Известия Российской академии наук. Серия математическая
RUS  ENG    ЖУРНАЛЫ   ПЕРСОНАЛИИ   ОРГАНИЗАЦИИ   КОНФЕРЕНЦИИ   СЕМИНАРЫ   ВИДЕОТЕКА   ПАКЕТ AMSBIB  
Общая информация
Последний выпуск
Скоро в журнале
Архив
Импакт-фактор
Правила для авторов
Загрузить рукопись

Поиск публикаций
Поиск ссылок

RSS
Последний выпуск
Текущие выпуски
Архивные выпуски
Что такое RSS



Изв. РАН. Сер. матем.:
Год:
Том:
Выпуск:
Страница:
Найти






Персональный вход:
Логин:
Пароль:
Запомнить пароль
Войти
Забыли пароль?
Регистрация


Известия Российской академии наук. Серия математическая, 2023, том 87, выпуск 5, страницы 71–91
DOI: https://doi.org/10.4213/im9404
(Mi im9404)
 

Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)

Two-weight estimates for Hardy–Littlewood maximal functions and Hausdorff operators on $p$-adic Herz spaces

Kieu Huu Dunga, Dao Van Duongb

a Faculty of Fundamental Sciences, Van Lang University, Ho Chi Minh City, Vietnam
b School of Mathematics, Mientrung University of Civil Engineering, Phu Yen, Vietnam
Список литературы:
Аннотация: This paper is concerned with some sufficient conditions for the boundedness of Hardy–Littlewood maximal functions, rough Hausdorff and matrix Hausdorff operators on two-weighted Herz spaces on $p$-adic fields through its atomic decomposition.
Ключевые слова: Hardy–Littlewood maximal operator, Hausdorff operator, Hardy operator, Herz space, atom, $p$-adic analysis.
Финансовая поддержка Номер гранта
Van Lang University, Vietnam
Kieu Huu Dung would like to thank Van Lang University, Vietnam for funding this work.
Поступило в редакцию: 02.08.2022
Исправленный вариант: 08.01.2023
Англоязычная версия:
Izvestiya: Mathematics, 2023, Volume 87, Issue 5, Pages 920–940
DOI: https://doi.org/10.4213/im9404e
Реферативные базы данных:
Тип публикации: Статья
УДК: 517.51
Язык публикации: английский

§ 1. Introduction

The $p$-adic analysis in the past decades has attracted much attention due to its great applications in mathematical physics as well as its necessity in sciences and technologies (see e.g. [1]–[4] and references therein). In recent years, harmonic analysis, pseudo-differential operators, quantum theory, and wavelet analysis have been intensively studied over $p$-adic fields (see [2], [5]–[11]).

In 2010, Volosivets [12] studied the Hausdorff operator on $p$-adic field defined by

$$ \begin{equation} \mathcal{H}^p_{\Psi,A}(f)(x)=\int_{\mathbb Q_p^n} \Psi(y)f(A(y)x)\, dy,\qquad x\in\mathbb Q^n_p, \end{equation} \tag{1.1} $$
where $\Psi$ is a locally integrable function on $\mathbb Q_p^n$ and $A(y)$ is an $n\times n$ invertible matrix for almost everywhere $y$ in the support of $\Psi$. It should be noted that the history of the Hausdorff operator on real field may be dated back to the works of Hurwitz and Silverman [13] and Hausdorff [14].

Clearly, the Hausdorff operator includes the weighted Hardy–Littlewood average studied by Rim and Lee [15] as follows:

$$ \begin{equation} \mathcal{H}^p_\varphi f(x)=\int_{\mathbb{Z}_p^*} f(yx)\varphi(y)\, dy, \qquad x\in\mathbb{Q}^n_p. \end{equation} \tag{1.2} $$
It is worth noticing that the solution of $p$-adic pseudo-differential equations studied by Kochubei [16] is closely connected with the weighted Hardy–Littlewood average. More details, Kochubei discussed the following Cauchy problem:
$$ \begin{equation*} \begin{cases} D^\alpha u+a(|x|_p)u=f(|x|_p), &x\in\mathbb{Q}_p, \\ u(0)=0, \end{cases} \end{equation*} \notag $$
where $D^\alpha$ is the Vladimirov operator of order $\alpha$ (see [4] for further detail). The solution of this problem is given by $u=\mathcal{R}^p_\alpha(f)$, where
$$ \begin{equation} \mathcal{R}^p_\alpha(f)(x)=\frac{1-p^{-\alpha}}{1-p^{\alpha-1}}\int_{|y|_p\leqslant |x|_p}\bigl(|x-y|^{\alpha-1}_p-|y|_p^{\alpha-1}\bigr)f(y)\, dy. \end{equation} \tag{1.3} $$
It is easy to see that
$$ \begin{equation*} \mathcal{R}^p_\alpha(f)(x)=|x|^\alpha_p \bigl(\mathcal{H}^p_{\varphi_1} f(x)-\mathcal{H}^p_{\varphi_2} f(x)\bigr), \end{equation*} \notag $$
with
$$ \begin{equation*} \varphi_1(y)=\frac{1-p^{-\alpha}}{1-p^{\alpha-1}}|1-y|_p^{\alpha-1}\quad\text{and}\quad \varphi_2(y)=\varphi_1(1-y). \end{equation*} \notag $$

Note that $\mathcal{H}^p_{\Psi, A}$ contains $p$-adic Hardy–Hilbert type integral operator studied in [17] as a special case. More precisely, $p$-adic Hardy–Hilbert operator is of the form

$$ \begin{equation*} \mathcal{T}^p (f)(x)=\int_{\mathbb Q^*_p} K(|x|_p, |y|_p)f(y)\, dy, \qquad x\in\mathbb Q^*_p, \end{equation*} \notag $$
where $K$ is the nonnegative function such that $K(t u, t v) = t^{-1}K(u, v)$, for every $u, v, t> 0$. It is plain to see that $\mathcal{T}^p$ can be rewritten by
$$ \begin{equation*} \mathcal{T}^p (f)(x)=\int_{\mathbb Q^*_p} {K}(1, |y|_p)f(yx)\, dy. \end{equation*} \notag $$
Obviously, $\mathcal{T}^p$ is a special case of $\mathcal{H}^p_{\Psi, A}$ by choosing $n\,{=}\,1$, $A(y)\,{=}\,y$ and $\Psi(y)=K(1, |y|_p)$. Let us now give some special cases of $K$ which are deduced to several important operators in harmonic analysis. By taking
$$ \begin{equation*} K(|x|_p, |y|_p)=\frac{1}{|x|_p+|y|_p}, \end{equation*} \notag $$
we have $p$-adic Hilbert operator defined by
$$ \begin{equation*} H^p (f)(x)=\int_{\mathbb Q^*_p}\frac{f(y)}{|x|_p+|y|_p}\, dy,\qquad x\in\mathbb Q^*_p. \end{equation*} \notag $$
Note that the term “$p$-adic Hilbert operator” was also assigned to a completely different operator, a singular integral of the Calderón–Zygmund type (see in [18]). Also, for
$$ \begin{equation*} K(|x|_p,|y|_p)=|x|_p^{-1}\chi_{\{|y|_p\leqslant |x|_p\}}(|y|_p), \end{equation*} \notag $$
$\mathcal{T}^p$ becomes $p$-adic Hardy operator
$$ \begin{equation*} \mathcal{H}^p(f)(x)=\frac{1}{|x|_p}\int_{|y|_p\leqslant|x|_p}f(y)\, dy, \qquad x\in\mathbb Q^*_p. \end{equation*} \notag $$
Moreover, by letting
$$ \begin{equation*} K(|x|_p, |y|_p)=\frac{1}{\max\{|x|_p, |y|_p\}}, \end{equation*} \notag $$
we have $p$-adic Hardy–Littlewood–Pólya operator defined by
$$ \begin{equation*} R^p(f)(x)=\int_{\mathbb{Q}^*_p}\frac{f(y)}{\max\{|x|_p, |y|_p\}} \, dy, \qquad x\in\mathbb Q^*_p. \end{equation*} \notag $$

In 2017 Volosivets [19] also studied the rough Hausdorff operators on $p$-adic Lorentz spaces defined by

$$ \begin{equation} H^p_{\Phi,\Omega}(f)(x)=\int_{\mathbb Q^n_p} \frac{\Phi(x|y|_p)}{|y|^n_p}\Omega(y|y|_p)f(y)\, dy. \end{equation} \tag{1.4} $$
Next, Chuong, Duong and Dung [20] studied a new class of $p$-adic rough multilinear Hausdorff operators on weighted Morrey–Herz spaces. Let us now recall the definition of $p$-adic rough Hausdorff operators (see [20] for more details). Let $\Phi\colon \mathbb Q_p\to [0, \infty) $ and $\Omega\colon S_0\to\mathbb C $ be measurable functions such that $\Omega(y)\neq 0$ for almost everywhere $y$ in $S_0$. The $p$-adic rough Hausdorff operator is defined by
$$ \begin{equation} \mathcal{H}^p_{\Phi,\Omega}(f)(x)=\sum_{\gamma\in\mathbb Z}\int_{S_0} \frac{\Phi(p^{\gamma})}{p^{\gamma}}\Omega(y) f(p^{\gamma}|x|_p^{-1}y)\, dy,\qquad x\in\mathbb Q^n_p. \end{equation} \tag{1.5} $$
It is easily seen that for $\Omega \equiv 1$ and $\Phi(y)= |y|_p^{n-1}\chi_{\{|y|_p\leqslant 1\}}$, $\mathcal{H}^p_{\Phi,\Omega}$ becomes $p$-adic Hardy operator defined by
$$ \begin{equation} \mathcal H^p(f)(x)=\frac{1}{|x|_p^n}\int_{B(0,|x|_p)}f(y)\, dy, \qquad x\in\mathbb Q^n_p\setminus\{ 0\}, \end{equation} \tag{1.6} $$
where $B(0,|x|_p)$ is a ball in $\mathbb Q^n_p$ with center at origin and radius $|x|_p$. For further details about $p$-adic Hardy operator, see [17], [21], and references therein.

In particular, the Hardy–Littlewood maximal operator is of fundamental importance in harmonic analysis since its significant applications (see [22] and references therein). For a locally integrable function $f$ on $\mathbb Q^n_p$, the Hardy–Littlewood maximal operator $\mathcal M^p$ is defined by

$$ \begin{equation} \mathcal M^p(f)(x)=\sup_{\gamma\in\mathbb Z}\frac{1}{p^{n\gamma}}\int_{B_\gamma(x)}|f(y)|\, dy, \qquad x\in\mathbb Q^n_p. \end{equation} \tag{1.7} $$
Note that $\mathcal M^p(f)(x)$ is potentially infinite for any given $x$. In addition, if $f\,{\in}\, L^{q}(\mathbb Q^n_p)$, $1\leqslant q\leqslant \infty$, then $\mathcal M^p$ is finite almost everywhere. More information on $\mathcal M^p$ can be found in [9], [23], and references therein.

