1 Introduction↩︎

Averaging operators play a fundamental role in harmonic analysis and it is often desirable to obtain sharp estimates for them. The fractional Hardy operator is very interesting since it is a useful tool to study the embedding properties of function spaces. Mizuta et al. [1] showed that the optimal bound of fractional Hardy operator implies the sharp embedding properties of the function spaces. There is much literature on the function spaces. Let \(f\) be a locally integrable function on \(\mathbb{R} ^n\). Then the \(n\)-dimensional fractional Hardy operator and its duality from can be started as \[H_{\alpha}f\left( x \right) =\frac{1}{\left| x \right|^{n-\alpha}}\int_{\left| y \right|<\left| x \right|}{f\left( y \right)}dy,\quad H_{\alpha}^{*}f\left( x \right) =\int_{\left| y \right|>\left| x \right|}{\frac{1}{\left| y \right|^{n-\alpha}}f\left( y \right)}dy,\] where \(0<\alpha<n\), \(x\in \mathbb{R} ^n\backslash\left\{ 0\right\}\). If \(\alpha\) = 0, the fractional Hardy operator is the classic Hardy operator. There is much literature on the boundness of these operators [2]–[5]. Among them, Lu et al. [3] obtained the following estimates. Suppose \[0<\alpha <n,\quad1<p\leqslant \frac{n}{\alpha},\quad\frac{1}{p}-\frac{1}{q}=\frac{\alpha}{n}.\] Then \[\left\| H_{\alpha}f \right\| _{L^q\left( \mathbb{R} ^n \right)}\leqslant C\left\| f \right\| _{L^p\left( \mathbb{R} ^n \right)},\] where \[\left( \frac{p}{q} \right) ^{1/q}\left( \frac{p}{p-1} \right) ^{1/q}\left( \frac{q}{q-1} \right) ^{1-1/q}\left( 1-\frac{p}{q} \right) ^{1/p-1/q}\left( \frac{\omega _n}{n} \right) ^{1-\alpha /n}\leqslant C\leqslant \left( \frac{p}{p-1} \right) ^{p/q}\left( \frac{\omega _n}{n} \right) ^{1-\alpha /n}.\] If \(p=1\), then \[\left\| H_{\alpha} \right\| _{L^1\left( \mathbb{R} ^n \right) \rightarrow L^{n/\left( n-\alpha \right) ,\infty}\left( \mathbb{R} ^n \right)}=\left( \frac{\omega _n}{n} \right) ^{1-\frac{\alpha}{n}}.\] The optimal \(L^p\rightarrow L^q\) estimate was later obtained in [5]: \[\left\| H_{\alpha} \right\| _{L^p\left( \mathbb{R} ^n \right) \rightarrow L^q\left( \mathbb{R} ^n \right)}=\left( \frac{p\prime}{q} \right) ^{1/q}\left( \frac{n}{q\alpha}\cdot B\left( \frac{n}{q\alpha},\frac{n}{q^{\prime}\alpha} \right) \right) ^{-\alpha /n}\left( \frac{\omega _n}{n} \right) ^{1-\frac{\alpha}{n}},\] where \[p^{\prime}=\frac{p}{p-1},\quad q^{\prime}=\frac{q}{q-1},\] and \(B\left( \cdot ,\cdot \right)\) is the Beta function defined by \[B\left( z,\omega \right) =\int_0^1{t^{z-1}\left( 1-t \right) ^{\omega -1}dt},\] here \(z\) and \(\omega\) are complex numbers with positive real parts. Comparing with the complicated bounds in the power weighted spaces, the sharp weak bounds for \(H_{\alpha}\) and \(H_{\alpha}^{*}\) seem easier to understand. Gao and Zhao [5] set up \[\left\| H_{\alpha}^{*} \right\| _{L^1(\mathbb{R} ^n)\rightarrow L^{n/(n-\alpha ),\infty}(\mathbb{R} ^n)}=\left( \frac{\omega _n}{n} \right) ^{1-\frac{\alpha}{n}},\quad\parallel H_{\alpha}^{*}\parallel _{L^p(\mathbb{R} ^n)\rightarrow L^{q,\infty}(\mathbb{R} ^n)}\,=\left( \frac{\omega _n}{n} \right) ^{\frac{1}{q}+\frac{1}{p^{\prime}}}\bigl( \frac{q}{p^{\prime}} \bigr) ^{1/p^{\prime}}.\]

It is then a nature problem to obtain the operator norms of \(H_\alpha\) and its dual operator \(H_\alpha^*\) in corresponding power weighted spaces. For the latter, Gao et al. [6] set up \[\left\|H_\alpha^*\right\|_{L_{|x|^ \rho}^1\left(\mathbb{R}^n\right) \rightarrow L_{|x|^\beta}^{(n+\beta) /(n-\alpha+\rho), \infty}\left(\mathbb{R}^n\right)}=\left(\frac{\omega_n}{n+\beta}\right)^{(n-\alpha+\rho) /(n+\beta)},\] and \[\left\|H_\alpha^*\right\|_{L_{|x|}^p \rho}\left(\mathbb{R}^n\right) \rightarrow L_{|x| \beta}^{q, \infty}\left(\mathbb{R}^n\right)=\left(\frac{\omega_n}{n+\beta}\right)^{\frac{1}{q}+\frac{1}{p^{\prime}}}\left(\frac{q}{p^{\prime}}\right)^{1 / p^{\prime}}.\] Then Yu et al. [7] set up \[\parallel H_{\alpha}\parallel _{L_{|x|^{\rho}}^{p}(\mathbb{R} ^n)\rightarrow L_{|x|^{\beta}}^{q,\infty}(\mathbb{R} ^n)}\,=\bigl( \frac{\omega _n}{n+\beta} \bigr) ^{1/q}\biggl( \frac{\omega _n}{n-\frac{\rho}{p-1}} \biggr) ^{1/p^{\prime}},\] and \[\parallel H_{\alpha}\parallel _{L^1(\mathbb{R} ^n)\rightarrow L_{|x|^{\beta}}^{(n+\beta )/(n-\alpha ),\infty}(\mathbb{R} ^n)}\,=\bigl( \frac{\omega _n}{n+\beta} \bigr) ^{(n-\alpha )/(n+\beta )}.\]

With the rapid development of mathematics in various fields, \(p\)-adic analysis is becoming more and more important in mathematical analysis, it has attracted attention from mathematical physics and other fields [8]–[13]. The harmonic analysis on \(p\)-adic field has received high attention [14]–[17]. Inspired by them, we will study the sharp weak bound for the \(p\)-adic fractional Hardy operator from \(L^p\) to \(L^{q,\infty}\), which plays an important role in several branches of mathematics. Now, allow us to introduce some basic knowledge about the \(p\)-adic fields which will be used later.

