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This article is cited in 1 scientific paper (total in 1 paper) Two-weight estimates for Hardy–Littlewood maximal functions and Hausdorff operators on 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
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2023, Volume 87, Issue 5, DOI: https://doi.org/10.4213/im9404 
Bibliographic databases:
    § 1. IntroductionThe $p$-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, for example, [1]–[4] and the 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 related to the weighted Hardy–Littlewood average. To be more precise, Kochubei discussed the 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 non-negative 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 $K$ related to several important operators in harmonic analysis. By taking we have the $p$-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 [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 the $p$-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 we have the $p$-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 centre at the origin and of radius $|x|_p$. For further details about $p$-adic Hardy operator, see [17], [21], and the references therein. In particular, the Hardy–Littlewood maximal operator is of fundamental importance in harmonic analysis due to its significant applications (see [22] and the 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 the 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 the 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 the 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 § 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 formulated and proved in § 3. § 2. Some notation and definitionsLet $p$ 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 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 the ball of radius $p^k$ with centre at $x_0\in\mathbb Q^n_p$. Similarly, we let denote the sphere with centre at $x_0$ and of radius $p^k$. We also set $B_k=B_k(0)$, $S_k=S_k(0)$, and denote the characteristic function of the sphere $S_k$ by $\chi_k$. The Haar measure on $\mathbb Q^n_p$ is normalized by 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 non-negative 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. In what follows, we also write $a \lesssim b$ to mean that there exists a constant $C>0$, independent of the main parameters, such that $a\leqslant Cb$. We write $a\simeq b$ is $a$ is equivalent to $b$ (that is, $C^{-1}a\leqslant b\leqslant Ca$). For any real number $q>0$, we denote by $q'$ the 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 also have
$$
\begin{equation}
\omega(B_{\gamma}) =\sum_{k\leqslant \gamma} \int_{S_k}p^{k\alpha}\, dx=\sum_{k\leqslant \gamma} p^{k(\alpha+n)}(1-p^{-n})= p^{\gamma(\alpha+n)}\frac{1-p^{-n}}{1-p^{-(\alpha+n)}}.
\end{equation}
\tag{2.3}
$$
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 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 Next, let us recall the Muckenhoupt weights on $p$-adic fields. For 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 $B$, We say that $\omega\in A_1(\mathbb Q^n_p)$ if there is a constant $C$ such that, for all balls $B$, 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 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 $\mathbb Q^n_p$. Then $f\in\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$ 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, where the infimum is taken over all decompositions of $f$ as above. Similarly, $f\in K_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$ if and only if $f=\sum_{k=0}^{\infty}\lambda_k b_k$ with $\sum_{k=0}^{\infty}|\lambda_k|^\ell< \infty$, and each $b_k$ is a central $(\beta,q,\omega_1,\omega_2)$-atom of restrict type with the support in $B_k$. Furthermore, where the infimum is taken over all decompositions of $f$ as above.§ 3. The main resultsIn 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 Theorem 3.3 in [20], 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 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 Moreover, in this case, Proof. (i) Let $f\in \dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$. By Theorem 2.7, 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
$$
\begin{equation*}
|p^{\gamma}|x|_p^{-1}y|_p=p^{-\gamma}|x|_p|y|_p=p^{-\gamma}|x|_p>p^{k}.