On the real field, the two-weighted Herz spaces $\dot{K}_{q}^{\alpha,\ell}(\omega_1,\omega_2)$ and $K_{q}^{\alpha,\ell}(\omega_1,\omega_2)$ were introduced by Lu and Yang [24]. In the case $\omega_1\equiv \omega_2\equiv 1$, $\dot{K}_{q}^{\alpha,\ell}(\omega_1,\omega_2)$ and $K_{q}^{\alpha,\ell}(\omega_1,\omega_2)$ are denoted by $\dot{K}_{q}^{\alpha,\ell}(\mathbb R^n)$ and $K_{q}^{\alpha,\ell}(\mathbb R^n)$. The Herz spaces $\dot{K}_{q}^{\alpha,\ell}(\mathbb R^n)$ and $K_{q}^{\alpha,\ell}(\mathbb R^n)$ have several applications in the researches such as partial differential equations [25], and Fourier transform [26]. Moreover, the authors [27] established the boundedness on the two-weighted Herz spaces $\dot{K}_{q}^{\alpha,\ell}(\omega_1,\omega_2)$ with $\omega_1$ and $\omega_2$ belonging to the class of Muckenhoupt weights for the sublinear operators $T$, generated by the Hardy–Littlewood maximal operators. Recently, the Hausdorff operators have been extensively studied on Herz spaces over real field, $p$-adic field, and Heisenberg group (see [20], [28]–[30] and references therein). As a natural development, the boundedness of Hardy–Littlewood maximal operators and Hausdorff operators on two-weighted Herz spaces needs to be discussed.

Motivated by the above, our paper aims to give some sufficient conditions for the boundedness of the operators $\mathcal M^p$ on two-weighted Herz spaces over $p$-adic fields with general Muckenhoupt weights. In particular, we establish necessary and sufficient conditions for the continuity of the operators $\mathcal{H}^p_{\Phi,\Omega}$ and $\mathcal{H}^p_{\Psi,A}$ on two-weighted Herz spaces $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$ with weights of the form $\omega(x) = |x|^{\alpha}_p$ and the case $\ell\in(0,1)$. As a consequence, we obtain the boundedness of the $p$-adic Hardy operators on such spaces.

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 two-weighted Herz spaces, and its atomic decomposition. Our main theorems are given and proved in Section 3.

§ 2. Some notations and definitions

Let $p$ be a prime number. The field of $p$-adic numbers $\mathbb Q_p$ is defined as the completion of the field of rational numbers $\mathbb Q$ with respect to the $p$-adic norm $|\,{\cdot}\,|_p$. This norm is defined as follows: $|0|_p\,{=}\,0$; $|x|_p\,{=}\,p^{-\alpha}$ if $x\,{=}\,p^\alpha (a_0+a_1p+a_2p^2+\cdots)$, where $\alpha\in\mathbb Z$, $a_k=0,1,\dots,p-1$, $a_0\ne 0$, $k=0,1,\dots$ .

Let us denote $\mathbb Q^n_p=\mathbb Q_p\times\dots\times\mathbb Q_p$. Then the $p$-adic norm of $\mathbb Q^n_p$ is defined by

$$ \begin{equation} |x|_p=\max_{1\leqslant j\leqslant n}|x_j|_p, \end{equation} \tag{2.1} $$
for all $x=(x_1,\dots,x_n)$. Let
$$ \begin{equation*} B_k(x_0)=\{x\in\mathbb Q^n_p \colon |x-x_0|_p\leqslant p^k\} \end{equation*} \notag $$
be a ball of radius $p^k$ with center at $x_0\in\mathbb Q^n_p$. Similarly, denote by
$$ \begin{equation*} S_k(x_0)=\{x\in\mathbb Q^n_p \colon |x-x_0|_p=p^k\} \end{equation*} \notag $$
the sphere with center at $x_0$ and radius $p^k$. Denote $B_k=B_k(0), S_k=S_k(0)$ and the characteristic function of the sphere $S_k$ by $\chi_k$.

The Haar measure on $\mathbb Q^n_p$ is normalized by

$$ \begin{equation*} \int_{B_0} dx=1. \end{equation*} \notag $$
Let $|E|$ denote the Haar measure of a measurable subset $E$ of $\mathbb Q^n_p$. It is easy to show that $|B_k(x_0)|=p^{nk}$, $|S_k(x_0)|=p^{nk}(1-p^{-n})$, for any $x_0\in\mathbb Q^n_p$. For $f\in L^1_{\mathrm{loc}}(\mathbb Q^n_p)$, we have
$$ \begin{equation*} \int_{\mathbb Q^n_p}f(t)\, dt=\lim_{k\to \infty}\int_{B_k} f(t)\, dt=\lim_{k\to \infty}\sum_{-\infty<m\leqslant k}\int_{S_m} f(t)\, dt. \end{equation*} \notag $$
For a more complete introduction to $p$-adic analysis, the interested readers may refer to [3], [4], and the references therein.

Let $A$ be an $n\times n$ matrix with entries $a_{ij}\in\mathbb Q_p$. Let us denote

$$ \begin{equation*} Ax=\biggl(\sum_{j=1}^n a_{1j}x_j, \dots, \sum_{j=1}^n a_{nj}x_j \biggr), \end{equation*} \notag $$
for $x=(x_1, \dots, x_n)\in\mathbb Q^n_p$. The norm of $A$ is given by
$$ \begin{equation*} \|A\|_p:=\max_{1\leqslant i\leqslant n} \max_{1\leqslant j\leqslant n}|a_{ij}|_p. \end{equation*} \notag $$
It is obvious that $|Ax|_p\leqslant \|A\|_p|x|_p$ for all $x\in\mathbb Q^n_p$. Moreover, if $A$ is invertible, we also have
$$ \begin{equation} \|A\|_p^{-n}\leqslant |{\det{(A^{-1})}}|_p\leqslant \|A^{-1}\|_p^n. \end{equation} \tag{2.2} $$

Let $\omega$ be a weight function, namely, it is a locally integrable nonnegative measurable function on $\mathbb Q^n_p$. Let $L^q_\omega(\mathbb Q^n_p)$ ($0 < q < \infty$) denote the space of all Haar measurable functions $f$ on $\mathbb Q^n_p$ such that

$$ \begin{equation*} \|f\|_{L^q_\omega(\mathbb Q^n_p)}=\biggl(\int_{\mathbb Q^n_p}|f(x)|^q\omega(x)\, dx\biggr)^{1/q}<\infty. \end{equation*} \notag $$
The space $L^q_{\omega, \mathrm{loc}}(\mathbb Q^n_p)$ is defined as the set of all measurable functions $f$ on $\mathbb Q^n_ p$ such that $\int_{U}|f(x)|^q\omega(x)\, dx<\infty$, for any compact subset $U$ of $\mathbb Q^n_p$. The space $L^q_{\omega,\mathrm{loc}}(\mathbb Q^n_p\setminus\{0\})$ is also defined in a similar way.

Throughout the forthcoming, we also write $a \lesssim b$ to mean that there exists a constant $C>0$, independent of the main parameters, such that $a\leqslant Cb$. The symbol $a\simeq b$, namely, $a$ is equivalent to $b$ (i.e., $C^{-1}a\leqslant b\leqslant Ca$). For any real number $q>0$, denote by $q'$ conjugate real number of $q$. Denote $\omega(E)^{\beta}=\bigl(\int_E{\omega(x)}\, dx\bigr)^{\beta}$, for any $\beta\in\mathbb R$. Note that $\omega(x) = |x|_p^{\alpha}$ for $\alpha >-n$, we then get

$$ \begin{equation} \omega(B_{\gamma}) =\sum_{k\leqslant \gamma} \int_{S_k}p^{k\alpha}\, dx=\sum_{k\leqslant \gamma} p^{k(\alpha+n)}(1-p^{-n})= p^{\gamma(\alpha+n)}\frac{1-p^{-n}}{1-p^{-(\alpha+n)}}. \end{equation} \tag{2.3} $$
Let us now give the definition of two-weighted Herz spaces on $p$-adic field.

Definition 2.1. Let $0<\beta<\infty$, $1\leqslant q<\infty$, $0<\ell<\infty$ and $\omega_1$, $\omega_2$ be two weight functions on $\mathbb{Q}_p^n$. The $p$-adic two-weighted homogeneous Herz space $\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$ is defined as the set of all measurable functions $f\in L^q_{\omega_2,\mathrm{loc}}(\mathbb{Q}_p^n\setminus\{0\})$ such that

$$ \begin{equation*} \|f\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)} =\biggl(\sum_{k=-\infty}^{\infty}\omega_1(B_k)^{\beta \ell/n} \|f\chi_k\|^{\ell}_{L^q_{\omega_2}(\mathbb{Q}_p^n)} \biggr)^{1/\ell}<\infty. \end{equation*} \notag $$
The $p$-adic two-weighted non-homogeneous Herz space $K_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$ is defined as the set of all measurable functions $f\in L^q_{\omega_2,\mathrm{loc}}(\mathbb{Q}_p^n\setminus\{0\})$ such that
$$ \begin{equation*} \|f\|_{{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)} =\biggl(\sum_{k=0}^{\infty}\omega_1(B_k)^{\beta \ell/n} \|f\chi_k\|^{\ell}_{L^q_{\omega_2}(\mathbb{Q}_p^n)} \biggr)^{1/\ell}<\infty. \end{equation*} \notag $$

Next, let us recall the Muckenhoupt weights on $p$-adic fields. On the local fields or homogeneous type spaces see [9], [31] for more details.

Definition 2.2. Let $1 < \ell < \infty$. We say that $\omega \in A_\ell(\mathbb Q^n_p)$ if there exists a constant $C$ such that for all balls $B$,

$$ \begin{equation*} \biggl(\frac{1}{|B|}\int_{B}\omega(x)\, dx\biggr) \biggl(\frac{1}{|B|}\int_{B}\omega(x)^{-1/(\ell-1)}\, dx\biggr)^{\ell-1}\leqslant C. \end{equation*} \notag $$
It is said that $\omega\in A_1(\mathbb Q^n_p)$ if there is a constant $C$ such that for all balls $B$, we have
$$ \begin{equation*} \frac{1}{|B|}\int_{B}\omega(x)\, dx\leqslant C\operatorname*{ess\,inf}_{x\in B}\omega(x). \end{equation*} \notag $$
Let us denote $A_{\infty}(\mathbb Q^n_p) = \bigcup_{1\leqslant \ell<\infty}A_\ell(\mathbb Q^n_p)$.

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

Proposition 2.3. (i) $A_\ell(\mathbb Q^n_p)\subsetneq A_q(\mathbb Q^n_p)$ for $1\leqslant \ell < q < \infty$.

(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 $\omega\in A_{\ell-\varepsilon}(\mathbb Q^n_p)$.

It is easy to check that $\omega(x)=|x|^{\alpha}_p\in A_1(\mathbb Q^n_p)$ if and only if $-n< \alpha\leqslant 0$, and $\omega(x)=|x|^{\alpha}_p\in A_\ell(\mathbb Q^n_p)$ ($1 < \ell< \infty$) if and only if $-n < \alpha < n(\ell-1)$.

Proposition 2.4. Let $\omega\in A_\ell(\mathbb Q^n_p)$ with $\ell\geqslant 1$. Thus, there exists a constant $C>0$ such that

$$ \begin{equation*} \frac{\omega(B)}{\omega(E)}\leqslant C\biggl(\frac{|B|}{|E|}\biggr)^\ell, \end{equation*} \notag $$
for any measurable subset $E$ of a ball $B$.

Theorem 2.5 (see Theorem 3.5 in [9]). Let $1 < \ell < \infty$. Thus, the operator $\mathcal M^p$ is bounded from $L^\ell_\omega(\mathbb Q^n_p)$ to itself if and only if $\omega\in A_\ell(\mathbb Q^n_p)$.

Let us present the notation of central atom and central atom of restrict type.