For a prime number \(p\), let \(\mathbb{Q} _p\) be the field of \(p\)-adic numbers, which is defined as the completion of the field of rartional numbers \(\mathbb{Q}\) with respect to the non-Archimedean \(p\)-adic norm \(\left| \cdot \right|_p\). This norm is defined as follows: \(\left| 0 \right|_p=0\). If any non-zero rational number \(x\) is represented as \(x=p^{\gamma}\frac{m}{n}\), where \(m\) and \(n\) are integers which are not divisible by \(p\), and \(\gamma\) is an integer. then \(\left| x \right|_p=p^{-\gamma}\). It is not difficult to show that the norm satisfies the following properties \[\left| xy \right|_p=\left| x \right|_p\left| y \right|_p,\quad\left| x+y \right|_p\leqslant \max \{ \left| x \right|_p,\left| y \right|_p \}\] and \[| \left| x \right|_py |_p=\left| x \right|_{p}^{-1}\left| y \right|_p,\quad x\in \mathbb{Q} _{p}^{n}.\] It follows from the second property that when \(\left| x \right|_p\ne \left| y \right|_p\), then \(\left| x+y \right|_p=\max \{ \left| x \right|_p,\left| y \right|_p \}\). From the standard \(p\)-adic analysis [13], we see that any non-zero \(p\)-adic number \(x\in \mathbb{Q} _p\) can be unique represented in the canonical series \[\begin{align} x=p^{\gamma}\sum_{j=0}^{\infty}{a_jp^j},\quad\gamma =\gamma \left( x \right) \in \mathbb{Z}, \end{align}\] where \(a_j\) are integers, \(0\leqslant a_j\leqslant p-1,a_0\ne 0\). The series (1.1) converges in the \(p\)-adic norm because \(\left| a_jp^j \right|_p=p^{-\gamma}\). The space \(\mathbb{Q} _{p}^{n}\) consists of points \(x=\left( x_1,x_2,...,x_n \right)\), where \(x_j\in \mathbb{Q} _p\), \(j=1,...,2\). The \(p\)-aidc norm on \(\mathbb{Q} _{p}^{n}\) is \[\left| x \right|_p:=\underset{1\leqslant j\leqslant n}{\max}\left| x_j \right|_p.\] Denoted by \(B_{\gamma}=\{ x\in \mathbb{Q} _{p}^{n}:\left| x-a \right|_p\leqslant p^{\gamma} \}\), the ball with center at \(a\in \mathbb{Q} _p\) and radius \(p^{\gamma}\), and by \(S_{\gamma}\left( a \right) :=\left\{ x\in \mathbb{Q} _{p}^{n}:|x-a|_p=p^{\gamma} \right\}\) the sphere with center at \(a\in \mathbb{Q} _{p}^{n}\) and radius \(p^{\gamma}\), \(\gamma \in \mathbb{Z}\). it is clear that \(S_{\gamma}\left( a \right) =B_{\gamma}\left( a \right) \backslash B_{\gamma -1}\left( a \right)\) and

\[B_{\gamma}\left( a \right) =\bigcup_{k\leqslant \gamma}{S_k\left( a \right)},\quad \{ x\in \mathbb{Q} _{p}^{n}:\left| x-a \right|_p<p^{\gamma} \} =\bigcup_{k<\gamma}{S_k\left( a \right)}.\] We set \(B_{\gamma}\left( 0 \right) =B_{\gamma}\) and \(S_{\gamma}\left( 0 \right) =S_{\gamma}\). Since \(\mathbb{Q} _{p}^{n}\) is a locally compact commutative group under addition, it follows from the standard analysis that there exists a unique Haar measure \(d\)x on \(\mathbb{Q} _{p}^{n}\)(up to positive constant multiple) which is translation invariant.We normalize the measure \(dx\) so that \[\int_{B_0\left( 0 \right)}{dx}=\left| B_0\left( 0 \right) \right|_H=1.\] Where \(\left| E \right|_H\) denotes the Haar measure of a measurable subset \(E\) of \(\mathbb{Q} _{p}^{n}\). From this integral theory, it is easy to obtain that \(\left| B_{\gamma}\left( a \right) \right|_H=p^{\gamma n}\) and \(\left| S_{\gamma}\left( a \right) \right|_H=p^{\gamma n}\left( 1-p^{-n} \right)\) for any \(a\in \mathbb{Q} _{p}^{n}\). For a more complete introduction to \(p\)-adic field, see [18].

To get the main conclusion, it is necessary to introduce some fundamental knowledge and definitions. These operators and spaces are the \(p\)-adic \(n\)-dimensional fractional Hardy operator, \(p\)-adic \(m\)-linear \(n\)-dimensional Hardy operator, \(p\)-adic \(m\)-linear \(n\)-dimensional Hilbert operator and \(p\)-adic \(m\)-linear \(n\)-dimensional Hausdorff operator, \(p\)-adic weighted \(L^p\), \(p\)-adic weighted \(L^{q,\infty}\) and \(p\)-adic weighted-type space \(H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )\).

Definition 1. Let \(f\) be a nonnegative locally integrable function on \(\mathbb{Q}^{n}_{p}\), \(0<\alpha <n\). The \(p\)-adic \(n\)-dimensional fractional Hardy operator is defined by \[\begin{align} \mathcal{H} _{\alpha}f\left( x \right) =\frac{1}{|x|_{p}^{n-\alpha}}\int_{|y|_p<|x|_p}{f\left( y \right) dy,} \end{align}\] where \(x\in \mathbb{Q}^{n}_{p}\backslash\left\{ 0\right\}\).

Definition 2. Let \(1\leqslant p<\infty\). The \(p\)-adic Lebesgue space \(L^p\left( \mathbb{Q} ^{n}_{p}\right)\) is defined by \[L^p\left( \mathbb{Q} ^{n}_{p}\right) =\{ f\in L_{loc}^{p}:\left\| f \right\| _{L^p\left( \mathbb{Q} ^{n}_{p} \right)}<\infty \},\] where \[\begin{align} \left\| f \right\| _{L^p\left( \mathbb{Q} ^{n}_{p} \right)}=\left( \int_{\mathbb{Q} ^{n}_{p}}{\left| f\left( x \right) \right|^p dx} \right) ^{1/p}. \end{align}\]

Definition 3. Let \(\omega : \mathbb{Q} ^{n}_{p}\rightarrow \left( 0,\infty \right)\) be a positive measurable function, \(1\leqslant p<\infty\). The \(p\)-adic weighted Lebesgue space \(L^p\left( \mathbb{Q} ^{n}_{p},\omega \right)\) is defined by \[L^p\left( \mathbb{Q} _{p}^{n},\omega \right) =\{f\in L_{loc}^{p}:\left\| f \right\| _{L^p\left( \mathbb{Q} _{p}^{n},\omega \right)}<\infty \},\] where \[\begin{align} \left\| f \right\| _{L^p\left( \mathbb{Q} _{p}^{n},\omega \right)}=\left( \int_{\mathbb{Q} _{p}^{n}}{\left| f\left( x \right) \right|^p\omega \left( x \right) dx} \right) ^{1/p}. \end{align}\]

Definition 4. Let \(\omega : \mathbb{Q} ^{n}_{p}\rightarrow \left( 0,\infty \right)\) be a positive measurable function, \(1\leqslant p<\infty\). The weighted \(p\)-adic weak type Lebesgue space \(L^p\left( \mathbb{Q} ^{n}_{p},\omega \right)\) is defined by \[L^{q,\infty}\left( \mathbb{Q} ^{n}_{p},\omega \right) =\left\{ f\in L_{loc}^{p}:\left\| f \right\| _{L^{q,\infty}\left( \mathbb{Q} ^{n}_{p},\omega \right)}<\infty \right\} ,\] where \[\begin{align} \left\| f \right\| _{L^{q,\infty}\left( \mathbb{Q} ^{n}_{p},\omega \right)}=\underset{\lambda >0}{\mathrm{sup}}\,\lambda \left( \int_{\mathbb{Q} ^{n}_{p}}{\chi _{\left\{ x:f\left( x \right) >\lambda \right\}}\left( x \right) \omega \left( x \right) dx} \right) ^{1/q}. \end{align}\]

Definition 5. Let \(m\) be a positive integer and \(f_1,\dots,f_m\) be nonnegative locally integrable functions on \(\mathbb{Q}_p^n\). The \(p\)-adic \(m\)-linear \(n\)-dimensional Hardy operator is defined by \[\begin{align} T_{1}^{p}\left( f_1,...,f_m \right) \left( x \right) =\frac{1}{\left| x \right|_{p}^{mn}}\int_{\left| \left( y_1,...,y_m \right) \right|_p\leqslant \left| x \right|_p}{f_1\left( y_1 \right) \cdots f_m\left( y_m \right) dy_1\cdots dy_m}, \end{align}\] where \(x\in \mathbb{Q}_p^n\backslash\left\{ 0\right\}\).