\end{equation*}
\notag
$$
Since $\operatorname{supp}(b_k)\subset B_k$, we have $b_k(p^{\gamma}|x|_p^{-1}y)=0$. By the Minkowski and Hölder inequalities,
$$
\begin{equation*}
\begin{aligned} \, \|c_{k,\gamma} \|_{L^q_{\omega_2}(\mathbb{Q}_p^n)}&\leqslant \int_{S_0} \frac{\Phi(p^{\gamma})}{p^{\gamma}}|\Omega(y)|\|b_k(p^{\gamma}|\,{\cdot}\,|_p^{-1}y)\|_{L^q_{\omega_2} (B_{k+\gamma})} \, dy \\ &\leqslant \|\Omega\|_{L^{q'}(S_0)}\frac{\Phi(p^{\gamma})}{p^{\gamma}} \biggl(\int_{S_0}\int_{B_{k+\gamma}} |b_k(p^{\gamma}|x|_p^{-1}y)|^q\omega_2(x)\, dx\, dy\biggr)^{1/q}. \end{aligned}
\end{equation*}
\notag
$$
On the other hand, we also get
$$
\begin{equation*}
\begin{aligned} \, &\int_{S_0}\int_{B_{k+\gamma}} |b_k(p^{\gamma}|x|_p^{-1}y)|^q\omega_2(x)\, dx\, dy=\int_{S_0}\sum_{r\leqslant k+\gamma}\int_{S_r}|b_k(p^{\gamma-r}y)|^qp^{r\beta_2}\, dx\, dy \\ &\ \simeq\int_{S_0}\sum_{r\leqslant k+\gamma}|b_k(p^{\gamma-r}y)|^qp^{r(\beta_2+n)}\, dy=\sum_{r\leqslant k+\gamma}\int_{S_{-\gamma+r}}|b_k(t)|^qp^{r(\beta_2+n)}p^{(\gamma-r)n}\, dt \\ &\ =p^{\gamma(\beta_2+n)}\sum_{\sigma\leqslant k}\int_{S_{\sigma}}|b_k(t)|^q p^{\sigma\beta_2}\, dt=p^{\gamma(\beta_2+n)}\|b_k\|^q_{L^{q}_{\omega_2}(B_k)}. \end{aligned}
\end{equation*}
\notag
$$
Hence, since $\|b_k\|_{L^q_{\omega_2}(B_k)}\lesssim \omega_1(B_k)^{{-\beta}/{n}}$ and (2.3), we readily have
$$
\begin{equation}
\begin{aligned} \, \|c_{k,\gamma} \|_{L^q_{\omega_2}(\mathbb{Q}_p^n)} &\lesssim \|\Omega\|_{L^{q'}(S_0)} \frac{\Phi(p^{\gamma})}{p^{\gamma-\gamma(\beta_2+n)/q}}\|b_k\|_{L^{q}_{\omega_2}(B_k)} \nonumber \\ &\lesssim\|\Omega\|_{L^{q'}(S_0)}\frac{\Phi(p^{\gamma})}{p^{\gamma-\gamma(\beta_2+n)/q}} \biggl(\frac{\omega_1(B_{k+\gamma})}{\omega_1(B_k)}\biggr)^{\beta/n} \omega_1(B_{k+\gamma})^{-\beta/n} \nonumber \\ &:= \|\Omega\|_{L^{q'}(S_0)}\cdot\mu_{\tau,\gamma}\cdot\omega_1(B_{k+\gamma})^{-\beta/n}, \end{aligned}
\end{equation}
\tag{3.3}
$$
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 $|\gamma|=|{\log_p|x|_p}|$. Hence
$$
\begin{equation*}
\begin{aligned} \, K_1&=\sum_{\gamma=-\infty}^{-1}|\gamma|^{\sigma_0} \frac{\Phi(p^{\gamma})}{p^{\gamma(1-\tau+n)}}p^{\gamma n}\simeq \sum_{\gamma=-\infty}^{-1}\int_{S_{\gamma}}|\gamma|^{\sigma_0}\frac{\Phi(|x|_p)}{|x|_p^{1-\tau+n}}\, dx \\ &= \sum_{\gamma=-\infty}^{-1}\int_{S_{\gamma}}\frac{\Phi(|x|_p)|{\log_p|x|_p}|^{\sigma_0}} {|x|_p^{1-\tau+n}}\, dx =\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)|{\log_p|x|_p}|^{\sigma_0} \chi_{B_{-1}}(x)}{|x|_p^{1-\tau+n}}\, dx. \end{aligned}
\end{equation*}
\notag
$$
A similar analysis also shows that
$$
\begin{equation*}
K_2=\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)|{\log_p|x|_p}|^{\sigma_0}\chi_{\mathbb Q^n_p\setminus B_{0}}(x)}{|x|_p^{1-\tau+n}}\, dx\quad\text{and}\quad \mu_{\tau,0}\simeq\int_{S_0}\frac{\Phi(|x|_p)}{|x|_p^{1-\tau+n}}\, dx.
