using the Minkowski inequality, we get

[TABLE]

Next, by applying the Hölder inequality, we have

[TABLE]

Thus, we deduce

[TABLE]

On the other hand, $\int_{S_{0}}\int_{B_{R}}|f_{i}(p^{\gamma}|x|_{p}^{-1}y)|^{q_{i}}\omega_{i}(x)dxdy$ is equaled by

[TABLE]

Hence, by (3.3), we have

[TABLE]

Now, we need to prove that

[TABLE]

where

[TABLE]

Indeed, we get

[TABLE]

and $\omega_{i}(B_{R-\gamma})=\frac{p^{(R-\gamma)(\alpha_{i}+n)}(1-p^{-n})}{1-p^{-(\alpha_{i}+n)}}$. Thus, by (3.1), we complete the proof for (3.6).

Applying (3.5) and (3.6), we imply

[TABLE]

Therefore, we have $\|{\mathcal{H}}^{p}_{\Phi,\Omega}(\vec{f})\|_{{\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})}\leq(1-p^{-n})\mathcal{K}.\mathcal{C}_{1}\|\Omega\|_{L^{q^{\prime}}(S_{0})}\prod\limits_{i=1}^{m}\|f_{i}\|_{{\mathop{B}\limits^{.}}^{q_{i},\lambda_{i}}_{\omega_{i}}(\mathbb{Q}^{n}_{p})}.$

To give the proof for necessary condition of the theorem, let us now choose

[TABLE]

Then, we get

[TABLE]

Thus, by the similar argument as (3.7), it is not hard to show that

[TABLE]

Now, $\|{\mathcal{H}}^{p}_{\Phi,\Omega}(\vec{f})\|_{{\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})}$ is equal to

[TABLE]

By estimating as (3.7) and (3), we obtain

[TABLE]

Thus, it follows from (3.9) that

[TABLE]

Since ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is bounded from ${\mathop{B}\limits^{.}}^{q_{1},\lambda_{1}}_{\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{B}\limits^{.}}^{{q_{m}},{\lambda_{m}}}_{\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})$, it immediately implies that $\mathcal{C}_{1}<\infty$. Thus, the proof of the theorem is finished. ∎

Next, we have the boundedness of $p$-adic rough multilinear Hausdorff operators on weighted $\lambda$-central Morrey $p$-adic spaces associated with the Muckenhoupt weights.

Theorem 3.2**.**

Let $1\leq q^{*},\zeta<\infty$, $\Omega\in L^{(\frac{q}{\zeta})^{\prime}}(S_{0}),$ $\lambda_{i}\in(\frac{-1}{q_{i}},0)$, for all $i=1,...,m$ and $\omega\in A_{\zeta}$ with the finite critical index $r_{\omega}$ for the reverse Hölder condition. Assume that $q>q^{*}\zeta r_{\omega}/(r_{\omega}-1)$, $\delta\in(1,r_{\omega})$ and the two following conditions hold:

[TABLE]

[TABLE]

Then, ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is bounded from ${\mathop{B}\limits^{.}}^{q_{1},\lambda_{1}}_{\omega}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{B}\limits^{.}}^{{q_{m}},{\lambda_{m}}}_{\omega}(\mathbb{Q}^{n}_{p})$ to ${\mathop{B}\limits^{.}}^{q^{*},\lambda^{*}}_{\omega}(\mathbb{Q}^{n}_{p})$.

Proof.

According to the Minkowski inequality, it implies that

[TABLE]

In view of the condition $q>q^{*}\zeta r_{\omega}/(r_{\omega}-1)$, there exists $r\in(1,r_{\omega})$ such that $q/\zeta=q^{*}r^{\prime}$. By the Hölder inequality and the reverse Hölder condition, we have

[TABLE]

Next, by using the Höder inequality and estimating as (3) and (3) above, we deduce

[TABLE]

Thus, by (3.12) and (3), we imply

[TABLE]

For $i=1,...,m$, by applying Proposition 2.10 again, we have

[TABLE]

It follows that

[TABLE]

Consequently, by (3.10), we have $\dfrac{1}{\omega^{\frac{1}{q*}+\lambda*}(B_{R})}\|{\mathcal{H}}^{p}_{\Phi,\Omega}(\vec{f})\|_{L_{\omega}^{q^{*}}(B_{R})}$ is controlled by