Definition 6. Let \(m\) be a positive integer and \(f_1,\dots,f_m\) be nonnegative locally integrable functions on \(\mathbb{Q}_p^n\). The \(p\)-adic \(m\)-linear \(n\)-dimensional Hilbert operator is defined by \[\begin{align} T_{2}^{p}\left( f_1,...,f_m \right) \left( x \right) =\int_{\mathbb{Q} _{p}^{n}}{\cdots}\int_{\mathbb{Q} _{p}^{n}}{\frac{f_1\left( y_1 \right) \cdots f_m\left( y_m \right)}{( \left| x \right|_{p}^{n}+\left| y_1 \right|_{p}^{n}+\cdots +\left| y_m \right|_{p}^{n} ) ^m}}dy_1\cdots dy_m, \end{align}\] where \(x\in \mathbb{Q}_p^n\backslash\left\{ 0\right\}\).

Definition 7. Let \(m\) be a positive integer, \(f_1,\dots,f_m\) be nonnegative locally integrable functions on \(\mathbb{Q}_p^n\), and \(\Phi\) be a nonnegative function on \(\mathbb{Q}_p^n\). The \(p\)-adic \(m\)-linear \(n\)-dimensional Hausdorff operator is defined by \[\begin{align} T_{\Phi}^{p}\left( f_1,...,f_m \right) \left( x \right) =\int_{\mathbb{Q} _{p}^{n}}{\cdots}\int_{\mathbb{Q} _{p}^{n}}{\frac{\Phi ( x/\left| y_1 \right|_p,...,x/\left| y_m \right|_p )}{\left| y_1 \right|_{p}^{n}\cdots \left| y_m \right|_{p}^{n}}}f_1\left( y_1 \right) \cdots f_m\left( y_m \right) dy_1\cdots dy_m, \end{align}\] where \(x\in \mathbb{Q}_p^n\backslash\left\{ 0\right\}\).

We present the definition of the weighted \(p\)-adic space \(H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )\) on \(\mathbb{Q} _{p}^{n}\), where \(\left| x \right|_{p}^{\alpha}\): \(\mathbb{Q} _{p}^{n}\rightarrow \left( 0,\infty \right)\) are positive measurable functions.

Definition 8. The \(p\)-adic weighted-type space \(H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )\) with \(\alpha>0\) consist of all measurable functions \(f\) satisfying \[\begin{align} \left\| f \right\| _{H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )}:=\mathrm{ess} \underset{x\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}\left| x \right|_{p}^{\alpha}\left| f\left( x \right) \right|<\infty. \end{align}\]

The function \(\left\| \cdot \right\| _{H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n})}\) is a norm on the space. Weighted-type spaces on various domains frequently appear in the literature and are quite suitable for invertigations, see [19].

Computation of the operator norm of integral operators is a challenging work in harmonic analysis. Batbold and Sawano [20] obtained that the norm of \(T_1^p\) on \(p\)-adic Lebesgue spaces and \(p\)-adic Morrey spaces, that is \[\left\| T_1^p \right\| _{L^{q_1}\left( \mathbb{Q} _p,\left| x \right|_{p}^{\alpha _1q_1/q} \right) \times \cdots \times L^{q_m}\left( \mathbb{Q} _p,\left| x \right|_{p}^{\alpha _mq_m/q} \right) \rightarrow L^q\left( \mathbb{Q} _p,\left| x \right|_{p}^{\alpha} \right)}=\frac{\left( 1-p^{-1} \right) ^m}{\prod\nolimits_{j=1}^m{( 1-p^{\left( 1/q_j \right) +\left( a_j/q \right) -1} )}}\] and \[\left\| T_1^p \right\| _{L^{q_1,\lambda _1}\left( \mathbb{Q} _p,\left| x \right|_{p}^{\beta _1q_1/q} \right) \times \cdots \times L^{q_m,\lambda _m}\left( \mathbb{Q} _p,\left| x \right|_{p}^{\beta _mq_m/q} \right) \rightarrow L^{q,\lambda}\left( \mathbb{Q} _p,\left| x \right|_{p}^{\beta} \right)}=\frac{\left( 1-p^{-1} \right) ^m}{\prod\nolimits_{i=1}^m{( 1-p^{\delta _i} )}}.\] Duong and Hong [21] obtained the norm of Multilinear Hausdorff operator on \(p\)-adic function space. In 2017, Batbold and Sawano [22] studied the one-dimensional \(m\)-linear Hilbert-type operators that includes Hardy-Littlewood-Pólya operator on weighted Morrey spaces, and they obtained the sharp bounds. Later, He et al [23] extended the results in [22] and obtain the sharp bound for the generalized Hardy-Littlewood-Pólya operator on power weighted central and noncentral homogenous Morrey space. In 2011, Wu and Fu [24] got the best estimate of the \(m\)-linear \(p\)-adic Hardy operator on Lebesgue spaces with power weights weights.

Inspired by the above, we first study the sharp estimate for the \(p\)-adic \(n\)-dimensional fractional Hardy operator from \(L^p\) to \(L^{q,\infty}\). Secondly, we study a more general operator which includes the \(p\)-adic Hardy and Hilbert operator as a special case and consider their operator norm on \(p\)-adic weighted function space. Finally, we also find the sharp bound for the \(p\)-adic Hausdorff operator on \(p\)-adic weighted function space, which generalizes the previous results.

2 Sharp weak bound for the \(\boldsymbol{p}\)-adic fractional Hardy operator↩︎

In this section, we will study the \(p\)-adic weighted \(L^p\) estimate for the \(p\)-adic fractional Hardy operator. For the \(p\)-adic \(n\)-dimensional fractional Hardy operator, our results have a restricted condition: \(\beta =0\) when \(p_1=1\) and \(\beta>0\) when \(p_1>1\). Removing this restrictive condition requires a more complicated argument, and it will be presented in a paper.

Theorem 1. Let \(1<p_1<\infty\), \(1\leqslant q<\infty\), \(\beta <n\left( p_1-1 \right)\), \(n+\gamma >0\), \(0\leqslant\alpha <\frac{\beta}{p_1-1}\), and \(\frac{1}{p_1}+\frac{1}{p_{1}^{\prime}}=1\).

If \[\frac{\gamma +n}{q}+\alpha =\frac{\beta +n}{p_1},\]

then \[\begin{align} \left\| \mathcal{H} _{\alpha} \right\| _{L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})\rightarrow \,\,L^{q,\infty}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma})}=\left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_{1}^{\prime}}}. \end{align}\]

Theorem 2. Let \(n+\gamma >0\), and \(0<\alpha <n\). Then \[\begin{align} \left\| \mathcal{H} _{\alpha} \right\| _{L^1\left( \mathbb{Q} _{p}^{n} \right) \rightarrow L^{\left( n+\gamma \right) /\left( n-\alpha \right) ,\infty}\left( \mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma} \right)}=\left( \frac{\left( 1-p^{-n} \right)}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{n-\alpha}{n+\gamma}}. \end{align}\]