\end{equation*}
\notag
$$
Therefore, for all $(1-\ell)/\ell<\sigma_0<(1+n_0^{-1})(1-\ell)/\ell$,
$$
\begin{equation*}
\biggl(\sum_{\gamma\in\mathbb{Z}}|\mu_{\tau,\gamma}|^\ell \biggr)^{1/\ell}\lesssim \int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)\max\{|{\log_p|x|_p}|^{\sigma_0},1\}}{|x|_p^{1-\tau+n}}\, dx.
\end{equation*}
\notag
$$
Using (3.4), it follows from $\mathcal C_{n_0,\ell,\tau,n}<\infty$ and the dominated convergence theorem of Lebesgue that
$$
\begin{equation}
\|\widetilde{\mathcal{H}}^p_{\Phi,\Omega}(b_k)\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell} (\mathbb{Q}_p^n)}\lesssim \|\Omega\|_{L^{q'}(S_0)}\cdot\mathcal C_{\infty,\ell,\tau,n}\quad \text{for all}\quad k\in\mathbb{Z}.
\end{equation}
\tag{3.5}
$$
This implies that
$$
\begin{equation*}
\begin{aligned} \, &\|\mathcal{H}^p_{\Phi,\Omega}(f)\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}\leqslant \biggl(\sum_{k=-\infty}^{\infty}|\lambda_k|^\ell\|\widetilde{\mathcal{H}}^p_{\Phi,\Omega}(b_k) \|^{\ell}_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}\biggr)^{1/\ell} \\ &\quad\! \lesssim \|\Omega\|_{L^{q'}(S_0)}\cdot\mathcal C_{\infty,\ell,\tau,n}\cdot \biggl(\sum_{k=-\infty}^{\infty}|\lambda_k|^\ell \biggr)^{1/\ell} \lesssim \|\Omega\|_{L^{q'}(S_0)}\cdot\mathcal C_{\infty,\ell,\tau,n}\cdot \|f\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}. \end{aligned}
\end{equation*}
\notag
$$
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. For $\eta\in\mathbb Z^+$, we set
$$
\begin{equation*}
f_{\eta}(x) = \begin{cases} |x|_p^{-\tau-p^{-\eta}} &\text{if }|x|_p\geqslant 1, \\ 0 &\text{otherwise}. \end{cases}
\end{equation*}
\notag
$$
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
$$
\begin{equation*}
\mathbb D=(1-p^{-n})^{(\beta/n+1/q)}\bigl(1-p^{-(\beta_1+n)}\bigr)^{-\beta/n}.