[TABLE]

Because of having (3.10), it also gives $\lambda^{*}<0$. Thus, by $\delta\in(1,r_{\omega})$ and Proposition 2.9, if $\gamma\geq 0$

[TABLE]

and otherwise,

[TABLE]

By (3.15), (3.16) and (3.17), we obtain

[TABLE]

This completes the proof of theorem. ∎

Theorem 3.3**.**

Let $\omega_{1}(x)=|x|^{\alpha_{1}}_{p},...,\omega_{m}(x)=|x|^{\alpha_{m}}_{p}$, $\omega(x)=|x|^{\alpha}_{p}$ and $\Omega\in L^{q^{\prime}}(S_{0})$. Then, we have that ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is bounded from ${K}^{\beta_{1},\ell_{1}}_{q_{1},\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{K}^{{\beta_{m},\ell_{m}}}_{q_{m},\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${K}^{\beta,\ell}_{q,\omega}(\mathbb{Q}^{n}_{p})$ if and only if

[TABLE]

Furthermore, {\big{\|}{\mathcal{H}}^{p}_{\Phi,\Omega}\big{\|}_{{K}^{\beta_{1},\ell_{1}}_{q_{1},\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{K}^{{\beta_{m},\ell_{m}}}_{q_{m},\omega_{m}}(\mathbb{Q}^{n}_{p})\to{K}^{\beta,\ell}_{q,\omega}(\mathbb{Q}^{n}_{p})}\simeq\mathcal{C}_{3}}.

Proof.

We begin with the proof for the sufficient condition of theorem. For $k\in\mathbb{Z}$, by using Minkowski inequality and Hölder inequality again and making as (3.3) above, we have

[TABLE]

Next, a simple calculation as (3) above follows us to have that

[TABLE]

Thus, we have

[TABLE]

Combining this with the Minkowski inequality, we have

[TABLE]

In view of the Hölder inequality and the definition of the Herz space, we obtain that

[TABLE]

which implies that $\|{\mathcal{H}}^{p}_{\Phi,\Omega}(\vec{f})\|_{K^{\beta,\ell}_{q,\omega}(\mathbb{Q}_{p}^{n})}\lesssim\mathcal{C}_{2}.\|\Omega\|_{L^{q^{\prime}}(S_{0})}.\prod\limits_{i=1}^{m}\|f_{i}\|_{K^{\beta_{i},\ell_{i}}_{q_{i},\omega_{i}}(\mathbb{Q}^{n}_{p})}.$

Conversely, suppose that ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is bounded from ${K}^{\beta_{1},\ell_{1}}_{q_{1},\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{K}^{{\beta_{m},\ell_{m}}}_{q_{m},\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${K}^{\beta,\ell}_{q,\omega}(\mathbb{Q}^{n}_{p})$. For $i=1,...,m$ and $r\in\mathbb{Z}^{+}$, let us choose the following functions

[TABLE]

It is clear to see that when $k<0$ then $f_{i,r}\chi_{k}=0$. Otherwise, we have

[TABLE]

Thus, it leads that $\|f_{i,r}\|_{K^{\beta_{i},\ell_{i}}_{q_{i},\omega_{i}}(\mathbb{Q}^{n}_{p})}>0$. Moreover,

[TABLE]

Now, we get

[TABLE]

Therefore, for $r\in\mathbb{Z}$, it follows immediately that

[TABLE]

Estimating as (3.20) above, it implies $\left\|g_{k}\right\|_{L^{q}_{\omega}(\mathbb{Q}^{n}_{p})}\simeq p^{-k(\beta+\frac{m}{p^{r}})}$. Thus,

[TABLE]

Hence, by (3), we have

[TABLE]

where $\mathcal{A}(r)=\frac{p^{\frac{-rm}{p^{r}}}\prod\limits_{i=1}^{m}(1-p^{\frac{-\ell_{i}}{p^{r}}})^{1/\ell_{i}}}{(1-p^{\frac{-m\ell}{p^{r}}})^{1/\ell}}.$ Note that, by lettting $r$ big enough, we get

[TABLE]

By having $\frac{1}{\ell_{1}}+\cdots+\frac{1}{\ell_{m}}=\frac{1}{\ell}$, and lettting $r\to+\infty$, we obtain

[TABLE]