Definition 2.6. Let $0<\beta<\infty$ and $1\leqslant q<\infty$ and $\omega_1$, $\omega_2$ be two weight functions on $\mathbb{Q}_p^n$. A function $b$ on $\mathbb{Q}_p^n$ is said to be a central $(\beta,q,\omega_1,\omega_2)$-atom (a central $(\beta,q,\omega_1,\omega_2)$-atom of restrict type, respectively) if there exists $k\in\mathbb{Z}$ ($k\in\mathbb N$, respectively) such that

(i) $\operatorname{supp}(b)\subset B_k$,

(ii) $\|b\|_{L^q_{\omega_2}(\mathbb Q^n_p)}\lesssim \omega_1(B_k)^{-\beta/n}$.

The following useful decomposition theorem allows us to obtain that the central atoms are the building atoms of Herz spaces. The reader may refer to [32], [33] for more details.

Theorem 2.7. Let $0\,{<}\,\beta$, $\ell\,{<}\,\infty$, $1\,{\leqslant}\, q\,{<}\,\infty$, $\omega_1\,{\in}\, A_1(\mathbb Q^n_p)$, and $\omega_2$ be a weight function on $\mathbb Q^n_p$. Thus, we have that $f{\kern1pt}{\in}{\kern1pt}\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$ if and only if $f\,{=}\sum_{k=-\infty}^{\infty}\lambda_k b_k$, where $\sum_{k=-\infty}^{\infty}|\lambda_k|^\ell<\infty$, and each $b_k$ is a central $(\beta,q,\omega_1,\omega_2)$-atom with the support in $B_k$. Moreover,

$$ \begin{equation*} \|f\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}\simeq \inf\biggl\{\sum_{k=-\infty}^{\infty}|\lambda_k|^\ell \biggr\}^{1/\ell}, \end{equation*} \notag $$
where the infimum is taken over all decompositions of $f$ as above.

Similarly, we have $f\in K_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$ if and only if $f=\sum_{k=0}^{\infty}\lambda_k b_k$, where $\sum_{k=0}^{\infty}|\lambda_k|^\ell<\infty$, and each $b_k$ is a central $(\beta,q,\omega_1,\omega_2)$-atom of restrict type with the support in $B_k$. Furthermore,

$$ \begin{equation*} \|f\|_{{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}\simeq \inf\biggl\{\sum_{k=0}^{\infty}|\lambda_k|^\ell \biggr\}^{1/\ell}, \end{equation*} \notag $$
where the infimum is taken over all decompositions of $f$ as above.

§ 3. The main results

In the case $\omega_1\equiv 1$ and $\omega_2\equiv \omega$, we denote $E_{q,\omega}^{\beta,\ell}(\mathbb Q^n_p):=\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb Q^n_p)$. As a consequence of [20; Theorem 3.3], the necessary and sufficient conditions for the boundedness of the rough Hausdorff operators $\mathcal{H}^p_{\Phi,\Omega}$ on weighted Herz spaces $E_{q,\omega}^{\beta,\ell}(\mathbb Q^n_p)$ with power weight are established. In more detail, under conditions $\ell, q\in[1,\infty)$, $\beta\in\mathbb R$, $\omega(x)=|x|^{\beta_2}_p$ with $\beta_2\in (-n,\infty)$ and $\Omega\in L^{q'}(S_0)$, the operator $\mathcal{H}^p_{\Phi,\Omega}$ is bounded on $E_{q,\omega}^{\beta,\ell}(\mathbb Q^n_p)$ if and only if

$$ \begin{equation*} [\mathcal E]=\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)}{|x|_p^{1-\beta-(n+\beta_2)/q}}<\infty. \end{equation*} \notag $$
Moreover, the operator norm is equivalent to $[\mathcal E]$. In the case $\ell\in(0,1)$, by using atomic decomposition, we give the necessary and sufficient conditions for the boundedness of the operators $\mathcal{H}^p_{\Phi,\Omega}$ on two-weighted Herz spaces $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$ with weights of the form $\omega(x) = |x|^{\alpha}_p$.

Theorem 3.1. Let $q\in[1,\infty)$, $\beta\in (0,\infty)$, $\omega_1(x)=|x|^{\beta_1}_p$, $\omega_2(x)=|x|^{\beta_2}_p$ with $\beta_1\in(-n,0]$, $\beta_2\in (-n,\infty)$ and $\Omega\in L^{q'}(S_0)$. Let $\ell\in(0,1)$ and $\tau=(\beta_2+n)/q+(\beta_1+n)\beta/n$.

(i) If there exists $n_0\in\mathbb Z^+$ such that

$$ \begin{equation*} \mathcal C_{n_0,\ell,\tau,n}=\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)\max \{|{\log_p|x|_p}|^{(1+n_0^{-1})(1-\ell)/\ell},1\}}{|x|_p^{1-\tau+n}}\, dx< \infty, \end{equation*} \notag $$
then $\mathcal{H}^p_{\Phi,\Omega}$ is a bounded operator from $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$ into itself. Furthermore,
$$ \begin{equation*} \|\mathcal{H}^p_{\Phi,\Omega}\|_{{\dot{K}}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)\to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}\lesssim \|\Omega\|_{L^{q'}(S_0)}\cdot\mathcal C_{\infty,\ell,\tau,n}, \end{equation*} \notag $$
where
$$ \begin{equation*} \mathcal C_{\infty,\ell,\tau,n} =\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)\max\{|{\log_p|x|_p}|^{(1-\ell)/\ell},1\}}{|x|_p^{1-\tau+n}}\, dx. \end{equation*} \notag $$

(ii) Conversely, if $\mathcal{H}^p_{\Phi,\Omega}$ is a bounded operator from $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$ into itself, then

$$ \begin{equation*} \mathcal C_{\tau,n} =(1-p^{-n})^{-1}\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)}{|x|_p^{1-\tau+n}}\, dx<\infty. \end{equation*} \notag $$
Moreover, in this case, we have
$$ \begin{equation*} \biggl|\int_{S_0}\Omega(y)\, dy\biggr|\cdot \mathcal C_{\tau,n}\leqslant \|\mathcal{H}^p_{\Phi,\Omega}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)\to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}. \end{equation*} \notag $$