\end{equation*}
\notag
$$
Hence, 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
$$
\begin{equation*}
\begin{aligned} \, &\|\mathcal{H}^p_{\Phi,\Omega}\|_{{\dot{K}}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)\to{\dot{K}}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)} \\ &\qquad\geqslant (1-p^{-n})^{-1}\biggl|\int_{S_0}\Omega(y)\, dy \biggr| \cdot\biggl(\int_{\mathbb Q^n_p}\frac{\Phi(|x|_p)(|x|_p)^{p^{-\eta}}p^{-\eta p^{-\eta}}}{|x|_p^{1-\tau+n}} \chi_{B_\eta}(x)\, dx\biggr). \end{aligned}
\end{equation*}
\notag
$$
Note 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 Thus, by the dominated convergence theorem of Lebesgue, we immediately obtain The proof of assertion (ii) is finished. This completes the proof of the theorem. By Theorem 3.1, we obtain the boundedness of the $p$-adic Hardy operator on two-weighted Herz spaces. Namely, the following result holds. 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 $\mathcal{H}^p$ 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 test for convergence of series, we obtain 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 Proof. Suppose that $f\in \dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)$. By Theorem 2.7, 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 further, for each ${k \in \mathbb{Z}}$, $b_k$ is an $(\beta,q,\omega_1,\omega_2)$-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
$$
\begin{equation*}
\begin{aligned} \, |\mathcal{H}^p_{\Psi,A}(f)(x)| &\leqslant \sum_{k=-\infty}^{\infty}|\lambda_k|\int_{\mathbb Q^n_p} \Psi(y)|b_k(A(y)x)|\, dy \\ &=:\sum_{k=-\infty}^{\infty}|\lambda_k|\widetilde{\mathcal{H}}^p_{\Psi,A}(b_k)(x). \end{aligned}
\end{equation*}
\notag
$$
By a similar argument as in the proof of Theorem 3.1, it suffices to prove that
$$
\begin{equation}
\begin{aligned} \, \|\widetilde{\mathcal{H}}^p_{\Psi,A}(b_k)\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell} (\mathbb{Q}_p^n)}\lesssim 1\quad\text{for all}\quad k\in\mathbb{Z}. \end{aligned}
\end{equation}
\tag{3.12}
$$
Note that
$$
\begin{equation}
\widetilde{\mathcal{H}}^p_{\Psi,A}(b_k)(x)=\sum_{j\in\mathbb{Z}}\int_{\|A^{-1}(y)\|_p=p^j} \Psi(y)|b_k(A(y)x)|\, dy=:\sum_{j\in\mathbb{Z}}d_{k,j}(x).
\end{equation}
\tag{3.13}
$$
Since $\operatorname{supp}(b_k)\subset B_k$ 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 $d_{k,j}(x)=0$. Hence 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
$$
\begin{equation*}
\begin{aligned} \, \|b_k(A(y)\,{\cdot}\,)\|_{L^q_{\omega_2}(\mathbb{Q}_p^n)}&=\biggl(\int_{\mathbb Q^n_p}|b_k(z)|^q|A^{-1}(y)z|^{\beta_2}_p|{\det A^{-1}(y)}|_p\, dz\biggr)^{1/q} \\ &\leqslant \|A^{-1}(y)\|_p^{\beta_2/q}|{\det A^{-1}(y)}|_p^{1/q}\|b_k\|_{L^q_{\omega_2}(\mathbb Q^n_p)} \\ &\lesssim\|A^{-1}(y)\|_p^{\beta_2/q}|{\det A^{-1}(y)}|_p^{1/q}\omega_1(B_k)^{-\beta/n}. \end{aligned}
\end{equation*}
\notag
$$
Now using (2.3), we have
$$
\begin{equation}
\begin{aligned} \, &\|d_{k,j} \|_{L^q_{\omega_2}(\mathbb{Q}_p^n)} \nonumber \\ &\ \leqslant \biggl(\int_{\|A^{-1}(y)\|_p=p^j} \Psi(y)\|A^{-1}(y)\|_p^{\beta_2/q}|{\det A^{-1}(y)}|_p^{1/q} \, dy\biggr) \nonumber \\ &\ \qquad\times\biggl(\frac{\omega_1(B_{k+j})}{\omega_1(B_k)}\biggr)^{\beta/n} \omega_1(B_{k+j})^{-\beta/n} \nonumber \\ &\ =\biggl(\int_{\|A^{-1}(y)\|_p=p^j} \Psi(y)\|A^{-1}(y)\|_p^{\beta_2/q+(\beta_1+n)\beta/n}|{\det A^{-1}(y)}|_p^{1/q} \, dy\biggr)\omega_1(B_{k+j})^{-\beta/n} \nonumber \\ &\ \lesssim\biggl(\int_{\|A^{-1}(y)\|_p=p^j} \Psi(y)\|A^{-1}(y)\|_p^{\tau}\, dy\biggr)\omega_1(B_{k+j})^{-\beta/n}=:\nu_{\tau,j}\cdot\omega_1(B_{k+j})^{-\beta/n}. \end{aligned}
\end{equation}
\tag{3.15}
$$
We set
$$
\begin{equation*}
\widetilde{d}_{k,j}= \begin{cases} \dfrac{d_{k,j}}{\nu_{\tau, j}} &\text{if }\nu_{\tau, j}\ne 0, \\ 0 &\text{otherwise}. \end{cases}
\end{equation*}
\notag
$$
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. Now Theorem 2.7 implies that
$$
\begin{equation*}
\|\widetilde{\mathcal{H}}^p_{\Psi,A}(b_k)\|_{\dot{K}_{q,\omega_1,\omega_2}^{\beta,\ell} (\mathbb{Q}_p^n)}\lesssim \biggl(\sum_{j\in\mathbb{Z}}|\nu_{\tau,j}|^\ell \biggr)^{1/\ell}.