Hence, by the dominated convergence theorem of Lebesgue and the boundedness of ${\mathcal{H}}^{p}_{\Phi,\Omega}$ on ${K}^{\beta_{1},\ell_{1}}_{q_{1},\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{K}^{{\beta_{m},\ell_{m}}}_{q_{m},\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${K}^{\beta,\ell}_{q,\omega}(\mathbb{Q}^{n}_{p})$, we get

[TABLE]

which finishes the proof of this theorem. ∎

Theorem 3.4**.**

Let $1\leq q^{*}<\infty$, $\beta^{*}\in\mathbb{R}$, $\beta_{1},...,\beta_{m}\in\mathbb{R}^{-}$ (set of all negative real numbers) such that

[TABLE]

*Suppose that $\omega\in A_{\zeta}$, $1\leq\zeta<\infty$, with the finite critical index $r_{\omega}$ for the reverse Hölder condition and $q>q^{*}\zeta r_{\omega}/(r_{\omega}-1)$, $\Omega\in L^{(\frac{q}{\zeta})^{\prime}}(S_{0}),\delta\in(1,r_{\omega})$.

$(\rm i)$ If $\dfrac{1}{q_{i}}+\dfrac{\alpha_{i}}{n}\geq 0$, for $i=1,...,m$, and*

[TABLE]

then ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is bounded from ${\mathop{K}\limits^{.}}^{\beta_{1},\ell_{1}}_{q_{1},\omega}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{K}\limits^{.}}^{\beta_{m},\ell_{m}}_{q_{m},\omega}(\mathbb{Q}^{n}_{p})$ to ${\mathop{K}\limits^{.}}^{\beta^{*},\ell}_{q^{*},\omega}(\mathbb{Q}^{n}_{p})$.

$\,(\rm ii)$* If $\dfrac{1}{q_{i}}+\dfrac{\alpha_{i}}{n}<0$, for $i=1,...,m$, and*

[TABLE]

then ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is bounded from ${\mathop{K}\limits^{.}}^{\beta_{1},\ell_{1}}_{q_{1},\omega}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{K}\limits^{.}}^{\beta_{m},\ell_{m}}_{q_{m},\omega}(\mathbb{Q}^{n}_{p})$ to ${\mathop{K}\limits^{.}}^{\beta^{*},\ell}_{q^{*},\omega}(\mathbb{Q}^{n}_{p}).$

Proof.

According to estimating as (3), we also have

[TABLE]

From this, by the Minkowski inequality, $\|{\mathcal{H}}^{p}_{\Phi,\Omega}(\vec{f})\|_{{\mathop{K}\limits^{.}}^{\beta^{*},\ell}_{q^{*},\omega}(\mathbb{Q}^{n}_{p})}$ is controlled by

[TABLE]

By the relation $\frac{1}{q^{*}}+\frac{\beta^{*}}{n}=\frac{1}{q}+\frac{\beta}{n}$ and the Hölder inequality, we get

[TABLE]

Thus, by (3), it follows that

[TABLE]

From $\beta_{i}<0$ and $\eta\leq-\gamma$, by Proposition 2.9 again, we see that

[TABLE]

When $\dfrac{1}{q_{i}}+\dfrac{\beta_{i}}{n}\geq 0$, Proposition 2.9 follows us to have

[TABLE]

When $\dfrac{1}{q_{i}}+\dfrac{\beta_{i}}{n}<0$, it is easy to see that Proposition 2.9 yields

[TABLE]

In the case $(i)$, by (3), (3.24) and (3.28), we obtain that

[TABLE]

Note that, by using the Minkowski inequality again, we deduce

[TABLE]

Therefore, by (3), we have

[TABLE]

which completes the proof for this case.

In this case $(\rm ii)$, from by (3), (3.24), (3.32) and a similar argument as above, we also have

[TABLE]

This implies that the proof of theorem is finished. ∎

Theorem 3.5**.**

Let $\omega_{i},\omega,\Omega$ be as Theorem 3.3, $\lambda_{1},...,\lambda_{m}>0$ and the hypothesis (3.10) in Theorem 3.2 holds. Then, ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is a bounded operator from ${MK}^{\beta_{1},\lambda_{1}}_{\ell_{1},q_{1},\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{MK}^{{\beta_{m},\lambda_{m}}}_{\ell_{m},q_{m},\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${MK}^{\beta,\lambda^{*}}_{\ell,q,\omega}(\mathbb{Q}^{n}_{p})$ if and only if

[TABLE]

Moreover, {\big{\|}{\mathcal{H}}^{p}_{\Phi,\Omega}\big{\|}_{{MK}^{\beta_{1},\lambda_{1}}_{\ell_{1},q_{1},\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{MK}^{{\beta_{m},\lambda_{m}}}_{\ell_{m},q_{m},\omega_{m}}(\mathbb{Q}^{n}_{p})\to{MK}^{\beta,\lambda^{*}}_{\ell,q,\omega}(\mathbb{Q}^{n}_{p})}\simeq\mathcal{C}_{5}}.