Proof of Theorem 1:. Noticing \(n-\frac{\beta}{p_1-1}>n-\frac{n\left( p_1-1 \right)}{p_1-1}=0\), by \(\mathrm{H}\ddot{\mathrm{o}}\mathrm{lder}\)’s inequality, we have \[\begin{align} |\mathcal{H} _{\alpha}f\left( x \right) | &=\left| \frac{1}{|x|_{p}^{n-\alpha}}\int_{|y|_p<|x|_p}{f\left( y \right) dy} \right|=\left|\frac{1}{|x|_{p}^{n-\alpha}}\int_{|y|_p<|x|_p}{|y|_{p}^{-\frac{\beta}{p_1}}f\left( y \right) |y|_{p}^{\frac{\beta}{p_1}}dy} \right| \\ &\leqslant \frac{1}{|x|_{p}^{n-\alpha}}\left( \int_{|y|_p<|x|_p}{|y|_{p}^{-\frac{\beta p_{1}^{\prime}}{p_1}}dy} \right) ^{\frac{1}{p_{1}^{\prime}}}\left( \int_{|y|_p<|x|_p}{\left| f\left( y \right) \right|^{p_1}\left| y \right|_{p}^{\beta}dy} \right) ^{\frac{1}{p_1}} \\ &\leqslant \frac{1}{|x|_{p}^{n-\alpha}}\left( \sum_{i=-\infty}^{\log _p|x|_p-1}{\int_{S_i}{\left| y \right|_{p}^{-\frac{\beta p_{1}^{\prime}}{p_1}}dy}} \right) ^{\frac{1}{p_{1}^{\prime}}}\left( \int_{\mathbb{Q} _{p}^{n}}{\left| f\left( y \right) \right|^{p_1}\left| y \right|_{p}^{\beta}dy} \right) ^{\frac{1}{p_1}} \\ &=|x|_{p}^{\alpha -n}\times \left( \sum_{i=-\infty}^{\log _p|x|_p-1}{p^{-\frac{i\beta p_{1}^{\prime}}{p_1}}\int_{S_i}{dy}} \right) ^{\frac{1}{p_{1}^{\prime}}}\left\| f \right\| _{L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})} \\ &=|x|_{p}^{\alpha -n}\left( \left( 1-p^{-n} \right) \sum_{i=-\infty}^{\log _p|x|_p-1}{p^{i(n-\frac{\beta}{p_1-1})}} \right) ^{\frac{1}{p_{1}^{\prime}}}\left\| f \right\| _{L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})} \\ &=\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_{1}^{\prime}}}\left\| f \right\| _{L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})}\left| x \right|_{p}^{-\frac{n}{p_1}-\frac{\beta}{p_1}+\alpha}=C_{p_1,n,\beta ,f}\left| x \right|_{p}^{-\frac{n}{p_1}-\frac{\beta}{p_1}+\alpha}, \end{align}\] where \[C_{p_1,n,\beta ,f}=\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_{1}^{\prime}}}\left\| f \right\| _{L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})}.\]

Noticing \(|\mathcal{H} _{\alpha}f\left( x \right) |\leqslant C_{p_1,n,\beta ,f}\left| x \right|_{p}^{-\frac{n}{p_1}-\frac{\beta}{p_1}+\alpha}\), we have \(\left\{ x:|\mathcal{H} _{\alpha}f\left( x \right) |>\lambda \right\} \subset \{ x:C_{p_1,n,\beta ,f}\left| x \right|_{p}^{-\frac{n}{p_1}-\frac{\beta}{p_1}+\alpha}>\lambda \}\).

Since \[n+\gamma >0 \,\,\,\text{and} \,\,\,\frac{\gamma +n}{q}+\alpha =\frac{\beta +n}{p_1},\] we have \[\begin{align} \left\| \mathcal{H} _{\alpha}f \right\| _{L^{q,\infty}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma})} &=\underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\left\{ x:|\mathcal{H} _{\alpha}f\left( x \right) |>\lambda \right\}}\left( x \right) \left| x \right|_{p}^{\gamma}dx} \right) ^{\frac{1}{q}} \\ &\leqslant \underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\{x:C_{p,n,\beta ,f}\left| x \right|_{p}^{-\frac{Q}{p_1}-\frac{\beta}{p_1}+\alpha}>\lambda \}}\left( x \right) \left| x \right|^{\gamma}_pdx} \right) ^{\frac{1}{q}} \\ &=\underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \int_{|x|_p<(\frac{C_{p_1,n,\beta ,f}}{\lambda})^{\frac{q}{n+\gamma}}}{\left| x \right|_{p}^{\gamma}dx} \right) ^{\frac{1}{q}}=\underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \prod_{i=-\infty}^{\log _p(\frac{C_{p_1,n,\beta ,f}}{\lambda})^{\frac{q}{n+\gamma}}-1}{p^{i\gamma}\int_{S_i}{dx}} \right) ^{\frac{1}{q}} \\ &=\underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \frac{\left( 1-p^{-n} \right) \left( C_{p_1,n,\beta ,f}/\lambda \right) ^q}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{1}{q}}=\,\,C_{p_1,n,\beta ,f}\times \left( \frac{\left( 1-p^{-n} \right)}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{1}{q}} \\ &=\left( \frac{\left( 1-p^{-n} \right)}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{1}{q}}\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_{1}^{\prime}}}\left\| f \right\| _{L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})}. \end{align}\] Thus \[\left\| \mathcal{H} _{\alpha} \right\| _{L^{p_1}\left( \mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta} \right) \rightarrow L^{q,\infty}\left( \mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma} \right)}\leqslant \left( \frac{\left( 1-p^{-n} \right)}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{1}{q}}\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_{1}^{\prime}}}.\] On the other hand, let \[f_0\left( x \right) =|x|_{p}^{-\frac{\beta}{p_1-1}}\chi _{\{x:\left| x \right|_p<1\}}\left( x \right) .\] Noticing \(n+\beta ( 1-\frac{p_1}{p_1-1} ) =n-\frac{\beta}{p_1-1}>0\), we have \[\begin{align} \left\| f_0 \right\| _{L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})} &=\left( \int_{\mathbb{Q} _{p}^{n}}{|\left| x \right|_{p}^{-\frac{\beta}{p_1-1}}\chi _{\{x:\left| x \right|_p<1\}}\left. \left( x \right) \right|^{p_1}\left| x \right|_{p}^{\beta}dx} \right) ^{\frac{1}{p_1}}=\left( \int_{|x|_p<1}{\left| x \right|_{p}^{-\frac{\beta p_1}{p_1-1}}\left| x \right|_{p}^{\beta}dx} \right) ^{\frac{1}{p_1}}\\ &=\left( \left( 1-p^{-n} \right) \sum_{i=-\infty}^{-1}{p^{i( n+\beta -\frac{\beta p_1}{p_1-1} )}} \right) ^{\frac{1}{p_1}}=\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_1}}<\infty . \end{align}\] So we have proved that \(f_0\in L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})\). Then we calculate \(\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right)\). \[\begin{align} \mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) &=\frac{1}{|x|_{p}^{n-\alpha}}\int_{|y|_p<|x|_p}{|y|_{p}^{-\frac{\beta}{p_1-1}}\chi _{\{y:\left| y \right|_p<1\}}\left( y \right) dy}\\ &=\frac{1}{|x|_{p}^{n-\alpha}}\left\{ \begin{array}{l} \int_{|y|_p<|x|_p}{|y|_{p}^{-\frac{\beta}{p_1-1}}dy,\,\left| x \right|_p\leqslant1}\\ \int_{|y|_h<1}{|y|_{p}^{-\frac{\beta}{p_1-1}}dy,\,\,\,\,\,\,\left| x \right|_p>1}\\ \end{array} \right. =\left\{ \begin{array}{l} \sum_{i=-\infty}^{\log _p|x|_p-1}{\int_{S_i}{|y|_{p}^{-\frac{\beta}{p_1-1}}dy,\,\left| x \right|_p\leqslant1}}\\ \sum_{i=-\infty}^{-1}{\int_{S_i}{|y|_{p}^{-\frac{\beta}{p_1-1}}dy,\,\,\,\,\,\,\,\,\,\,\left| x \right|_p>1}}\\ \end{array} \right. \\ &=\left( 1-p^{-n} \right) \begin{cases} \sum_{i=-\infty}^{\log _p|x|_p-1}{p^{i( n-\frac{\beta}{p_1-1})},\,\left| x \right|_p\leqslant1}\\ \sum_{i=-\infty}^{-1}{p^{i( n-\frac{\beta}{p_1-1} )},\,\,\,\,\,\,\,\,\,\,\left| x \right|_p>1}\\ \end{cases}=\frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}}\begin{cases} |x|_{p}^{\alpha -\frac{\beta}{p_1-1}},\,\left| x \right|_p\leqslant1\\ |x|_{p}^{\alpha -n},\,\,\,\,\,\,\,\,\,\left| x \right|_p>1\\ \end{cases}. \end{align}\] For convenience, we set \(C_{p_1,n,\beta}=\frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}}\), we have \[\left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\} =\{\left| x \right|_p\leqslant 1:C_{p_1,n,\beta}\left| x \right|_{p}^{\alpha -\frac{\beta}{p_1-1}}>\lambda \}\cup \{\left| x \right|_p>1:C_{p_1,n,\beta}\left| x \right|_{p}^{\alpha -n}>\lambda \}.\]