\end{equation*}
\notag
$$
Since $\ell\in(0,1)$ and since $\sigma>(1-\ell)/{\ell}$, 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 inequality (3.12) holds, and so $\mathcal{H}^p_{\Psi,A}$ is bounded on $\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$. By the Lebesgue dominated convergence theorem,
$$
\begin{equation}
\|\mathcal{H}^p_{\Psi,A}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)\to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}\lesssim \mathcal K_{(1-\ell)/\ell}.
\end{equation}
\tag{3.16}
$$
To complete 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 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 that $\mathbb D=(1-p^{-n})^{(\beta/n+1/q)}(1-p^{-(\beta_1+n)})^{-\beta/n}$ as in the proof of Theorem 3.1. Note that from 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
$$
\begin{equation*}
|A_i(y)x|_p\geqslant \|A_i^{-1}(y)\|^{-1}_p\,|x|_p =\frac{\|A_i(y)\|_p\, |x|_p}{\|A_i^{-1}(y)\|_p\,\|A_i(y)\|_p}\geqslant p^{-\xi_{A}}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
V_\eta :=\bigl\{y\in\mathbb Q^n_p\colon \|A_i(y)\|_p\geqslant p^{-\eta}\bigr\}\subset \bigl\{y\in\mathbb Q_p^n\colon |A_i(y)x|_p\geqslant p^{-\xi_{A}}\bigr\}:=W_x
\end{equation*}
\notag
$$
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
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|\,|\,{\cdot}\,|_p^{-\tau-p^{-\eta}}\chi_{\mathbb Q^n_p\setminus B_{\eta-1}}\bigr\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}= \mathbb D\biggl(\sum_{k=\eta}^{\infty}p^{-k\ell p^{-\eta}}\biggr)^{1/\ell} \\ &\qquad= \mathbb D\cdot p^{-\eta p^{-\eta}}\bigl(1-p^{-\ell p^{-\eta}}\bigr)^{-1/\ell}= p^{-\eta p^{-\eta}-\xi_Ap^{-\eta}} \|g_{\eta}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}. \end{aligned}
\end{equation*}
\notag
$$
Now by (3.18) we have
$$
\begin{equation*}
\begin{aligned} \, &\|\mathcal{H}^p_{\Psi,A}(g_\eta)\|_{\dot K^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}\geqslant \biggl(\sum_{k=\eta}^{\infty}\omega_1(B_k)^{\beta\ell/n} \|\mathcal{H}^p_{\Psi,A}(g_\eta)\chi_k\|_{L^q_{\omega_2}(\mathbb Q^n_p)}^{\ell}\biggr)^{1/\ell} \\ &\qquad\geqslant p^{-\xi_A\tau-2\xi_Ap^{-\eta}-\eta p^{-\eta}} \biggl(\int_{V_\eta} \Psi(y) \|A^{-1}(y)\|_p^{\tau+p^{-\eta}}\, dy\biggr) \|g_{\eta}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)}, \end{aligned}
\end{equation*}
\notag
$$
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 $y\in V_\eta$, we have by (3.10)
$$
\begin{equation*}
\begin{aligned} \, p^{-2\xi_Ap^{-\eta}-\eta p^{-\eta}}{\|A^{-1}(y)\|_p}^{p^{-\eta}}&= p^{-2\xi_Ap^{-\eta}}\bigl(p^{-\eta}{\|A^{-1}(y)\|_p}\bigr)^{p^{-\eta}} \\ &\leqslant p^{-2\xi_Ap^{-\eta}}\bigl(\|A(y)\|_p {\|A^{-1}(y)\|_p}\bigr)^{p^{-\eta}} \leqslant p^{-\xi_{A}p^{-\eta}}\lesssim 1. \end{aligned}
\end{equation*}
\notag
$$
Note that
$$
\begin{equation*}
\lim_{\eta\to \infty}p^{-2\xi_Ap^{-\eta}-\eta p^{-\eta}} =1.