Proof. (i) Let $f\in \dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$. By Theorem 2.7, we have
$$ \begin{equation*} f=\sum_{k=-\infty}^{\infty}\lambda_kb_k, \end{equation*} \notag $$
where
$$ \begin{equation*} \biggl(\sum_{k=-\infty}^{\infty}|\lambda_k|^\ell \biggr)^{{1}/{\ell}}\lesssim \|f\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}, \end{equation*} \notag $$
and for $k\in\mathbb{Z}, $ $b_k$ is a central $(\beta,q,\omega_1,\omega_2)$-atom such that
$$ \begin{equation*} \operatorname{supp}(b_k)\subset B_k \quad\text{and} \quad \|b_k\|_{L^{q}_{\omega_2}(\mathbb{Q}_p^n)}\lesssim \omega_1(B_k)^{-{\beta}/{n}}. \end{equation*} \notag $$
It is easy to prove that
$$ \begin{equation*} \begin{aligned} \, |\mathcal{H}^p_{\Phi,\Omega}(f)(x)| &\leqslant \sum_{k=-\infty}^{\infty}|\lambda_k|\sum_{\gamma\in\mathbb Z}\int_{S_0} \frac{\Phi(p^{\gamma})}{p^{\gamma}}|\Omega(y)||b_k(p^{\gamma}|x|_p^{-1}y)|\, dy \\ &=:\sum_{k=-\infty}^{\infty}|\lambda_k|\widetilde{\mathcal{H}}^p_{\Phi,\Omega}(b_k)(x), \qquad x\in\mathbb Q^n_p. \end{aligned} \end{equation*} \notag $$
For $k\in\mathbb Z$, let us compose
$$ \begin{equation} \widetilde{\mathcal{H}}^p_{\Phi,\Omega}(b_k)(x)=\sum_{\gamma\in\mathbb{Z}}c_{k,\gamma}(x), \end{equation} \tag{3.1} $$
where
$$ \begin{equation*} \sum_{\gamma\in\mathbb{Z}}c_{k,\gamma}(x)=\sum_{\gamma\in\mathbb Z}\int_{S_0} \frac{\Phi(p^{\gamma})}{p^{\gamma}}|\Omega(y)||b_k(p^{\gamma}|x|_p^{-1}y)|\, dy. \end{equation*} \notag $$
Note that we can show that
$$ \begin{equation} \operatorname{supp}(c_{k,\gamma})\subset B_{k+\gamma}. \end{equation} \tag{3.2} $$
Indeed, for $x\in\mathbb Q^n_p$ and $k,\gamma\in\mathbb Z$ such that $|x|_p>p^{k+\gamma}$, we have
$$ \begin{equation*} |p^{\gamma}|x|_p^{-1}y|_p=p^{-\gamma}|x|_p|y|_p=p^{-\gamma}|x|_p>p^{k}. \end{equation*} \notag $$
Thus, by $\operatorname{supp}(b_k)\subset B_k$, it follows that $b_k(p^{\gamma}|x|_p^{-1}y)=0$. By the Minkowski inequality and the Hölder inequality, one has
$$ \begin{equation*} \begin{aligned} \, \|c_{k,\gamma} \|_{L^q_{\omega_2}(\mathbb{Q}_p^n)}&\leqslant \int_{S_0} \frac{\Phi(p^{\gamma})}{p^{\gamma}}|\Omega(y)|\|b_k(p^{\gamma}|\,{\cdot}\,|_p^{-1}y)\|_{L^q_{\omega_2} (B_{k+\gamma})} \, dy \\ &\leqslant \|\Omega\|_{L^{q'}(S_0)}\frac{\Phi(p^{\gamma})}{p^{\gamma}} \biggl(\int_{S_0}\int_{B_{k+\gamma}} |b_k(p^{\gamma}|x|_p^{-1}y)|^q\omega_2(x)\, dx\, dy\biggr)^{1/q}. \end{aligned} \end{equation*} \notag $$
On the other hand, we also get
$$ \begin{equation*} \begin{aligned} \, &\int_{S_0}\int_{B_{k+\gamma}} |b_k(p^{\gamma}|x|_p^{-1}y)|^q\omega_2(x)\, dx\, dy=\int_{S_0}\sum_{r\leqslant k+\gamma}\int_{S_r}|b_k(p^{\gamma-r}y)|^qp^{r\beta_2}\, dx\, dy \\ &\ \simeq\int_{S_0}\sum_{r\leqslant k+\gamma}|b_k(p^{\gamma-r}y)|^qp^{r(\beta_2+n)}\, dy=\sum_{r\leqslant k+\gamma}\int_{S_{-\gamma+r}}|b_k(t)|^qp^{r(\beta_2+n)}p^{(\gamma-r)n}\, dt \\ &\ =p^{\gamma(\beta_2+n)}\sum_{\sigma\leqslant k}\int_{S_{\sigma}}|b_k(t)|^q p^{\sigma\beta_2}\, dt=p^{\gamma(\beta_2+n)}\|b_k\|^q_{L^{q}_{\omega_2}(B_k)}. \end{aligned} \end{equation*} \notag $$
From this, by $\|b_k\|_{L^q_{\omega_2}(B_k)}\lesssim \omega_1(B_k)^{{-\beta}/{n}}$ and (2.3), it follows immediately that
$$ \begin{equation} \begin{aligned} \, \|c_{k,\gamma} \|_{L^q_{\omega_2}(\mathbb{Q}_p^n)} &\lesssim \|\Omega\|_{L^{q'}(S_0)} \frac{\Phi(p^{\gamma})}{p^{\gamma-\gamma(\beta_2+n)/q}}\|b_k\|_{L^{q}_{\omega_2}(B_k)} \nonumber \\ &\lesssim\|\Omega\|_{L^{q'}(S_0)}\frac{\Phi(p^{\gamma})}{p^{\gamma-\gamma(\beta_2+n)/q}} \biggl(\frac{\omega_1(B_{k+\gamma})}{\omega_1(B_k)}\biggr)^{\beta/n} \omega_1(B_{k+\gamma})^{-\beta/n} \nonumber \\ &:= \|\Omega\|_{L^{q'}(S_0)}\cdot\mu_{\tau,\gamma}\cdot\omega_1(B_{k+\gamma})^{-\beta/n}, \end{aligned} \end{equation} \tag{3.3} $$
where
$$ \begin{equation*} \mu_{\tau,\gamma}=\frac{\Phi(p^{\gamma})}{p^{\gamma(1-\tau)}}. \end{equation*} \notag $$
For any $x\in\mathbb {Q}_p^n\setminus\{0\}$, denote
$$ \begin{equation*} \widetilde{c}_{k,\gamma}(x)= \begin{cases} \dfrac{c_{k,\gamma}(x)}{\mu_{\tau,\gamma}\|\Omega\|_{L^{q'}(S_0)}} &\text{if } c_{k,\gamma}(x)\ne 0, \\ 0 &\text{otherwise}. \end{cases} \end{equation*} \notag $$
Hence, by (3.1), we come to
$$ \begin{equation*} \widetilde{\mathcal{H}}^p_{\Phi,\Omega}(b_k)(x) =\sum_{\gamma\in\mathbb{Z}}\|\Omega\|_{L^{q'}(S_0)}\mu_{\tau,\gamma}\widetilde{c}_{k,\gamma}(x). \end{equation*} \notag $$
By (3.2) and (3.3) and the definition of $\widetilde{c}_{k,\gamma}$, we deduce
$$ \begin{equation*} \|\widetilde{c}_{k,\gamma}\|_{L^q_{\omega_2}(\mathbb{Q}_p^n)}\lesssim \omega_1(B_{k+\gamma})^{-{\beta}/{n}}\quad\text{and}\quad \operatorname{supp}(\widetilde{c}_{k,\gamma})\subset B_{k+\gamma}. \end{equation*} \notag $$
Thus, $\widetilde{c}_{k,\gamma}$ is a central $(\beta,q,\omega_1,\omega_2)$-atom for all $\gamma\in\mathbb{Z}$. By Theorem 2.7, we have
$$ \begin{equation} \|\widetilde{\mathcal{H}}^p_{\Phi,\Omega}(b_k)\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell} (\mathbb{Q}_p^n)}\lesssim \|\Omega\|_{L^{q'}(S_0)}\biggl(\sum_{\gamma\in\mathbb{Z}}|\mu_{\tau,\gamma}|^\ell \biggr)^{1/\ell}. \end{equation} \tag{3.4} $$
Note that by letting $\sigma_0$ such that $(1-\ell)/\ell<\sigma_0<(1+n_0^{-1})(1-\ell)/\ell$ and the Hölder inequality, we get
$$ \begin{equation*} \begin{aligned} \, \biggl(\sum_{\gamma\in\mathbb{Z}}|\mu_{\tau,\gamma}|^\ell \biggr)^{1/\ell}&\lesssim \sum_{\gamma=-\infty}^{-1}|\gamma|^{\sigma_0} \mu_{\tau,\gamma}+\sum_{\gamma=1}^{\infty}\gamma^{\sigma_0} \mu_{\tau,\gamma}+\mu_{\tau,0} \\ &=:K_1+K_2+\mu_{\tau,0}. \end{aligned} \end{equation*} \notag $$
Let us now estimate $K_1$. It is easy to see that for $|x|_p=p^{\gamma}$, we have $|\gamma|=|{\log_p|x|_p}|$. Thus,
$$ \begin{equation*} \begin{aligned} \, K_1&=\sum_{\gamma=-\infty}^{-1}|\gamma|^{\sigma_0} \frac{\Phi(p^{\gamma})}{p^{\gamma(1-\tau+n)}}p^{\gamma n}\simeq \sum_{\gamma=-\infty}^{-1}\int_{S_{\gamma}}|\gamma|^{\sigma_0}\frac{\Phi(|x|_p)}{|x|_p^{1-\tau+n}}\, dx \\ &= \sum_{\gamma=-\infty}^{-1}\int_{S_{\gamma}}\frac{\Phi(|x|_p)|{\log_p|x|_p}|^{\sigma_0}} {|x|_p^{1-\tau+n}}\, dx =\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)|{\log_p|x|_p}|^{\sigma_0} \chi_{B_{-1}}(x)}{|x|_p^{1-\tau+n}}\, dx. \end{aligned} \end{equation*} \notag $$
Similarly, we also obtain
$$ \begin{equation*} K_2=\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)|{\log_p|x|_p}|^{\sigma_0}\chi_{\mathbb Q^n_p\setminus B_{0}}(x)}{|x|_p^{1-\tau+n}}\, dx\quad\text{and}\quad \mu_{\tau,0}\simeq\int_{S_0}\frac{\Phi(|x|_p)}{|x|_p^{1-\tau+n}}\, dx. \end{equation*} \notag $$
Therefore, for all $(1-\ell)/\ell<\sigma_0<(1+n_0^{-1})(1-\ell)/\ell$, one has
$$ \begin{equation*} \biggl(\sum_{\gamma\in\mathbb{Z}}|\mu_{\tau,\gamma}|^\ell \biggr)^{1/\ell}\lesssim \int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)\max\{|{\log_p|x|_p}|^{\sigma_0},1\}}{|x|_p^{1-\tau+n}}\, dx. \end{equation*} \notag $$
By (3.4) and $\mathcal C_{n_0,\ell,\tau,n}<\infty$ and the dominated convergence theorem of Lebesgue, we obtain
$$ \begin{equation} \|\widetilde{\mathcal{H}}^p_{\Phi,\Omega}(b_k)\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell} (\mathbb{Q}_p^n)}\lesssim \|\Omega\|_{L^{q'}(S_0)}\cdot\mathcal C_{\infty,\ell,\tau,n}\quad \text{for all}\quad k\in\mathbb{Z}. \end{equation} \tag{3.5} $$
This implies that
$$ \begin{equation*} \begin{aligned} \, &\|\mathcal{H}^p_{\Phi,\Omega}(f)\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}\leqslant \biggl(\sum_{k=-\infty}^{\infty}|\lambda_k|^\ell\|\widetilde{\mathcal{H}}^p_{\Phi,\Omega}(b_k) \|^{\ell}_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}\biggr)^{1/\ell} \\ &\ \lesssim \|\Omega\|_{L^{q'}(S_0)}\cdot\mathcal C_{\infty,\ell,\tau,n}\cdot \biggl(\sum_{k=-\infty}^{\infty}|\lambda_k|^\ell \biggr)^{1/\ell} \lesssim \|\Omega\|_{L^{q'}(S_0)}\cdot\mathcal C_{\infty,\ell,\tau,n}\cdot \|f\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}. \end{aligned} \end{equation*} \notag $$
Thus, $\mathcal{H}^p_{\Phi,\Omega}$ is bounded from $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$ into itself, and
$$ \begin{equation} \|\mathcal{H}^p_{\Phi,\Omega}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)\to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}\lesssim \|\Omega\|_{L^{q'}(S_0)}\cdot\mathcal C_{\infty,\ell,\tau,n}. \end{equation} \tag{3.6} $$
(ii) Conversely, suppose that $\mathcal{H}^p_{\Phi,\Omega}$ is bounded from $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$ into itself. Then for $\eta\in\mathbb Z^+$, let us choose
$$ \begin{equation*} f_{\eta}(x) = \begin{cases} |x|_p^{-\tau-p^{-\eta}} &\text{if }|x|_p\geqslant 1, \\ 0 &\text{otherwise}. \end{cases} \end{equation*} \notag $$
If $k<0$, then $f_{\eta}\chi_k=0$. Otherwise,
$$ \begin{equation*} \begin{aligned} \, \|f_{\eta}\chi_k\|_{L^{q}_{\omega_2}(\mathbb Q^n_p)}^{q}&=\int_{S_k}|x|_p^{-(\tau +p^{-\eta})q+\beta_2}\, dx = p^{-k(\tau+p^{-\eta})q+k\beta_2}p^{kn}(1-p^{-n}) \\ &=p^{-kq(\tau+p^{-\eta}-(\beta_2+n)/q)}(1-p^{-n}). \end{aligned} \end{equation*} \notag $$
Hence, by (2.3), we have
$$ \begin{equation} \begin{aligned} \, 0<\|f_{\eta}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)} &=\biggl(\sum_{k=0}^{\infty}\omega_1(B_k)^{\beta\ell/n}\|f_{\eta}\chi_k\|^{\ell}_{L^{q}_{\omega_2} (\mathbb Q_p^n)}\biggr)^{1/\ell} \nonumber \\ &= \mathbb D\biggl(\sum_{k=0}^{\infty}p^{-k\ell p^{-\eta}}\biggr)^{1/\ell}= \mathbb D\biggl(1-p^{-\ell p^{-\eta}}\biggr)^{-1/\ell}<\infty, \end{aligned} \end{equation} \tag{3.7} $$
where
$$ \begin{equation*} \mathbb D=(1-p^{-n})^{(\beta/n+1/q)}\bigl(1-p^{-(\beta_1+n)}\bigr)^{-\beta/n}. \end{equation*} \notag $$
Then, for $\eta\in\mathbb Z$, we have
$$ \begin{equation*} \begin{aligned} \, |\mathcal{H}^p_{\Phi,\Omega}(f_\eta)(x)| &=\biggl|\sum_{\gamma\in\mathbb Z} \int_{S_0}\frac{\Phi(p^{-\gamma})}{p^{-\gamma}}\Omega(y)f_{\eta}(p^{-\gamma}|x|_p^{-1}y)\, dy\biggr| \\ &=\biggl|\int_{S_0}\Omega(y)\, dy\biggr|\biggl(\sum_{\gamma\geqslant -\log_p|x|_p} \frac{\Phi(p^{-\gamma})}{p^{-\gamma(1-\tau-p^{-\eta})}}\biggr)|x|_p^{-(\tau+p^{-\eta})}. \\ &\geqslant \biggr|\int_{S_0}\Omega(y)\, dy\biggr|\biggl(\sum_{\gamma\geqslant -\eta} \frac{\Phi(p^{-\gamma})}{p^{-\gamma(1-\tau-p^{-\eta})}}\biggr)|x|_p^{-(\tau+p^{-\eta})} \chi_{\mathbb Q^n_p\setminus B_{\eta-1}}(x). \end{aligned} \end{equation*} \notag $$
From this, by (3.7) above, we write
$$ \begin{equation} \begin{aligned} \, &\|\mathcal{H}^p_{\Phi,\Omega}(f_\eta)\|_{\dot K^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}\geqslant \biggl(\sum_{k=\eta}^{\infty}\omega_1(B_k)^{\beta\ell/n} \|\mathcal{H}^p_{\Phi,\Omega}(f_\eta)\chi_k\|_{L^q_{\omega_2}(\mathbb Q^n_p)}^{\ell}\biggr)^{1/\ell} \nonumber \\ &\geqslant \biggl|\int_{S_0}\Omega(y)\, dy\biggr|\biggl(\sum_{\gamma\geqslant -\eta} \frac{\Phi(p^{-\gamma})}{p^{-\gamma(1-\tau-p^{-\eta})}}\biggr) \biggl(\sum_{k=\eta}^{\infty}\omega_1(B_k)^{\beta\ell/n}\bigl\||x|_p^{-(\tau+p^{-\eta})}\chi_k \bigr\|_{L^q_{\omega_2}(\mathbb Q^n_p)}^{\ell}\biggr)^{1/\ell} \nonumber \\ &=\biggl|\int_{S_0}\Omega(y)\, dy \biggr|\biggl(\sum_{\gamma\geqslant -\eta} \frac{\Phi(p^{-\gamma})}{p^{-\gamma(1-\tau-p^{-\eta})}}\biggr)\mathbb D\cdot p^{-\eta p^{-\eta}}\biggl(1-p^{-\ell p^{-\eta}}\biggr)^{-1/\ell} \nonumber \\ &=\biggl|\int_{S_0}\Omega(y)\, dy \biggr|\biggl(\sum_{\gamma\leqslant \eta} \frac{\Phi(p^{\gamma})}{p^{\gamma(1-\tau-p^{-\eta})}}\biggr)p^{-\eta p^{-\eta}} \|f_{\eta}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}. \end{aligned} \end{equation} \tag{3.8} $$
It is remarkable to see that
$$ \begin{equation*} \begin{aligned} \, \sum_{\gamma\leqslant\eta}\frac{\Phi(p^{\gamma})}{p^{\gamma(1-\tau-p^{-\eta})}} &=(1-p^{-n})^{-1}\sum_{\gamma\leqslant\eta}\int_{S_\gamma}\frac{\Phi(|x|_p)}{|x|_p^{1-\tau-p^{-\eta}+n}}\, dx \\ &=(1-p^{-n})^{-1}\int_{B_\eta}\frac{\Phi(|x|_p)(|x|_p)^{p^{-\eta}}}{|x|_p^{1-\tau+n}}\, dx. \end{aligned} \end{equation*} \notag $$
Hence, by (3.8), we infer
$$ \begin{equation*} \begin{aligned} \, &\|\mathcal{H}^p_{\Phi,\Omega}\|_{{\dot{K}}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)\to{\dot{K}}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)} \\ &\qquad\geqslant (1-p^{-n})^{-1}\biggl|\int_{S_0}\Omega(y)\, dy \biggr| \cdot\biggl(\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)(|x|_p)^{p^{-\eta}}p^{-\eta p^{-\eta}}}{|x|_p^{1-\tau+n}} \chi_{B_\eta}(x)\, dx\biggr). \end{aligned} \end{equation*} \notag $$
We observe that
$$ \begin{equation*} (|x|_p)^{p^{-\eta}}p^{-\eta p^{-\eta}}\leqslant 1\quad \text{for all}\quad x\in B_{\eta}, \end{equation*} \notag $$
and
$$ \begin{equation*} \lim_{\eta\to \infty}(|x|_p)^{p^{-\eta}}p^{-\eta p^{-\eta}} =1\quad\text{for all}\quad x\neq 0. \end{equation*} \notag $$
Thus, by the dominated convergence theorem of Lebesgue, we immediately obtain
$$ \begin{equation*} \big\|\mathcal{H}^p_{\Phi,\Omega}\big\|_{{\dot{K}}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p) \to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}\geqslant \biggl|\int_{S_0}\Omega(y)\, dy \biggr|\cdot\mathcal C_{\tau,n}. \end{equation*} \notag $$
The proof of part (ii) is finished. The theorem is completely proved.