Proof.

By the estimation (3.19) and the definition of the Morrey-Herz $p$-adic space, we have

[TABLE]

Using the Hölder inequality and the relation (3.10), we get

[TABLE]

Therefore, we obtain

[TABLE]

which implies that ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is a bounded operator from ${MK}^{\beta_{1},\lambda_{1}}_{\ell_{1},q_{1},\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{MK}^{{\beta_{m},\lambda_{m}}}_{\ell_{m},q_{m},\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${MK}^{\beta,\lambda^{*}}_{\ell,q,\omega}(\mathbb{Q}^{n}_{p})$ if $\mathcal{C}_{5}<\infty.$

Conversely, by a similar argument, we also choose

[TABLE]

Thus, $\|f_{i}\chi_{k}\|^{q_{i}}_{L^{q_{i}}_{\omega_{i}}(\mathbb{Q}^{n}_{p})}=\int_{S_{k}}|x|_{p}^{-\beta_{i}q_{i}-n+\lambda_{i}q_{i}}dx\simeq p^{kq_{i}(\lambda_{i}-\beta_{i})}$, which implies that

[TABLE]

In addition, it is easy to show that

[TABLE]

Hence, we estimate

[TABLE]

Since ${\mathcal{H}}^{p}_{\Phi,\Omega}$ is a bounded operator from ${MK}^{\beta_{1},\lambda_{1}}_{\ell_{1},q_{1},\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{MK}^{{\beta_{m},\lambda_{m}}}_{\ell_{m},q_{m},\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${MK}^{\beta,\lambda^{*}}_{\ell,q,\omega}(\mathbb{Q}^{n}_{p})$, it follows that

[TABLE]

which finishes the proof. ∎

4. The main results about the boundness of ${\mathcal{H}}^{p}_{\Phi,\Omega,\vec{b}}$

Before stating our next results, we introduce some notations which will be used throughout this section. Let $q$ and $q_{i}\in[1,\infty)$, $\alpha,\beta,\ell$, $\alpha_{i},\beta_{i},\ell_{i},r_{i}$ are real numbers such that $\alpha_{i}\in(-n,\infty)$, $\ell,\ell_{i}\in[1,\infty)$, $r_{i}\in[1,\infty)$, $i=1,2,...,m$ and denote

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Theorem 4.1**.**

*Let \lambda_{i}\in\big{(}\frac{-1}{q_{i}},0\big{)}, $\omega_{i}(x)=|x|_{p}^{\alpha_{i}}$, $b_{i}\in{\mathop{CMO}\limits^{.}}^{r_{i}}_{\omega_{i}}(\mathbb{Q}^{n}_{p})$ for all $i=1,...,m$, $\omega(x)=|x|_{p}^{\alpha}$, $\Omega\in L^{q^{\prime}}(S_{0})$, and the hypothesis (3.1) in Theorem 3.1 holds.

(i) If*

[TABLE]

*then we have ${\mathcal{H}}^{p}_{\Phi,\Omega,\vec{b}}$ is bounded from ${\mathop{B}\limits^{.}}^{q_{1},\lambda_{1}}_{\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{B}\limits^{.}}^{{q_{m}},{\lambda_{m}}}_{\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})$.

(ii) If ${\mathcal{H}}^{p}_{\Phi,\Omega,\vec{b}}$ is bounded from ${\mathop{B}\limits^{.}}^{q_{1},\lambda_{1}}_{\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{B}\limits^{.}}^{{q_{m}},{\lambda_{m}}}_{\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})$ and*

[TABLE]

then $\mathcal{C}^{*}_{6}$ is finite. Furthermore,

[TABLE]

Proof.