\end{equation*}
\notag
$$
Therefore, another appeal to the Lebesgue dominated convergence theorem shows that
$$
\begin{equation*}
\|\mathcal{H}^p_{\Psi,A}\|_{\dot{K}^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p) \to \dot{K}^{\beta,\ell}_{q, \omega_1,\omega_2}(\mathbb Q^n_p)}\geqslant \mathcal K_{0}.
\end{equation*}
\notag
$$
This completes the proof of Theorem 3.3. Next, estimating as in Theorem 1 of [27], we prove 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: 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 above hypotheses of Theorem 3.4 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 the 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,
$$
\begin{equation*}
f=\sum_{k=0}^{\infty}\lambda_kb_k\quad\text{and}\quad\biggl(\sum_{k=0}^{\infty}|\lambda_k|^\ell \biggr)^{1/\ell}\lesssim \|f\|_{{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}.
\end{equation*}
\notag
$$
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
$$
\begin{equation*}
\|b_k\|_{L^{q}_{\omega_2}(\mathbb{Q}_p^n)}\lesssim \omega_1(B_k)^{-\beta/n}.
\end{equation*}
\notag
$$
Hence, 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 verify that
$$
\begin{equation}
\|\mathcal{M}^p(b_k)\|_{{K}_{q,\omega_1,\omega_2}^{\beta,\ell}(\mathbb{Q}_p^n)}\lesssim 1 \quad\text{for all}\quad k\in\mathbb{N}.
\end{equation}
\tag{3.22}
$$
Since $\mathbb Q^n_p= B_k\cup \bigl(\bigcup_{\gamma=1}^{\infty}S_{k+\gamma}\bigr)$, we have
$$
\begin{equation}
\mathcal{M}^p(b_k)=\chi_{B_k}\mathcal M^p(b_k)+\sum_{\gamma=1}^{\infty}\chi_{{k+\gamma}}\mathcal M^p(b_k):=\sum_{\gamma=0}^{\infty}e_{k,\gamma}.
\end{equation}
\tag{3.23}
$$
Consequently,
$$
\begin{equation}
\operatorname{supp}(e_{k,\gamma})\subset B_{k+\gamma}.