By Theorem 3.1, we obtain the boundedness of the $p$-adic Hardy operator on two-weighted Herz spaces. Namely, the following is true.

Corollary 3.2. Let $q\in[1,\infty)$, $\beta\in (0,\infty)$, $\omega_1(x)=|x|^{\beta_1}_p$, $\omega_2(x)=|x|^{\beta_2}_p$ such that $\beta_1\in(-n,0]$, $\beta_2\in (-n,\infty)$. Let $\tau=(\beta_2+n)/q+(\beta_1+n)\beta/n>2n$ and $\ell\in (0,1)$. Then we have that $\mathcal{H}^p$ is bounded on $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$.

Proof. For $\zeta>0$, by taking $\Phi(y)= |y|_p^{n-1}\chi_{\{|y|_p\leqslant 1\}}$ and (2.3), we have
$$ \begin{equation} \begin{aligned} \, \int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)\max\{|{\log_p|x|_p}|^{\zeta},1\}}{|x|_p^{1-\tau+n}}\, dx &=\int_{B_0}\frac{\max\{|{\log_p|x|_p}|^{\zeta},1\}}{|x|_p^{2n-\tau}}\, dx \nonumber \\ &=\sum_{k\leqslant 0} \frac{\max\{|k|^{\zeta},1\}}{p^{k(2n-\tau)}}=1+\sum_{k\geqslant 1} k^{\zeta}p^{k(2n-\tau)}. \end{aligned} \end{equation} \tag{3.9} $$
By the condition $\tau>2n$, we get
$$ \begin{equation*} \lim_{k\to \infty}\frac{(k+1)^{\zeta}p^{(k+1)(2n-\tau)}}{(k)^{\zeta}p^{k(2n-\tau)}} =p^{2n-\tau}<1. \end{equation*} \notag $$
As a consequence, by the D’Alembert criterion for convergence of series, we obtain
$$ \begin{equation*} 1+\sum_{k\geqslant 1}k^{\zeta}p^{k(2n-\tau)}<\infty. \end{equation*} \notag $$
Hence, by (3.9) and Theorem 3.1, $\mathcal{H}^p$ is bounded on $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$. Moreover,
$$ \begin{equation*} \|\mathcal{H}^p\|_{{\dot{K}}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)\to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}\lesssim 1+\sum_{k\geqslant 1}k^{(1-\ell)/\ell}p^{k(2n-\tau)}. \end{equation*} \notag $$
Corollary 3.2 is proved.

Let us now discuss the boundedness of the matrix Hausdorff operators on two-weighted Herz spaces associated to the case of matrices having the important property as follows: there exists $\xi_A\in\mathbb N$ such that

$$ \begin{equation} \|A(y)\|_p\cdot\|A^{-1}(y)\|_p \leqslant p^{\xi_{A}}, \end{equation} \tag{3.10} $$
for almost everywhere $y \in \mathbb Q^n_p$. Thus, by the property of invertible matrices, it is easy to show that
$$ \begin{equation} |A(y)x|^\sigma_p \geqslant p^{\xi_A\sigma}\|A^{-1}(y)\|^{-\sigma}_p|x|^{\sigma}_p \quad\text{for all} \quad \sigma<0,\ \ x\in \mathbb Q^n_p\setminus\{0\}. \end{equation} \tag{3.11} $$

Theorem 3.3. Let $q\in[1,\infty)$, $\beta\in (0,\infty)$, $\omega_1(x)=|x|^{\beta_1}_p$, $\omega_2(x)=|x|^{\beta_2}_p$ for $\beta_1\in(-n,0]$, $\beta_2\in[0,\infty)$, $\tau=(\beta_2+n)/q+(\beta_1+n)\beta/n$ . Assume that $\ell\in (0,1)$, $\sigma>{(1-\ell)}/{\ell}$ and

$$ \begin{equation*} \mathcal K_{\sigma}=\int_{\mathbb Q^n_p}{\Psi(y)}\|A^{-1}(y)\|_p^{\tau} \max\{|{\log_p\|A^{-1}(y)\|_p}|^\sigma,1\} \, dy< \infty. \end{equation*} \notag $$
Then we have that $\mathcal{H}^p_{\Psi,A}$ is bounded on $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$. Moreover, when (3.10) holds, we have
$$ \begin{equation*} \mathcal K_{0}\leqslant \|\mathcal{H}^p_{\Psi,A}\|_{{\dot{K}}^{\beta,\ell}_{q,\omega_1,\omega_2} (\mathbb Q^n_p)\to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}\lesssim \mathcal K_{(1-\ell)/\ell}, \end{equation*} \notag $$
where
$$ \begin{equation*} \begin{aligned} \, \mathcal K_{0}&= p^{-\xi_A.\tau}\int_{\mathbb Q^n_p}{\Psi(y)}\|A^{-1}(y)\|_p^{\tau}\, dy, \\ \mathcal K_{(1-\ell)/\ell}&=\int_{\mathbb Q^n_p}{\Psi(y)}\|A^{-1}(y)\|_p^{\tau}\max\{|{\log_p\|A^{-1}(y)}\|_p |^{(1-\ell)/\ell},1\} \, dy. \end{aligned} \end{equation*} \notag $$