When \(0<\lambda <C_{p_1,n,\beta}\), noticing \(\alpha <\frac{\beta}{p_1-1}\,\,\) and \(\beta <n\left( p_1-1 \right)\), we have \(\alpha <n\) and \[\begin{align} \left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\} &=\left\{\left| x \right|_p\leqslant 1:\left| x \right|_{p}^{\frac{\beta}{p_1-1}-\alpha}<\frac{C_{p_1,n,\beta}}{\lambda}\right\}\cup \left\{ \left| x \right|_p>1:\left| x \right|_p<\left( \frac{C_{p_1,n,\beta}}{\lambda} \right) ^{\frac{1}{n-\alpha}} \right\} \\ &=\{\left| x \right|_p\leqslant 1\}\cup \left\{ \left| x \right|_p>1:\left| x \right|_p<\left( \frac{C_{p_1,n,\beta}}{\lambda} \right) ^{\frac{1}{n-\alpha}} \right\} \\ &=\left\{ x:\left| x \right|_p<\left( \frac{C_{p_1,n,\beta}}{\lambda} \right) ^{\frac{1}{n-\alpha}} \right\} . \end{align}\]

When \(\lambda \geqslant C_{p_1,n,\beta}\), noticing \(\alpha <\frac{\beta}{p_1-1}\) and \(\beta <n\left( p_1-1 \right)\), we have \(\alpha <n\) and \[\begin{align} \left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\} &=\left\{\left| x \right|_p\leqslant 1:\left| x \right|_{p}^{\frac{\beta}{p_1-1}-\alpha}<\frac{C_{p_1,n,\beta}}{\lambda}\right\}\cup \varnothing \\ &=\left\{ x:\left| x \right|_p<\left( \frac{C_{p_1,n,\beta}}{\lambda} \right) ^{\frac{1}{\frac{\beta}{p_1-1}-\alpha}} \right\} . \end{align}\] Based on the above analysis, we have \[\begin{align} &\left\| \mathcal{H} _{\alpha}\left( f_0 \right) \right\| _{L^{q,\infty}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma})} \\ &=\max \left\{ \underset{0<\lambda <C_{p_1,n,\beta}}{\mathrm{sup}}\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\}}\left( x \right) |x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{1}{q}},\underset{C_{p_1,n,\beta}\leqslant \lambda}{\mathrm{sup}}\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\}}\left( x \right) |x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{1}{q}} \right\} \\ &=:\max \left\{ M_1,M_2 \right\} . \end{align}\] Now we first calculate \(M_1\). Since \[\left\| f_0 \right\| _{L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})}=\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_1}},\quad \gamma >-n\] and \[1-\frac{n+\gamma}{\left( n-\alpha \right) q}=1-\frac{1}{n-\alpha}( \frac{\beta +n}{p_1}-\alpha ) =\frac{n\left( p_1-1 \right) -\beta}{p_1\left( n-\alpha \right)}>0,\] we have \[\begin{align} M_1&=\underset{0<\lambda <C_{p_1,n,\beta}}{\mathrm{sup}}\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\}}\left( x \right) |x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{1}{q}}=\underset{0<\lambda <C_{p_1,n,\beta}}{\mathrm{sup}}\lambda \left( \int_{\left| x \right|_p<(\frac{C_{p_1,n,\beta}}{\lambda})^{\frac{1}{n-\alpha}}}{|x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{1}{q}} \\ &=\underset{0<\lambda <C_{p_1,n,\beta}}{\mathrm{sup}}\lambda \left( \left( 1-p^{-n} \right) \sum_{i=-\infty}^{\frac{1}{n-\alpha}\log _p\frac{C_{p_1,n,\beta}}{\lambda}-1}{p^{i\left( n+\gamma \right)}} \right) ^{\frac{1}{q}} \\ &=\underset{0<\lambda <C_{p_1,n,\beta}}{\mathrm{sup}}\lambda \left( \frac{1-p^{-n}}{1-p^{-n-\gamma}}\times p^{( \frac{1}{n-\alpha}\log _p\frac{C_{p_1,n,\beta}}{\lambda}-1 ) \left( n+\gamma \right)} \right) ^{\frac{1}{q}} \\ &=\underset{0<\lambda <C_{p_1,n,\beta}}{\mathrm{sup}}\left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}\left( C_{p_1,n,\beta} \right) ^{\frac{n+\gamma}{\left( n-\alpha \right) q}}\lambda ^{1-\frac{n+\gamma}{\left( n-\alpha \right) q}} \\ &=\left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}C_{p_1,Q,\beta}=\left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_1}+\frac{1}{p_1^{\prime}}} \\ &=\left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_1^{\prime}}}\left\| f_0 \right\| _{L^p(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})}. \end{align}\] Then we calculate \(M_2\), noticing \(\left\| f_0 \right\| _{L^p(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})}=\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_1}}\), \(\gamma >-n\) and \[1-\frac{n+\gamma}{( \frac{\beta}{p_1-1}-\alpha) q}=1-\frac{1}{\frac{\beta}{p_1-1}-\alpha}( \frac{\beta +n}{p_1}-\alpha ) =\frac{\beta -n\left( p_1-1 \right)}{p_1(\frac{\beta}{p_1-1}-\alpha )(p_1-1)}<0,\] we have \[\begin{align} M_2&=\underset{C_{p_1,n,\beta}\geqslant \lambda}{\mathrm{sup}}\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\}}\left( x \right) |x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{1}{q}}=\underset{C_{p_1,n,\beta}\geqslant \lambda}{\mathrm{sup}}\lambda \left( \int_{\left| x \right|_p<(\frac{C_{p_1,n,\beta}}{\lambda})^{\frac{1}{\frac{\beta}{p_1-1}-\alpha}}}{|x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{1}{q}} \\ &=\underset{C_{p_1,n,\beta}\geqslant \lambda}{\mathrm{sup}}\lambda \left( \left( 1-p^{-n} \right) \sum_{i=-\infty}^{\frac{1}{(\frac{\beta}{p_1-1}-\alpha )}\log _p\frac{C_{p_1,n,\beta}}{\lambda}-1}{p^{i\left( n+\gamma \right)}} \right) ^{\frac{1}{q}} \\ &=\underset{C_{p_1,n,\beta}\geqslant \lambda}{\mathrm{sup}}\lambda \left( \frac{1-p^{-n}}{1-p^{-n-\gamma}}\times p^{(\frac{1}{(\frac{\beta}{p_1-1}-\alpha )}\log _p\frac{C_{p_1,n,\beta}}{\lambda}-1)\left( n+\gamma \right)} \right) ^{\frac{1}{q}} \\ &=\underset{C_{p_1,n,\beta}\geqslant \lambda}{\mathrm{sup}}\lambda \left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}\left( C_{p_1,n,\beta} \right) ^{\frac{n+\gamma}{(\frac{\beta}{p_1-1}-\alpha )q}}\lambda ^{1-\frac{n+\gamma}{(\frac{\beta}{p_1-1}-\alpha )q}} \\ &=\left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}C_{p_1,n,\beta}=\left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_1}+\frac{1}{p_1^{\prime}}} \\ &=\left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_1^{\prime}}}\left\| f_0 \right\| _{L^p(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})}. \end{align}\] Its easy to see that \(M_1=M_2\), and then \[\left\| \mathcal{H} _{\alpha} \right\| _{L^{p_1}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\beta})\rightarrow \,\,L^{q,\infty}(\mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma})}=\left( \frac{\left( 1-p^{-n} \right) p^{-n-\gamma}}{1-p^{-n-\gamma}} \right) ^{\frac{1}{q}}\left( \frac{\left( 1-p^{-n} \right) p^{\frac{\beta}{p_1-1}-n}}{1-p^{\frac{\beta}{p_1-1}-n}} \right) ^{\frac{1}{p_{1}^{\prime}}}.\] This finishes the proof of Theorem 1. ◻