By applying the Minkowski inequality, it implies that

[TABLE]

According to the Hölder inequality, we get

[TABLE]

Therefore, by using the Hölder inequality again, $\|{\mathcal{H}}^{p}_{\Phi,\Omega,\vec{b}}(\vec{f})\|_{L^{q}_{\omega}(B_{R})}$ is controlled by

[TABLE]

Next, we can show that

[TABLE]

Indeed, we estimate

[TABLE]

In view of the definition of the space ${\mathop{CMO}\limits^{.}}_{\omega_{i}}^{r_{i}}(\mathbb{Q}^{n}_{p})$, we deduce

[TABLE]

Next, we observe that

[TABLE]

which implies that

[TABLE]

Thus,

[TABLE]

Now, we see that

[TABLE]

In the case $\gamma<0$, by using the Hölder inequality, it follows that

[TABLE]

Otherwise, by estimating as above, we also have

[TABLE]

By (4), (4) and (4.9), we obtain that

[TABLE]

From this, by (4) and (4), we finish the proof of the inequality (4.2). In view of the relations (3), (4) and (4.2), we imply

[TABLE]

By having the hypothesis (3.1), it deduces $\frac{1}{\omega^{\frac{1}{q}+\lambda}(B_{R})}\simeq\prod\limits_{i=1}^{m}\frac{p^{\frac{-R(n+\alpha_{i})}{r_{i}}-\gamma(n+\alpha_{i})(\frac{1}{q_{i}}+\lambda_{i})}}{\omega_{i}^{\frac{1}{q_{i}}+\lambda_{i}}(B_{R-\gamma})}.$ Thus, by (4) and the definition of the $\lambda$-central Morrey $p$-adic space, we have

[TABLE]

which completes the proof for the first part of theorem.

Next, we will prove the second part of theorem. Let us take that ${\mathcal{H}}^{p}_{\Phi,\Omega,\vec{b}}$ is bounded from ${\mathop{B}\limits^{.}}^{q_{1},\lambda_{1}}_{\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{B}\limits^{.}}^{{q_{m}},{\lambda_{m}}}_{\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})$. We choose

[TABLE]

Note that, by Lemma 6.1 in [17] (also see Lemma 2.1 in [32]), we imply $b_{i}\in{\rm BMO}_{\omega}(\mathbb{Q}^{n}_{p})\subset{\rm CMO}^{r_{i}}_{\omega}(\mathbb{Q}^{n}_{p})$. From (3.9), we have $f_{i}\in{\mathop{B}\limits^{.}}^{q_{i},\lambda_{i}}_{\omega_{i}}(\mathbb{Q}^{n}_{p})$ and $|x|_{p}^{(n+\alpha)\lambda}\in{\mathop{B}\limits^{.}}^{q,\lambda}_{\omega}(\mathbb{Q}^{n}_{p})$.

Thus, we get

[TABLE]

Therefore, we deduce

[TABLE]

This leads that $\mathcal{C}_{4}^{*}$ is finite. The proof of theorem is ended. ∎

Theorem 4.2**.**

Let $1\leq q,r_{1}^{*},...,r_{m}^{*},q_{1}^{*},...,q_{m}^{*},\zeta<\infty$, $\omega\in A_{\zeta}$ with the finite critical index $r_{\omega}$ for the reverse Hölder condition, $\delta\in(1,r_{\omega})$, $\Omega\in L^{q^{\prime}}(S_{0})$, \lambda_{i}\in\big{(}\frac{-1}{q^{*}_{i}},0\big{)} and $b_{i}\in{\mathop{CMO}\limits^{.}}^{r_{i}^{*}}_{\omega}(\mathbb{Q}^{n}_{p})$ for all $i=1,...,m$. Assume that the hypothesis (3.10) in Theorem 3.2 holds and the two following conditions are true:

[TABLE]

[TABLE]

Then, ${\mathcal{H}}^{p}_{\Phi,\Omega,\vec{b}}$ is bounded from ${\mathop{B}\limits^{.}}^{q_{1}^{*},\lambda_{1}}_{\omega}(\mathbb{Q}^{n}_{p})\times\cdots\times{\mathop{B}\limits^{.}}^{{q_{m}^{*}},{\lambda_{m}}}_{\omega}(\mathbb{Q}^{n}_{p})$ to ${\mathop{B}\limits^{.}}^{q,\lambda^{*}}_{\omega}(\mathbb{Q}^{n}_{p})$.