Proof. Suppose that $f\in \dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$. By Theorem 2.7, we have
$$ \begin{equation*} f=\sum_{k=-\infty}^{\infty}\lambda_kb_k, \end{equation*} \notag $$
where $\bigl(\sum_{k=-\infty}^{\infty}|\lambda_k|^\ell \bigr)^{1/\ell}{\lesssim}\, \|f\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}$, and for each $k\,{\in}\,\mathbb{Z},b_k$ is the $(\beta,q,\omega_1,\omega_2)$- atom such that
$$ \begin{equation*} \operatorname{supp}(b_k)\subset B_k \quad\text{and} \quad \|b_k\|_{L^{q}_{\omega_2}(\mathbb{Q}_p^n)}\lesssim \omega_1(B_k)^{-\beta/n}. \end{equation*} \notag $$
We have
$$ \begin{equation*} \begin{aligned} \, |\mathcal{H}^p_{\Psi,A}(f)(x)| &\leqslant \sum_{k=-\infty}^{\infty}|\lambda_k|\int_{\mathbb Q^n_p} \Psi(y)|b_k(A(y)x)|\, dy \\ &=:\sum_{k=-\infty}^{\infty}|\lambda_k|\widetilde{\mathcal{H}}^p_{\Psi,A}(b_k)(x). \end{aligned} \end{equation*} \notag $$
Similar argument as in the proof of Theorem 3.1, it suffices to prove that
$$ \begin{equation} \begin{aligned} \, \|\widetilde{\mathcal{H}}^p_{\Psi,A}(b_k)\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell} (\mathbb{Q}_p^n)}\lesssim 1\quad\text{for all}\quad k\in\mathbb{Z}. \end{aligned} \end{equation} \tag{3.12} $$
It is useful to see that
$$ \begin{equation} \widetilde{\mathcal{H}}^p_{\Psi,A}(b_k)(x)=\sum_{j\in\mathbb{Z}}\int_{\|A^{-1}(y)\|_p=p^j} \Psi(y)|b_k(A(y)x)|\, dy=:\sum_{j\in\mathbb{Z}}d_{k,j}(x). \end{equation} \tag{3.13} $$
From $\operatorname{supp}(b_k)\subset B_k$ and $\|A^{-1}(y)\|_p=p^{j}$, we have
$$ \begin{equation*} |A(y)x|_p\geqslant \|A^{-1}(y)\|_p^{-1}|x|_p\geqslant p^{-j}\cdot p^{k+j+1}=p^{k+1}, \end{equation*} \notag $$
for all $x\in\mathbb Q_p^n\setminus B_{k+j}$, which gives $d_{k,j}(x)=0$. Thus, we infer
$$ \begin{equation} \operatorname{supp}(d_{k,j})\subset B_{k+j}. \end{equation} \tag{3.14} $$
It follows from the Minkowski inequality that
$$ \begin{equation*} \|d_{k,j} \|_{L^q_{\omega_2}(\mathbb{Q}_p^n)}\leqslant \int_{\|A^{-1}(y)\|_p=p^j} {\Psi(y)}\|b_k(A(y)\,{\cdot}\,)\|_{L^q_{\omega_2}(\mathbb{Q}_p^n)} \, dy. \end{equation*} \notag $$
Using the change of variables formula, by $\beta_2\geqslant 0$ and $\|b_k\|_{L^q_{\omega_2}(\mathbb Q^n_p)}\lesssim \omega_1(B_k)^{-\beta/n}$, it implies that
$$ \begin{equation*} \begin{aligned} \, \|b_k(A(y)\,{\cdot}\,)\|_{L^q_{\omega_2}(\mathbb{Q}_p^n)}&=\biggl(\int_{\mathbb Q^n_p}|b_k(z)|^q|A^{-1}(y)z|^{\beta_2}_p|{\det A^{-1}(y)}|_p\, dz\biggr)^{1/q} \\ &\leqslant \|A^{-1}(y)\|_p^{\beta_2/q}|{\det A^{-1}(y)}|_p^{1/q}\|b_k\|_{L^q_{\omega_2}(\mathbb Q^n_p)} \\ &\lesssim\|A^{-1}(y)\|_p^{\beta_2/q}|{\det A^{-1}(y)}|_p^{1/q}\omega_1(B_k)^{-\beta/n}. \end{aligned} \end{equation*} \notag $$
Then, by (2.3), we deduce
$$ \begin{equation} \begin{aligned} \, &\|d_{k,j} \|_{L^q_{\omega_2}(\mathbb{Q}_p^n)} \nonumber \\ &\ \leqslant \biggl(\int_{\|A^{-1}(y)\|_p=p^j} \Psi(y)\|A^{-1}(y)\|_p^{\beta_2/q}|{\det A^{-1}(y)}|_p^{1/q} \, dy\biggr) \nonumber \\ &\ \qquad\times\biggl(\frac{\omega_1(B_{k+j})}{\omega_1(B_k)}\biggr)^{\beta/n} \omega_1(B_{k+j})^{-\beta/n} \nonumber \\ &\ =\biggl(\int_{\|A^{-1}(y)\|_p=p^j} \Psi(y)\|A^{-1}(y)\|_p^{\beta_2/q+(\beta_1+n)\beta/n}|{\det A^{-1}(y)}|_p^{1/q} \, dy\biggr)\omega_1(B_{k+j})^{-\beta/n} \nonumber \\ &\ \lesssim\biggl(\int_{\|A^{-1}(y)\|_p=p^j} \Psi(y)\|A^{-1}(y)\|_p^{\tau}\, dy\biggr)\omega_1(B_{k+j})^{-\beta/n}=:\nu_{\tau,j}\cdot\omega_1(B_{k+j})^{-\beta/n}. \end{aligned} \end{equation} \tag{3.15} $$
Denote
$$ \begin{equation*} \widetilde{d}_{k,j}= \begin{cases} \dfrac{d_{k,j}}{\nu_{\tau, j}} &\text{if }\nu_{\tau, j}\ne 0, \\ 0 &\text{otherwise}. \end{cases} \end{equation*} \notag $$
By (3.13), we then have
$$ \begin{equation*} \widetilde{\mathcal{H}}^p_{\Psi,A}(b_k)=\sum_{j\in\mathbb{Z}}\nu_{\tau, j}\widetilde{d}_{k,j}. \end{equation*} \notag $$
Note that by (3.14), (3.15), and the definition of $\widetilde{d}_{k,j}$, it is easy to see that $\widetilde{d}_{k,j}$ is a $(\beta,q,\omega_1,\omega_2)$-atom. Thus, Theorem 2.7 implies that
$$ \begin{equation*} \|\widetilde{\mathcal{H}}^p_{\Psi,A}(b_k)\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell} (\mathbb{Q}_p^n)}\lesssim \biggl(\sum_{j\in\mathbb{Z}}|\nu_{\tau,j}|^\ell \biggr)^{1/\ell}. \end{equation*} \notag $$
By $\ell\in(0,1)$ and $\sigma>(1-\ell)/{\ell}$, it follows from the Hölder inequality that
$$ \begin{equation*} \begin{aligned} \, \biggl(\sum_{j\in\mathbb{Z}}|\nu_{\tau,j}|^\ell \biggr)^{{1}/{\ell}}&\lesssim \sum_{j\in\mathbb Z\setminus\{0\}}|j|^\sigma \nu_{\tau,j}+\nu_{\tau,0} \\ &\lesssim \sum_{j\in\mathbb Z\setminus\{0\}}\int_{\|A^{-1}(y)\|_p= p^j}{\Psi(y)}\|A^{-1}(y)\|_p^{\tau}\bigl|\log_p\|A^{-1}(y)\|_p \bigr|^\sigma \, dy \\ &\qquad+ \int_{\|A^{-1}(y)\|_p= 1}{\Psi(y)}\|A^{-1}(y)\|_p^{\tau} \, dy = \mathcal K_{\sigma}. \end{aligned} \end{equation*} \notag $$
This shows that the inequality (3.12) holds, and hence we have that $\mathcal{H}^p_{\Psi,A}$ is bounded on $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$. By the dominated convergence theorem of Lebesgue, we obtain
$$ \begin{equation} \|\mathcal{H}^p_{\Psi,A}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)\to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}\lesssim \mathcal K_{(1-\ell)/\ell}. \end{equation} \tag{3.16} $$
To end the proof, it remains to show that $\|\mathcal{H}^p_{\Psi,A}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)\to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)} \geqslant {\mathcal K}_0$. Let us take $g_\eta$ as follows
$$ \begin{equation*} g_{\eta}(x) = \begin{cases} 0 &\text{if }|x|_p\leqslant p^{-\xi_{A}-1}, \\ |x|_p^{-\tau-p^{-\eta}} &\text{otherwise}. \end{cases} \end{equation*} \notag $$
Similarly to (3.7) above, we also have
$$ \begin{equation} \begin{aligned} \, 0 &<\|g_{\eta}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}=\biggl(\sum_{k=-\xi_A}^{\infty}\omega_1(B_k)^{\beta\ell/n} \|g_{\eta}\chi_k\|^{\ell}_{L^{q}_{\omega_2}(\mathbb Q_p^n)}\biggr)^{1/\ell} \nonumber \\ &= \mathbb D\biggl(\sum_{k=-\xi_A}^{\infty}p^{-k\ell p^{-\eta}}\biggr)^{1/\ell}= \mathbb D\cdot p^{\xi_Ap^{-\eta}}\bigl(1-p^{-\ell p^{-\eta}}\bigr)^{-1/\ell}<\infty. \end{aligned} \end{equation} \tag{3.17} $$
Here we recall again $\mathbb D=(1-p^{-n})^{(\beta/n+1/q)}(1-p^{-(\beta_1+n)})^{-\beta/n}$ as in the proof of Theorem 3.1. Note that by the inequality (3.10), for $x\in\mathbb Q^n_p\setminus B_{\eta-1}$ and $y\in\mathbb Q^n_p$ such that $\|A(y)\|_p\geqslant p^{-\eta}$, we get
$$ \begin{equation*} |A_i(y)x|_p\geqslant \|A_i^{-1}(y)\|^{-1}_p\,|x|_p =\frac{\|A_i(y)\|_p\, |x|_p}{\|A_i^{-1}(y)\|_p\,\|A_i(y)\|_p}\geqslant p^{-\xi_{A}}. \end{equation*} \notag $$
From this, one has
$$ \begin{equation*} V_\eta :=\bigl\{y\in\mathbb Q^n_p\colon \|A_i(y)\|_p\geqslant p^{-\eta}\bigr\}\subset \bigl\{y\in\mathbb Q_p^n\colon |A_i(y)x|_p\geqslant p^{-\xi_{A}}\bigr\}:=W_x \end{equation*} \notag $$
for all $x\in\mathbb Q^n_p\setminus B_{\eta-1}$. Thus, we have
$$ \begin{equation*} \mathcal H^p_{\Psi,A}(g_\eta)(x)\geqslant \int_{W_{x}} \Psi(y)|A(y)x|_p^{-\tau-p^{-\eta}} \, dy \geqslant \int_{V_\eta} \Psi(y)|A(y)x|_p^{-\tau -p^{-\eta}}\, dy \end{equation*} \notag $$
for all $x\in\mathbb Q_p^n\setminus B_{\eta-1}$. Combining this with (3.11), we come to
$$ \begin{equation} \mathcal H^p_{\Psi,A}(g_\eta)(x) \geqslant p^{-\xi_A(\tau+p^{-\eta})} \biggl(\int_{V_\eta}\Psi(y)\|A^{-1}(y)\|_p^{\tau+p^{-r}}\, dy\biggr) |x|_p^{-\tau-p^{-\eta}}\chi_{\mathbb Q^n_p\setminus B_{\eta-1}}(x). \end{equation} \tag{3.18} $$
It is easily seen that
$$ \begin{equation*} \begin{aligned} \, &\bigl\|\,|\,{\cdot}\,|_p^{-\tau-p^{-\eta}}\chi_{\mathbb Q^n_p\setminus B_{\eta-1}}\bigr\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}= \mathbb D\biggl(\sum_{k=\eta}^{\infty}p^{-k\ell p^{-\eta}}\biggr)^{1/\ell} \\ &\qquad= \mathbb D\cdot p^{-\eta p^{-\eta}}\bigl(1-p^{-\ell p^{-\eta}}\bigr)^{-1/\ell}= p^{-\eta p^{-\eta}-\xi_Ap^{-\eta}} \|g_{\eta}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}. \end{aligned} \end{equation*} \notag $$
Then, by (3.18), we infer
$$ \begin{equation*} \begin{aligned} \, &\|\mathcal{H}^p_{\Psi,A}(g_\eta)\|_{\dot K^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}\geqslant \biggl(\sum_{k=\eta}^{\infty}\omega_1(B_k)^{\beta\ell/n} \|\mathcal{H}^p_{\Psi,A}(g_\eta)\chi_k\|_{L^q_{\omega_2}(\mathbb Q^n_p)}^{\ell}\biggr)^{1/\ell} \\ &\qquad\geqslant p^{-\xi_A\tau-2\xi_Ap^{-\eta}-\eta p^{-\eta}} \biggl(\int_{V_\eta} \Psi(y) \|A^{-1}(y)\|_p^{\tau+p^{-\eta}}\, dy\biggr) \|g_{\eta}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}, \end{aligned} \end{equation*} \notag $$
which implies that
$$ \begin{equation} \begin{aligned} \, &\|\mathcal{H}^p_{\Psi,A}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p) \to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)} \nonumber \\ &\qquad\geqslant p^{-\xi_A\tau}\biggl(\int_{\mathbb Q^n_p} \Psi(y) \|A^{-1}(y)\|_p^{\tau+p^{-\eta}}\chi_{V_\eta}(y)p^{-2\xi_Ap^{-\eta}-\eta p^{-\eta}}\, dy\biggr). \end{aligned} \end{equation} \tag{3.19} $$
For $\eta$ sufficiently large and $y\in V_\eta$, by (3.10), we have
$$ \begin{equation*} \begin{aligned} \, p^{-2\xi_Ap^{-\eta}-\eta p^{-\eta}}{\|A^{-1}(y)\|_p}^{p^{-\eta}}&= p^{-2\xi_Ap^{-\eta}}\bigl(p^{-\eta}{\|A^{-1}(y)\|_p}\bigr)^{p^{-\eta}} \\ &\leqslant p^{-2\xi_Ap^{-\eta}}\bigl(\|A(y)\|_p {\|A^{-1}(y)\|_p}\bigr)^{p^{-\eta}} \leqslant p^{-\xi_{A}p^{-\eta}}\lesssim 1. \end{aligned} \end{equation*} \notag $$
Note that
$$ \begin{equation*} \lim_{\eta\to \infty}p^{-2\xi_Ap^{-\eta}-\eta p^{-\eta}} =1. \end{equation*} \notag $$
Therefore, by the dominated convergence theorem of Lebesgue again, we immediately obtain
$$ \begin{equation*} \|\mathcal{H}^p_{\Psi,A}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p) \to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}\geqslant \mathcal K_{0}. \end{equation*} \notag $$
Thus, the proof of Theorem 3.3 is finished.