Proof of Theorem 2:. It is easy to see that \[\left| \mathcal{H} _{\alpha}f\left( x \right) \right|=\left| \frac{1}{|x|_{p}^{n-a}}\int_{|y|_p<|x|_p}{f\left( y \right) dy} \right|\leqslant \left| \frac{1}{|x|_{p}^{n-a}}\int_{\mathbb{Q} _{p}^{n}}{f\left( y \right) dy} \right|=|x|_{p}^{\alpha -n}\left\| f \right\| _{L^1\left( \mathbb{Q} _{p}^{n} \right)}.\] Notice \(\left| \mathcal{H} _{\alpha}f\left( x \right) \right|\leqslant |x|_{p}^{\alpha -n}\left\| f \right\| _{L^1( \mathbb{Q} _{p}^{n})}\), and we have \(\left\{ x:\left| \mathcal{H} _{\alpha}f\left( x \right) \right|>\lambda \right\} \subset \{x:|x|_{p}^{\alpha -n}\left\| f \right\| _{L^1( \mathbb{Q} _{p}^{n} )}>\lambda \}\). Since \(n-\alpha >0\) and \(n+\gamma >0\), we have \[\begin{align} &\left\| \mathcal{H} _{\alpha}f \right\| _{L^{\left( n+\gamma \right) /\left( n-\alpha \right) ,\infty}\left( \mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma} \right)}=\underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\left\{ x:|\mathcal{H} _{\alpha}f\left( x \right) |>\lambda \right\}}\left( x \right) |x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{n-\alpha}{n+\gamma}} \\ &\leqslant \underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\{x:|x|_{p}^{\alpha -n}\left\| f \right\| _{L^1( \mathbb{Q} _{p}^{n} )}>\lambda \}}\left( x \right) |x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{n-\alpha}{n+\gamma}}=\underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \int_{\left| x \right|_p<(\left\| f \right\| _{L^1( \mathbb{Q} _{p}^{n} )}/\lambda )^{\frac{1}{n-\alpha}}}{|x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{n-\alpha}{n+\gamma}} \\ &=\underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \sum_{i=-\infty}^{\frac{1}{n-\alpha}\log _p\frac{\left\| f \right\| _{L^1( \mathbb{Q} _{p}^{n} )}}{\lambda}-1}{\int_{S_i}{|x|_{p}^{\gamma}dx}} \right) ^{\frac{n-\alpha}{n+\gamma}}=\underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \left( 1-p^{-n} \right) \sum_{i=-\infty}^{\frac{1}{n-\alpha}\log _p\frac{\left\| f \right\| _{L^1( \mathbb{Q} _{p}^{n} )}}{\lambda}-1}{p^{i\left( n+\gamma \right)}} \right) ^{\frac{n-\alpha}{n+\gamma}} \\ &=\underset{\lambda >0}{\mathrm{sup}}\,\,\lambda \left( \left( 1-p^{-n} \right) \frac{p^{(\frac{1}{n-\alpha}\log _p\frac{\left\| f \right\| _{L^1(\mathbb{Q} _{p}^{n})}}{\lambda}-1)\left( n+\gamma \right)}}{1-p^{-n-\gamma}} \right) ^{\frac{n-\alpha}{n+\gamma}}=\left( \frac{1-p^{-n}}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{n-\alpha}{n+\gamma}}\left\| f \right\| _{L^1(\mathbb{Q} _{p}^{n})}. \end{align}\] Thus \[\left\| \mathcal{H} _{\alpha}f\left( x \right) \right\| _{L^{( n+\gamma) /\left( n-\alpha \right) ,\infty}( \mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma} )}\leqslant \left( \frac{1-p^{-n}}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{n-\alpha}{n+\gamma}}\left\| f \right\| _{L^1( \mathbb{Q} _{p}^{n})}.\] On the other hand, let \(f_0\left( x \right) =\chi _{\{ x:\left| x \right|_p< 1 \}}\left( x \right)\), we have \[\left\| f_0 \right\| _{L^1(\mathbb{Q} _{p}^{n})}=\int_{\mathbb{Q} _{p}^{n}}{\chi _{\{x:\left| x \right|_p<1\}}\left( x \right) dx=\int_{\left| x \right|_p<1}{dx}}=\sum_{i=-\infty}^{-1}{\int_{S_i}{dx}}=\left( 1-p^{-n} \right) \sum_{i=-\infty}^{-1}{p^{in}}=p^{-n}<\infty ,\] thus \(f_0\in L^1( \mathbb{Q} _{p}^{n})\) and \[\begin{align} \mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) &=\frac{1}{|x|_{p}^{n-a}}\int_{|y|_p<|x|_p}{\chi _{\left\{ y:|y|_p<1 \right\}}\left( y \right) dy} \\ &=\frac{1}{|x|_{p}^{n-a}}\left\{ \begin{array}{l} \int_{|y|_p<|x|_p}{dy,\,|x|_p}\leqslant 1\\ \int_{|y|_p<1}{dy,\,\,\,\,\,\,|x|_p}>1\\ \end{array} \right. =p^{-n}\left\{ \begin{array}{l} |x|_{p}^{\alpha},\,\,\,\,\,\,\,\,|x|_p\leqslant 1\\ |x|_{p}^{\alpha -n},\,\,|x|_p>1\\ \end{array} \right. . \end{align}\] Denote \[\left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\} =\left\{ |x|_p\leqslant 1:|x|_{p}^{\alpha}p^{-n}>\lambda \right\} \cup \{|x|_p>1:|x|_{p}^{\alpha -n}p^{-n}>\lambda \}.\]

When \(\lambda \geqslant p^{-n}\), noticing \(0<\alpha <n\), we have \[\left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\} =\left\{ |x|_p\leqslant 1:|x|_p>\left( \frac{\lambda}{p^{-n}} \right) ^{\frac{1}{\alpha}} \right\} \cup \left\{ |x|_p>1:|x|_p<\left( \frac{p^{-n}}{\lambda} \right) ^{\frac{1}{n-\alpha}} \right\} =\varnothing.\]