\end{equation}
\tag{3.24}
$$
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 $B_\theta(x)\cap B_k=\varnothing$. Hence, since $\operatorname{supp}(b_k)\subset B_k$, 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, since $\beta_2<n(q-1)$, we have $-q'\beta_2/q>-n$. Hence from (2.3) and since $\|b_k\|_{L^q_{\omega_2}(B_k)}\lesssim \omega_1(B_k)^{-\beta/n}$ it follows from the Hölder inequality that
$$
\begin{equation*}
\begin{aligned} \, \|e_{k,\gamma}\|_{L^q_{\omega_2}(\mathbb Q_p^n)}&\leqslant \omega_2(B_{k+\gamma})^{1/q} \frac{1}{p^{kn}(p^\gamma-1)^n}\, \|b_k\|_{L^q_{\omega_2}(B_k)} \biggl(\int_{B_k}|x|_p^{-q'\beta_2/q}\, dx\biggr)^{1/q'} \\ &\lesssim p^{(k+\gamma)(\beta_2+n)/q} \frac{1}{p^{kn}(p^\gamma-1)^n} \, \omega_1(B_k)^{-\beta/n}p^{k(-\beta_2/q+n/q')} \\ &=\frac{p^{\gamma(\beta_2+n)/q}}{(p^\gamma-1)^n} \biggl(\frac{\omega_1(B_{k+\gamma})}{\omega_1(B_k)}\biggr)^{\beta/n} \omega_1(B_{k+\gamma})^{-\beta/n}. \end{aligned}
\end{equation*}
\notag
$$
Using the condition $\omega_1\in A_1(\mathbb Q^n_p)$ and Proposition 2.4, we obtain
$$
\begin{equation*}
\biggl(\frac{\omega_1(B_{k+\gamma})}{\omega_1(B_k)}\biggr)^{\beta/n}\lesssim \biggl(\frac{|B_{k+\gamma}|}{|B_k|}\biggr)^{\beta/n}= p^{\gamma\beta},
\end{equation*}
\notag
$$
which implies
$$
\begin{equation}
\|e_{k,\gamma}\|_{L^q_{\omega_2}(\mathbb Q_p^n)}\lesssim \frac{p^{\gamma\widetilde{\tau}}}{(p^\gamma-1)^n}\, \omega_1(B_{k+\gamma})^{-\beta/n} =:\vartheta_{\widetilde{\tau},\gamma}\omega_1(B_{k+\gamma})^{-\beta/n}.
\end{equation}
\tag{3.26}
$$
We set
$$
\begin{equation*}
\widetilde{e}_{k,\gamma}=\frac{e^{k,\gamma}}{\vartheta_{\widetilde{\tau},\gamma}}\quad \text{for all}\quad \gamma\in\mathbb N.
\end{equation*}
\notag
$$
Now applying (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
$$
\begin{equation*}
\lim_{\gamma\to \infty} \frac{p^{(\gamma+1)\widetilde{\tau}}(p^{\gamma+1}-1)^{-n}}{p^{\gamma\widetilde{\tau}} (p^\gamma-1)^{-n}} =p^{\widetilde{\tau}-n}<1.
\end{equation*}
\notag
$$
Now, using the D’Alembert test 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
$$
\begin{equation*}
\biggl(\sum_{\gamma=0}^{\infty}|\vartheta_{\widetilde{\tau},\gamma}|^\ell \biggr)^{1/\ell} \lesssim \sum_{\gamma=1}^{\infty}\gamma^\sigma \vartheta_{\widetilde{\tau},\gamma}+\vartheta_{\widetilde{\tau},0} =\sum_{\gamma=1}^{\infty}\gamma^{\sigma}p^{\gamma\widetilde{\tau}}(p^\gamma-1)^{-n}+ 1<\infty.
\end{equation*}
\notag
$$
So, for $0<\ell\leqslant 1$, we obtain
$$
\begin{equation*}
\biggl(\sum_{\gamma=0}^{\infty}|\vartheta_{\widetilde{\tau},\gamma}|^\ell \biggr)^{1/\ell}<\infty.
\end{equation*}
\notag
$$
Hence, using (3.27), we complete the proof of inequality (3.22). Hence $\mathcal{M}^p$ 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 met. Then $\mathcal{H}^p$ is bounded on $K^{\beta,\ell}_{q,\omega_1,\omega_2}(\mathbb Q^n_p)$. This paper is dedicated to the 100th anniversary of 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. 
Citation in format AMSBIB
\Bibitem{DunDuo23} |
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