Next, by estimating as in Theorem 1 of the paper [27], we give the boundedness of the Hardy–Littlewood maximal functions on two-weighted Herz spaces $K_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$ with $\omega_1$ and $\omega_2$ belonging to the class of Muckenhoupt weights.

Theorem 3.4. Let $\omega_1\in A_{q_{\omega_1}}(\mathbb Q^n_p)$, $\omega_2\in A_{q_{\omega_2}}(\mathbb Q^n_p)$, $\ell\in(0,1]$, $q\in(1,\infty)$, $\beta\in (0,\infty)$. Assume that $\omega_1$, $\omega_2$ such that one of the following cases holds true:

$$ \begin{equation} \omega_1=\omega_2,\quad 1\leqslant q_{\omega_1}\leqslant q\quad\textit{and}\quad -n\frac{q_{\omega_1}}{q}<\beta q_{\omega_1}< n\biggl(1-\frac{q_{\omega_1}}{q}\biggr); \end{equation} \tag{3.20} $$
$$ \begin{equation} 1\leqslant q_{\omega_1}<\infty,\quad 1\leqslant q_{\omega_2}\leqslant q\quad\textit{and}\quad 0<\beta q_{\omega_1}< n\biggl(1-\frac{q_{\omega_2}}{q}\biggr). \end{equation} \tag{3.21} $$
Then $\mathcal{M}^p$ is bounded on $K^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$.

The proof is left to the reader. Note that the hypotheses of Theorem 3.4 above require that the two weights have a “close” relationship. If the two weights are independent, we establish the boundedness of Hardy–Littlewood maximal functions on two-weighted Herz spaces $K_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$ through its atomic decomposition. On the real field, for the boundedness of the Hardy–Littlewood maximal functions on block spaces with variable exponent, see [34].

Theorem 3.5. Let $q\in (1,\infty)$, $\beta\in (0,\infty)$, $\omega_1\in A_1(\mathbb Q^n_p)$, $\omega_2(x)=|x|^{\beta_2}_p$ with $\beta_2\in (-n, n(q-1))$. If $\widetilde{\tau}=\beta+(\beta_2+n)/q<n$ and $\ell\in (0,1]$, then $\mathcal{M}^p$ is bounded on $K^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$.

Proof. Let $f\in {K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$. By Theorem 2.7, one has
$$ \begin{equation*} f=\sum_{k=0}^{\infty}\lambda_kb_k\quad\text{and}\quad\biggl(\sum_{k=0}^{\infty}|\lambda_k|^\ell \biggr)^{1/\ell}\lesssim \|f\|_{{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}. \end{equation*} \notag $$
Here, for any $k\in\mathbb{N}$, $b_k$ is a central $(\beta,q,\omega_1,\omega_2)$-atom of restrict type such that $\operatorname{supp}(b_k)$ in $B_k$ and
$$ \begin{equation*} \|b_k\|_{L^{q}_{\omega_2}(\mathbb{Q}_p^n)}\lesssim \omega_1(B_k)^{-\beta/n}. \end{equation*} \notag $$
Thus, by the definition of Hardy–Littlewood maximal operator, we have
$$ \begin{equation*} \mathcal{M}^p(f)(x) \leqslant\sum_{k=0}^{\infty}|\lambda_k|\mathcal{M}^p(b_k)(x),\qquad x\in\mathbb Q_p^n. \end{equation*} \notag $$
To complete the proof, it suffices to prove that
$$ \begin{equation} \|\mathcal{M}^p(b_k)\|_{{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}\lesssim 1 \quad\text{for all}\quad k\in\mathbb{N}. \end{equation} \tag{3.22} $$
By $\mathbb Q^n_p= B_k\cup \bigl(\bigcup_{\gamma=1}^{\infty}S_{k+\gamma}\bigr)$, we see that
$$ \begin{equation} \mathcal{M}^p(b_k)=\chi_{B_k}\mathcal M^p(b_k)+\sum_{\gamma=1}^{\infty}\chi_{{k+\gamma}}\mathcal M^p(b_k):=\sum_{\gamma=0}^{\infty}e_{k,\gamma}. \end{equation} \tag{3.23} $$
Consequently, we deduce
$$ \begin{equation} \operatorname{supp}(e_{k,\gamma})\subset B_{k+\gamma}. \end{equation} \tag{3.24} $$
Case 1: $\gamma=0$. Note that $\omega_2\in A_q(\mathbb Q^n_p)$. Combining this with Theorem 2.5, we have
$$ \begin{equation} \begin{aligned} \, \|e_{k,0}\|_{L^q_{\omega_2}(\mathbb Q^n_p)} &\leqslant \|\mathcal M^p(b_k)\|_{L^q_{\omega_2}(\mathbb Q^n_p)} \lesssim \|b_k\|_{L^q_{\omega_2}(\mathbb Q^n_p)} \nonumber \\ &\lesssim \omega_1(B_k)^{-\beta/n} =:\vartheta_{\widetilde{\tau},0} \omega_1(B_{k+\gamma})^{-\beta/n}. \end{aligned} \end{equation} \tag{3.25} $$
Case 2: $\gamma\in\mathbb Z^+$. Note that if we let $x\in S_{k+\gamma}$ and $\theta\in\mathbb Z$ such that $p^k(p^{\gamma}-1)>p^{\theta}$, then $B_\theta(x)\cap B_k=\varnothing$. Thus, by $\operatorname{supp}(b_k)\subset B_k$, we have
$$ \begin{equation*} \begin{aligned} \, e_{k,\gamma}(x)&=\chi_{k+\gamma}(x)\mathcal M^p(b_k)(x)=\chi_{k+\gamma}(x)\sup_{\theta\in\mathbb Z} \frac{1}{p^{n\theta}}\int_{B_\theta(x)\cap B_k}|b_k(y)|\, dy \\ &=\chi_{k+\gamma}(x)\sup_{\theta\in\mathbb Z\colon p^k(p^\gamma-1)\leqslant p^\theta} \frac{1}{p^{n\theta}} \int_{B_\theta(x)\cap B_k}|b_k(y)|\, dy \\ &\leqslant \chi_{k+\gamma}(x)\frac{1}{p^{kn}(p^\gamma-1)^n}\int_{B_k}|b_k(y)|\, dy. \end{aligned} \end{equation*} \notag $$
On the other hand, by $\beta_2<n(q-1)$, one has $-q'\beta_2/q>-n$. From these, by (2.3) and $\|b_k\|_{L^q_{\omega_2}(B_k)}\lesssim \omega_1(B_k)^{-\beta/n}$, it follows from the Hölder inequality that
$$ \begin{equation*} \begin{aligned} \, \|e_{k,\gamma}\|_{L^q_{\omega_2}(\mathbb Q_p^n)}&\leqslant \omega_2(B_{k+\gamma})^{1/q} \frac{1}{p^{kn}(p^\gamma-1)^n}\, \|b_k\|_{L^q_{\omega_2}(B_k)} \biggl(\int_{B_k}|x|_p^{-q'\beta_2/q}\, dx\biggr)^{1/q'} \\ &\lesssim p^{(k+\gamma)(\beta_2+n)/q} \frac{1}{p^{kn}(p^\gamma-1)^n} \, \omega_1(B_k)^{-\beta/n}p^{k(-\beta_2/q+n/q')} \\ &=\frac{p^{\gamma(\beta_2+n)/q}}{(p^\gamma-1)^n} \biggl(\frac{\omega_1(B_{k+\gamma})}{\omega_1(B_k)}\biggr)^{\beta/n} \omega_1(B_{k+\gamma})^{-\beta/n}. \end{aligned} \end{equation*} \notag $$
By the condition $\omega_1\in A_1(\mathbb Q^n_p)$ and Proposition 2.4, one has
$$ \begin{equation*} \biggl(\frac{\omega_1(B_{k+\gamma})}{\omega_1(B_k)}\biggr)^{\beta/n}\lesssim \biggl(\frac{|B_{k+\gamma}|}{|B_k|}\biggr)^{\beta/n}= p^{\gamma\beta}, \end{equation*} \notag $$
which implies
$$ \begin{equation} \|e_{k,\gamma}\|_{L^q_{\omega_2}(\mathbb Q_p^n)}\lesssim \frac{p^{\gamma\widetilde{\tau}}}{(p^\gamma-1)^n}\, \omega_1(B_{k+\gamma})^{-\beta/n} =:\vartheta_{\widetilde{\tau},\gamma}\omega_1(B_{k+\gamma})^{-\beta/n}. \end{equation} \tag{3.26} $$
Set
$$ \begin{equation*} \widetilde{e}_{k,\gamma}=\frac{e^{k,\gamma}}{\vartheta_{\widetilde{\tau},\gamma}}\quad \text{for all}\quad \gamma\in\mathbb N. \end{equation*} \notag $$
Then by (3.23), we immediately have
$$ \begin{equation*} \mathcal M^p(b_k)=\sum_{\gamma=0}^{\infty}\vartheta_{\widetilde{\tau},\gamma} \widetilde{e}_{k,\gamma}. \end{equation*} \notag $$
On the other hand, by (3.24)(3.26), it is clear to see that
$$ \begin{equation*} \|\widetilde{e}_{k,\gamma}\|_{L^q_{\omega_2}(\mathbb{Q}_p^n)}\lesssim \omega_1(B_{k+\gamma})^{-\beta/n}\quad\text{and}\quad \operatorname{supp}(\widetilde{e}_{k,\gamma})\subset B_{k+\gamma}. \end{equation*} \notag $$
From this, for all $\gamma\in\mathbb{N}$, ${\widetilde e}_{k,\gamma}$ is a central $(\beta,q,\omega_1,\omega_2)$-atom of restrict type. Consequently, by Theorem 2.7, we get
$$ \begin{equation} \|\mathcal{M}^p(b_k)\|_{K_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)} \lesssim\biggl(\sum_{\gamma=0}^{\infty}|\vartheta_{\widetilde{\tau},\gamma}|^\ell \biggr)^{1/\ell}. \end{equation} \tag{3.27} $$
For $\ell=1$, by the condition $\widetilde{\tau}<n$, we calculate
$$ \begin{equation*} \lim_{\gamma\to \infty} \frac{p^{(\gamma+1)\widetilde{\tau}}(p^{\gamma+1}-1)^{-n}}{p^{\gamma\widetilde{\tau}} (p^\gamma-1)^{-n}} =p^{\widetilde{\tau}-n}<1. \end{equation*} \notag $$
Thus, using the D’Alembert criterion for convergence of series, we have
$$ \begin{equation*} \sum_{\gamma=0}^{\infty}|\vartheta_{\widetilde{\tau},\gamma}| =\sum_{\gamma=1}^{\infty}\vartheta_{\widetilde{\tau},\gamma} +\vartheta_{\widetilde{\tau},0} =\sum_{\gamma=1}^{\infty}{p^{\gamma\widetilde{\tau}}}(p^\gamma-1)^{-n}+1<\infty. \end{equation*} \notag $$
Next, let us consider the case $\ell\in(0,1)$. By choosing $\sigma$ such that $\sigma>(1-\ell)/\ell$, using the Hölder inequality and estimating as the case $\ell=1$ above, we also have
$$ \begin{equation*} \biggl(\sum_{\gamma=0}^{\infty}|\vartheta_{\widetilde{\tau},\gamma}|^\ell \biggr)^{1/\ell} \lesssim \sum_{\gamma=1}^{\infty}\gamma^\sigma \vartheta_{\widetilde{\tau},\gamma}+\vartheta_{\widetilde{\tau},0} =\sum_{\gamma=1}^{\infty}\gamma^{\sigma}p^{\gamma\widetilde{\tau}}(p^\gamma-1)^{-n}+ 1<\infty. \end{equation*} \notag $$
Thus, for $0<\ell\leqslant 1$, we obtain
$$ \begin{equation*} \biggl(\sum_{\gamma=0}^{\infty}|\vartheta_{\widetilde{\tau},\gamma}|^\ell \biggr)^{1/\ell}<\infty. \end{equation*} \notag $$
From this, by (3.27), the proof of the inequality (3.22) is completed, hence we have that $\mathcal{M}^p$ is bounded on $K^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$. Theorem 3.5 is proved.