When \(0<\lambda <p^{-n}\), noticing \(0<\alpha <n\), we have \[\left\{ x:|\mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) |>\lambda \right\} =\left\{ x:\left( \frac{\lambda}{p^{-n}} \right) ^{\frac{1}{\alpha}}<\left| x \right|_p<\left( \frac{p^{-n}}{\lambda} \right) ^{\frac{1}{n-\alpha}} \right\} .\] We have \[\begin{align} &\left\| \mathcal{H} _{\alpha}\left( f_0 \right) \left( x \right) \right\| _{L^{\left( n+\gamma \right) /\left( n-\alpha \right) ,\infty}\left( \mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma} \right)} \\ &=\max \left\{ \underset{0<\lambda <p^{-n}}{\mathrm{sup}}\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\left\{ x:|\mathcal{H} _{\alpha}f_0\left( x \right) |>\lambda \right\}}\left( x \right) |x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{n-\alpha}{n+\gamma}},\underset{\lambda \geqslant p^{-n}}{\mathrm{sup}}\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\left\{ x:|\mathcal{H} _{\alpha}f_0\left( x \right) |>\lambda \right\}}\left( x \right) |x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{n-\alpha}{n+\gamma}} \right\} \\ &=:\max \left\{ M_3,M_4 \right\} . \end{align}\] When \(\lambda \geqslant p^{-n}\), then \(\left\{ x:|\mathcal{H} _{\alpha}f_0\left( x \right) |>\lambda \right\} =\varnothing\), and we have \(M_4=0\), then we only need to calculate \(M_3\). In addition, noticing \[n+\gamma >0,\quad 0<\alpha <n,\quad \left\| f_0 \right\| _{L^1( \mathbb{Q} _{p}^{n} )}=p^{-n},\] we have \[\begin{align} M_3&=\underset{0<\lambda <p^{-n}}{\mathrm{sup}}\lambda \left( \int_{\mathbb{Q} _{p}^{n}}{\chi _{\left\{ x:|\mathcal{H} _{\alpha}f_0\left( x \right) |>\lambda \right\}}\left( x \right) |x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{n-\alpha}{n+\gamma}} =\underset{0<\lambda <p^{-n}}{\mathrm{sup}}\lambda \left( \int_{(\lambda p^n)^{\frac{1}{\alpha}}<\left| x \right|_p<(\frac{1}{p^n\lambda})^{\frac{1}{n-\alpha}}}{|x|_{p}^{\gamma}dx\,\,} \right) ^{\frac{n-\alpha}{n+\gamma}} \\ &=\underset{0<\lambda <p^{-n}}{\mathrm{sup}}\lambda \left( \sum_{i=\log _p( \lambda p^n ) ^{\frac{1}{\alpha}}+1}^{\log _p( \frac{1}{\lambda p^n}) ^{\frac{1}{n-\alpha}}-1}{\int_{S_i}{|x|_{p}^{\gamma}dx}} \right) ^{\frac{n-\alpha}{n+\gamma}}=\underset{0<\lambda <p^{-n}}{\mathrm{sup}}\lambda \left( \left( 1-p^{-n} \right) \sum_{i=\log _p(\lambda p^n)^{\frac{1}{\alpha}}+1}^{\log _p(\frac{1}{\lambda p^n})^{\frac{1}{n-\alpha}}-1}{p^{i\left( n+\gamma \right)}} \right) ^{\frac{n-\alpha}{n+\gamma}} \\ &=\underset{0<\lambda <p^{-n}}{\mathrm{sup}}\lambda \left( \left( 1-p^{-n} \right) \frac{p^{(\log _p(\frac{1}{\lambda p^n})^{\frac{1}{n-\alpha}}-1)\left( n+\gamma \right)}(1-\left( p^{-n-\gamma} \right) ^{\log _p(\frac{1}{\lambda p^n})^{\frac{1}{n-\alpha}}-1-(\log _p(\lambda p^n)^{\frac{1}{\alpha}}+1)+1})}{1-p^{-n-\gamma}} \right) ^{\frac{n-\alpha}{n+\gamma}} \\ &=\underset{0<\lambda <p^{-n}}{\mathrm{sup}}\lambda \left( \left( 1-p^{-n} \right) \times \frac{( \frac{1}{\lambda p^n} ) ^{\frac{n+\gamma}{n-\alpha}}( 1-\left( p^{-n-\gamma} \right) ^{\log _p\left( \lambda p^n \right) ^{\frac{1}{\alpha -n}-\frac{1}{\alpha}}-1} )}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{n-\alpha}{n+\gamma}} \\ &=\underset{0<\lambda <p^{-n}}{\mathrm{sup}}\left( 1-\frac{\left( \lambda p^n \right) ^{\frac{n}{\alpha}}}{p^{-n-\gamma}} \right) \left( \frac{\left( 1-p^{-n} \right)}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{n-\alpha}{n+\gamma}}p^{-n}=\left( \frac{\left( 1-p^{-n} \right)}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{n-\alpha}{n+\gamma}}\left\| f_0 \right\| _{L^1\left( \mathbb{Q} _{p}^{n} \right)}. \end{align}\] Thus \[\left\| \mathcal{H} _{\alpha} \right\| _{L^1\left( \mathbb{Q} _{p}^{n} \right) \rightarrow L^{\left( n+\gamma \right) /\left( n-\alpha \right) ,\infty}\left( \mathbb{Q} _{p}^{n},\left| x \right|_{p}^{\gamma} \right)}=\left( \frac{\left( 1-p^{-n} \right)}{\left( 1-p^{-n-\gamma} \right) p^{n+\gamma}} \right) ^{\frac{n-\alpha}{n+\gamma}}.\] This finishes the proof of Theorem 2. Notice that theorem 2 no longer holds when \(\alpha=0\). ◻

3 Sharp bounds for the \(\boldsymbol{p}\)-adic \(\boldsymbol{m}\)-linear \(\boldsymbol{n}\)-dimensional Hardy and Hilbert operators on \(\boldsymbol{p}\)-adic function space↩︎

In this section, we will study the \(p\)-adic \(m\)-linear \(n\)-dimensional integral operator with a kernel. Let \(K:\mathbb{Q} _{p}^{n}\times \cdots \times \mathbb{Q} _{p}^{n}\rightarrow \left( 0,\infty \right)\) be a measurable kernel, it satisfies that \[\begin{align} \label{2461} C^p=\int_{\mathbb{Q} _{p}^{n}}{\cdots \int_{\mathbb{Q} _{p}^{n}}{K\left( y_1,...,y_m \right) \prod_{i=1}^m{\left| y_i \right|_{p}^{-\alpha _i}}}}dy_1\cdots dy_m<\infty, \end{align}\tag{1}\] where \(\alpha_j\) is pre-defined indicator and some fixed indices, \(j=1,2,...,m\). The \(p\)-adic \(m\)-linear \(n\)-dimensional integral operator with a kernel is defined by \[\begin{align} T^p\left( f_1,...,f_m \right) \left( x \right) =\int_{\mathbb{Q} _{p}^{n}}{\cdots}\int_{\mathbb{Q} _{p}^{n}}{K\left( y_1,...,y_m \right) f_1( \left| x \right|_{p}^{-1}y_1 ) \cdots}f_m( \left| x \right|_{p}^{-1}y_m ) dy_1\cdots dy_m, \end{align}\] where \(x\in \mathbb{Q}_p ^n\backslash \left\{ 0 \right\}\) and \(f_j\) is a measurable function on \(\mathbb{Q} _{p}^{n}\) with \(j=1,2,...,m\). Note that \(T^p\) is in fact an integral operator having a homogeneous kernel \(K\) of degree \(-mn\).

We will give the sharp bounds for the \(p\)-adic \(m\)-linear \(n\)-dimensional integral operator with a kernel on \(p\)-adic weighted space \(H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )\). Finally, by taking a particular kernel \(K\) in operator \(C^p\) defined by (1 ), we can obtain the sharp bounds for the \(p\)-adic Hardy and Hilbert operators. Our results in this section are as follows.