Proof.

From the inequality (4.11), there exist $r_{1},...,r_{m},q_{1},...,q_{m}$ such that

[TABLE]

[TABLE]

and

[TABLE]

Because of $\sum\limits_{i=1}^{m}\dfrac{1}{r_{i}}+\dfrac{1}{q_{i}}=\dfrac{1}{q}$ and making a similar argument as (4), we also have

[TABLE]

To prove this theorem, we need to show the following result,

[TABLE]

Actually, we compose

[TABLE]

By estimating as (4) above, we deduce

[TABLE]

Since $\frac{1}{r_{i}}>\frac{\zeta}{r_{i}^{*}}\frac{r_{\omega}}{r_{\omega}-1}$, there exists $\beta_{i,0}\in(1,r_{\omega})$ rewarding $\frac{r_{i}^{*}}{\zeta}=r_{i}\beta_{i,0}^{\prime}$. By the Hölder inequality and the reverse Hölder condition again, we infer

[TABLE]

By evaluating as (4) and Proposition 2.10, we have

[TABLE]

This deduces that

[TABLE]

By the reasons as (4), (4) and (4.9) above, we estimate

[TABLE]

Because of giving $\omega\in A_{\zeta}$ and using Proposition 2.9, we have

[TABLE]

Hence, we have

[TABLE]

Note that, from the inequality $r_{i}<r^{*}_{i}$, we get $\|b_{i}\|_{{\mathop{CMO}\limits^{.}}_{\omega}^{r_{i}}(\mathbb{Q}^{n}_{p})}\lesssim\|b_{i}\|_{{\mathop{CMO}\limits^{.}}_{\omega}^{r^{*}_{i}}(\mathbb{Q}^{n}_{p})}.$ Therefore, by (4.16), (4) and (4.26), we obtain the proof of the inequality (4.14).

On the other hand, because of having $\frac{1}{q_{i}}>\frac{1}{q_{i}^{*}}\zeta\dfrac{r_{\omega}}{r_{\omega}-1}$, there exists $\eta_{i}\in(1,r_{\omega})$ such that $\frac{q_{i}^{*}}{\zeta}=q_{i}\eta_{i}^{\prime}$. By estimating as (4) above, we infer

[TABLE]

From this, by (4) and (4.14), one has

[TABLE]

As a consequence, by (3.10), for all $R\in\mathbb{Z}$, we have

[TABLE]

Hence, by (3.16) and (3.17), we have

[TABLE]

which achieves the desired result. ∎

Theorem 4.3**.**

Let $\omega_{i}(x)=|x|_{p}^{\alpha_{i}}$, $b_{i}\in{\mathop{CMO}\limits^{.}}^{r_{i}}_{\omega_{i}}(\mathbb{Q}^{n}_{p})$, $\lambda_{i}\geq 0$ for all $i=1,...,m$, $\omega(x)=|x|_{p}^{\alpha}$, $\Omega\in L^{q^{\prime}}(S_{0})$. Assume that the hypothesis (3.10) in Theorem 3.2 is true and the following conditions hold:

[TABLE]

[TABLE]

Then, we have

[TABLE]

Proof.

By estimating as (4) above, it yields that

[TABLE]

From this, by the definition of the Morrey-Herz $p$-adic space and the relation (4.27), we obtain that

[TABLE]

Thus, by (3) above, we get

[TABLE]

which finishes the proof of Theorem 4.3. ∎

It is well known that the Herz $p$-adic space is a special case of Morrey-Herz $p$-adic space. From this and Theorem 4.3, we also obtain the desired result as follows.

Corollary 4.4**.**

The assumptions of Theorem 4.3 are true with $\lambda_{1}=\cdots=\lambda_{m}=0$ and the following condition holds:

[TABLE]

Then, we have ${\mathcal{H}}^{p}_{\Phi,\Omega,\vec{b}}$ is bounded from ${K}^{\beta_{1},\ell_{1}}_{q_{1},\omega_{1}}(\mathbb{Q}^{n}_{p})\times\cdots\times{K}^{{\beta_{m},\ell_{m}}}_{q_{m},\omega_{m}}(\mathbb{Q}^{n}_{p})$ to ${K}^{\beta^{*},\ell}_{q,\omega}(\mathbb{Q}^{n}_{p})$.