Since $\mathcal{H}^p(f)(x)\leqslant \mathcal M^p(f)(x)$ for all $x\in\mathbb Q^n_p\setminus \{0\}$, we obtain the following result.

Corollary 3.6. Let the assumptions of Theorem 3.4 or Theorem 3.5 be fulfilled. Then we have that $\mathcal{H}^p$ is bounded on $K^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$.

This paper is dedicated to the 100th anniversary of the birth of Professor Vasilii Sergeevich Vladimirov. The authors are grateful to the anonymous referee for the valuable suggestions and comments which have led to the improvement of the paper.

Список литературы

1. V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev, V. A. Osipov, “$p$-adic models of ultrametric diffusion constrained by hierarchical energy landscapes”, J. Phys. A, 35:2 (2002), 177–189  crossref  mathscinet  zmath  adsnasa
2. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, “On $p$-adic mathematical physics”, $p$-Adic Numbers Ultrametric Anal. Appl., 1:1 (2009), 1–17  crossref  mathscinet  zmath
3. A. Yu. Khrennikov, $p$-adic valued distributions in mathematical physics, Math. Appl., 309, Kluwer Acad. Publ., Dordrecht, 1994, xvi+264 pp.  crossref  mathscinet  zmath
4. B. C. Владимиров, И. В. Волович, Е. И. Зеленов, $p$-адический анализ и математическая физика, Наука, М., 1994, 352 с.  mathscinet  zmath; англ. пер.: V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, $p$-adic analysis and mathematical physics, Ser. Soviet East European Math., 1, World Sci. Publ., River Edge, NJ, 1994, xx+319 с.  crossref  mathscinet  zmath
5. S. Albeverio, A. Yu. Khrennikov, V. M. Shelkovich, “Harmonic analysis in the $p$-adic Lizorkin spaces: fractional operators, pseudo-differential equations, $p$-adic wavelets, Tauberian theorems”, J. Fourier Anal. Appl., 12:4 (2006), 393–425  crossref  mathscinet  zmath
6. Nguyen Minh Chuong, Pseudodifferential operators and wavelets over real and $p$-adic fields, Springer, Cham, 2018, xi+368 pp.  crossref  mathscinet  zmath
7. Harmonic, wavelet and $p$-adic analysis, eds. N. M. Chuong, Yu. V. Egorov, A. Khrennikov, Y. Meyer, D. Mumford, World Sci. Publ., Hackensack, NJ, 2007, x+381 pp.  crossref  mathscinet  zmath
8. Nguyen Minh Chuong, Nguyen Van Co, “The Cauchy problem for a class of pseudodifferential equations over $p$-adic field”, J. Math. Anal. Appl., 340:1 (2008), 629–645  crossref  mathscinet  zmath  adsnasa
9. Nguyen Minh Chuong, Ha Duy Hung, “Maximal functions and weighted norm inequalities on local fields”, Appl. Comput. Harmon. Anal., 29:3 (2010), 272–286  crossref  mathscinet  zmath
10. Shai Haran, “Riesz potentials and explicit sums in arithmetic”, Invent. Math., 101:3 (1990), 697–703  crossref  mathscinet  zmath  adsnasa
11. С. В. Козырев, “Методы и приложения ультраметрического и $p$-адического анализа: от теории всплесков до биофизики”, Совр. пробл. матем., 12, МИАН, М., 2008, 3–168  mathnet  crossref  zmath; англ. пер.: S. V. Kozyrev, “Methods and applications of ultrametric and $p$-adic analysis: from wavelet theory to biophysics”, Proc. Steklov Inst. Math., 274, suppl. 1 (2011), S1–S84  crossref
12. S. S. Volosivets, “Multidimensional Hausdorff operator on $p$-adic field”, $p$-Adic Numbers Ultrametric Anal. Appl., 2:3 (2010), 252–259  crossref  mathscinet  zmath
13. W. A. Hurwitz, L. L. Silverman, “On the consistency and equivalence of certain definitions of summability”, Trans. Amer. Math. Soc., 18:1 (1917), 1–20  crossref  mathscinet  zmath
14. F. Hausdorff, “Summationsmethoden und Momentfolgen. I”, Math. Z., 9:1-2 (1921), 74–109  crossref  mathscinet  zmath
15. Kyung Soo Rim, Jaesung Lee, “Estimates of weighted Hardy–Littlewood averages on the $p$-adic vector space”, J. Math. Anal. Appl., 324:2 (2006), 1470–1477  crossref  mathscinet  zmath  adsnasa
16. A. N. Kochubei, “Radial solutions of non-Archimedean pseudodifferential equations”, Pacific J. Math., 269:2 (2014), 355–369  crossref  mathscinet  zmath
17. Huabing Li, Jianjun Jin, “On a $p$-adic Hilbert-type integral operator and its applications”, J. Appl. Anal. Comput., 10:4 (2020), 1326–1334  crossref  mathscinet  zmath
18. K. L. Phillips, “Hilbert transforms for the $p$-adic and $p$-series fields”, Pacific J. Math., 23:2 (1967), 329–347  crossref  mathscinet  zmath
19. S. S. Volosivets, “Weak and strong estimates for rough Hausdorff type operator defined on $p$-adic linear space”, $p$-Adic Numbers Ultrametric Anal. Appl., 9:3 (2017), 236–241  crossref  mathscinet  zmath
20. N. M. Chuong, D. V. Duong, K. H. Dung, “Some estimates for $p$-adic rough multilinear Hausdorff operators and commutators on weighted Morrey–Herz type spaces”, Russ. J. Math. Phys., 26:1 (2019), 9–31  crossref  mathscinet  zmath  adsnasa
21. Zun Wei Fu, Qing Yan Wu, Shan Zhen Lu, “Sharp estimates of $p$-adic Hardy and Hardy–Littlewood–Pólya operators”, Acta Math. Sin. (Engl. Ser.), 29:1 (2013), 137–150  crossref  mathscinet  zmath
22. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., 43, Princeton Univ. Press, Princeton, NJ, 1993, xiv+695 pp.  mathscinet  zmath
23. Nguyen Minh Chuong, Dao Van Duong, Kieu Huu Dung, “Weighted estimates for maximal operators, Riesz potential operators and commutators on $p$-adic Lebesgue and Morrey spaces”, $p$-Adic Numbers Ultrametric Anal. Appl., 11:2 (2019), 123–134  crossref  mathscinet  zmath
24. Shanzhen Lu, Dachun Yang, “The decomposition of the weighted Herz space on $\mathbb R^n$ and its applications”, Sci. China Ser. A., 38:2 (1995), 147–158  mathscinet  zmath
25. M. A. Ragusa, “Homogeneous Herz spaces and regularity results”, Nonlinear Anal., 71:12 (2009), e1909–e1914  crossref  mathscinet  zmath
26. H. G. Feichtinger, F. Weisz, “Herz spaces and summability of Fourier transforms”, Math. Nachr., 281:3 (2008), 309–324  crossref  mathscinet  zmath
27. Shanzhen Lu, K. Yabuta, Dachun Yang, “Boundedness of some sublinear operators in weighted Herz-type spaces”, Kodai Math. J., 23:3 (2000), 391–410  crossref  mathscinet  zmath
28. Jiecheng Chen, Dashan Fan, Jun Li, “Hausdorff operators on function spaces”, Chinese Ann. Math. Ser. B, 33:4 (2012), 537–556  crossref  mathscinet  zmath
29. K. H. Dung, D. V. Duong, N. D. Duyet, “Weighted Triebel–Lizorkin and Herz spaces estimates for $p$-adic Hausdorff type operator and its applications”, Anal. Math., 48:3 (2022), 717–740  crossref  mathscinet  zmath
30. Jianmiao Ruan, Dashan Fan, Qingyan Wu, “Weighted Herz space estimates for Hausdorff operators on the Heisenberg group”, Banach J. Math. Anal., 11:3 (2017), 513–535  crossref  mathscinet  zmath
31. T. Hytönen, C. Pérez, E. Rela, “Sharp reverse Hölder property for $A_{\infty}$ weights on spaces of homogeneous type”, J. Funct. Anal., 263:12 (2012), 3883–3899  crossref  mathscinet  zmath
32. Shanzhen Lu, Dachun Yang, “The decomposition of Herz spaces on local fields and its applications”, J. Math. Anal. Appl., 196:1 (1995), 296–313  crossref  mathscinet  zmath
33. Shanzhen Lu, Dachun Yang, Guoen Hu, Herz type spaces and their applications, Mathematics Monograph Series, 10, Sci. Press, Beijing, 2008, 221 pp.
34. Ka Luen Cheung, Kwok-Pun Ho, “Boundedness of Hardy–Littlewood maximal operator on block spaces with variable exponent”, Czechoslovak Math. J., 64(139):1 (2014), 159–171  crossref  mathscinet  zmath

Образец цитирования: Kieu Huu Dung, Dao Van Duong, “Two-weight estimates for Hardy–Littlewood maximal functions and Hausdorff operators on $p$-adic Herz spaces”, Изв. РАН. Сер. матем., 87:5 (2023), 71–91; Izv. Math., 87:5 (2023), 920–940
Цитирование в формате AMSBIB
\RBibitem{DunDuo23}
\by Kieu~Huu~Dung, Dao~Van~Duong
\paper Two-weight estimates for Hardy--Littlewood maximal functions and Hausdorff operators on $p$-adic Herz spaces
\jour Изв. РАН. Сер. матем.
\yr 2023
\vol 87
\issue 5
\pages 71--91
\mathnet{http://mi.mathnet.ru/im9404}
\crossref{https://doi.org/10.4213/im9404}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4666681}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2023IzMat..87..920D}
\transl
\jour Izv. Math.
\yr 2023
\vol 87
\issue 5
\pages 920--940
\crossref{https://doi.org/10.4213/im9404e}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001101882800004}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85177081816}
Образцы ссылок на эту страницу:
  • https://www.mathnet.ru/rus/im9404
  • https://doi.org/10.4213/im9404
  • https://www.mathnet.ru/rus/im/v87/i5/p71
  • Эта публикация цитируется в следующих 2 статьяx:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
    Статистика просмотров:
    Страница аннотации:582
    PDF русской версии:8
    PDF английской версии:68
    HTML русской версии:69
    HTML английской версии:236
    Список литературы:115
    Первая страница:10
     
      Обратная связь:
     Пользовательское соглашение  Регистрация посетителей портала  Логотипы © Математический институт им. В. А. Стеклова РАН, 2024