Theorem 3. Let \(m\in \mathbb{N}\), \(\alpha \in \mathbb{R}\) and \(\alpha=\alpha_1+\cdots+\alpha_m\) with \(\alpha _j\in \mathbb{R}\) \((j=1,2,...,m)\). \(f_j\) be a measurable function in \(H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )\). Then \[\begin{align} \left\| T^p\left( f_1,...,f_m \right) \left( x \right) \right\| _{\prod\nolimits_{j=1}^m{H_{\alpha _j}^{\infty}(\mathbb{Q} _{p}^{n})}\rightarrow H_{\alpha}^{\infty}(\mathbb{Q} _{p}^{n})}=C^p, \end{align}\] where \(C^p\) is the constant defined by (1 ).

Corollary 1. Assume that the real paramenters \(\alpha\), \(\alpha_j\) with \(j=1,2,...,m\) as same as in Theorem 1, \(f_j\) be a measurable function in \(H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )\). Assume also that \(\alpha_j<n\), then \[\begin{align} \left\| T_{1}^{p}\left( f_1,...,f_m \right) \left( x \right) \right\| _{\prod\nolimits_{j=1}^m{H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )}\rightarrow H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )}=\frac{\left( 1-p^{-n} \right) ^m}{\prod\nolimits_{j=1}^m{\left( 1-p^{\alpha _j-n} \right)}}. \end{align}\]

Corollary 2. Assume that the real paramenters \(\alpha\), \(\alpha_j\) with \(j=1,2,...,m\) as same as in Theorem 1, \(f_j\) be a measurable function in \(H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n})\). Assume also that \(\alpha_j<n\) and \(\alpha>0\), then \[\begin{align} &\left\| T_{2}^{p}\left( f_1,...,f_m \right) \left( x \right) \right\| _{\prod\nolimits_{j=1}^m{H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )}\rightarrow H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )} \\ &=(1-p^{-n})^m\sum_{k_1=-\infty}^{+\infty}{\sum_{k_2=-\infty}^{+\infty}{\cdots}\sum_{k_m=-\infty}^{+\infty}{\frac{1}{(1+p^{k_1n}+\cdots +p^{k_mn})^m}}}\prod_{j=1}^m{p^{k_j(-\alpha _j+n)}} \\ &\leqslant \frac{\left( 1-p^{-n} \right) ^m\left( 1-p^{-mn} \right)}{\left( 1-p^{-\alpha} \right) \prod\nolimits_{j=1}^m{\left( 1-p^{\alpha _j-n} \right)}}<\infty. \end{align}\]

3.1 Sharp bound for \(\boldsymbol{p}\)-adic integral operator with a kernel↩︎

Proof of Theroem 3:. Since the proof of the case when \(m=1\) is similar to and even simpler than of the case when \(m>1\), for simplicity, we will only give the proof of case when \(m>1\). Using the definition of weighted-type space \(H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )\), we can infer that \[\begin{align} &\left\| T^p\left( f_1,...,f_m \right) \left( x \right) \right\| _{H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )} \\ &=\mathrm{ess} \underset{x\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}\left| x \right|_{p}^{\alpha}\left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) f_1( \left| x \right|_{p}^{-1}y_1 ) \cdots f_m( \left| x \right|_{p}^{-1}y_m ) dy_1\cdots dy_m} \right| \\ &=\mathrm{ess} \underset{x\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}\left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) \left| x \right|_{p}^{\alpha _1}f_1( \left| x \right|_{p}^{-1}y_1 ) \cdots \left| x \right|_{p}^{\alpha _m}f_m( \left| x \right|_{p}^{-1}y_m ) dy_1\cdots dy_m} \right| \\ &=\mathrm{ess} \underset{x\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}\left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) \prod_{j=1}^m{\frac{1}{\left| y_j \right|_{p}^{\alpha _j}}| \left| x \right|_{p}^{-1}y_j |_{p}^{\alpha _j}f_j( \left| x \right|_{p}^{-1}y_j )}dy_1\cdots dy_m} \right| \\ &\leqslant \left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) \left( \prod_{j=1}^m{\left| y_j \right|_{p}^{-\alpha _j}} \right) \left( \prod_{j=1}^m{\mathrm{ess} \underset{x_j,y_j\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}| | x_j |_{p}^{-1}y_j |_{p}^{\alpha _j}| f_j( \left| x_j \right|_{p}^{-1}y_j ) |} \right) dy_1\cdots dy_m} \right| \\ &=\left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) \left( \prod_{j=1}^m{\left| y_j \right|_{p}^{-\alpha _j}} \right) \left( \prod_{j=1}^m{\mathrm{ess} \underset{t_j\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}\left| t_j \right|_{p}^{\alpha _j}\left| f_j\left( t_j \right) \right|} \right) dy_1\cdots dy_m} \right| \\ &=\left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) \prod_{j=1}^m{\left| y_j \right|_{p}^{-\alpha _j}}dy_1\cdots dy_m} \right|\times \prod_{j=1}^m{\left\| f_j \right\| _{H_{\alpha _j}^{\infty}\left( \mathbb{Q} _{p}^{n} \right)}}=C^p\prod_{j=1}^m{\left\| f_j \right\| _{H_{\alpha _j}^{\infty}\left( \mathbb{Q} _{p}^{n} \right)}}, \end{align}\] for ever \(f_j \in \prod\nolimits_{j=1}^m{H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )}\), from which by taking the supremum in \(H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )\), we can infer that \[\begin{align} \left\| T^p \right\| _{\prod\nolimits_{j=1}^m{H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )}\rightarrow H_{\alpha}^{\infty}\left( \mathbb{Q} _{p}^{n} \right)}\leqslant C^p \end{align}\] and consequently the boundedness of the operator.

On the other hand, by taking \[f_j\left( x \right) =\begin{cases} 1/\left| x \right|_{p}^{\alpha _j}, x\ne 0.\\ 0, x=0.\\ \end{cases},\] then it is clear that for \(j=1,2,...,m\), we have \[\begin{align} \left\| f_j\left( x \right) \right\| _{H_{\alpha _j}^{\infty}\left( \mathbb{Q} _{p}^{n} \right)}=\mathrm{ess} \underset{x\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}\left| x \right|_{p}^{\alpha _j}\times \frac{1}{\left| x \right|_{p}^{\alpha _j}}=1<\infty. \end{align}\] After performing some straightforward calculations, it follows that \[\begin{align} &\left\| T^p\left( f_1,...,f_m \right) \left( x \right) \right\| _{H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )} \\ &=\mathrm{ess} \underset{x\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}\left| x \right|_{p}^{\alpha}\left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) | \left| x \right|_{p}^{-1}y_1 |_{p}^{-\alpha _1}\cdots | \left| x \right|_{p}^{-1}y_m |_{p}^{-\alpha _m}dy_1\cdots dy_m} \right| \\ &=\mathrm{ess} \underset{x\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}\left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) \left| x \right|_{p}^{\alpha}\left| x \right|_{p}^{-\alpha _1}\left| y_1 \right|_{p}^{-\alpha _1}\cdots \left| x \right|_{p}^{-\alpha _m}\left| y_m \right|_{p}^{-\alpha _m}dy_1\cdots dy_m} \right| \\ &=\left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) \left| y_1 \right|_{p}^{-\alpha _1}\cdots \left| y_m \right|_{p}^{-\alpha _m}dy_1\cdots dy_m} \right| \\ &=\left| \int_{\mathbb{Q} _{p}^{mn}}{K\left( y_1,...,y_m \right) \prod_{j=1}^m{\left| y_j \right|_{p}^{-\alpha _j}}dy_1\cdots dy_m} \right|\times \prod_{j=1}^m{\mathrm{ess} \underset{x\in \mathbb{Q} _{p}^{n}}{\mathrm{sup}}\left| x \right|_{p}^{\alpha _1}\times \left| \frac{1}{\left| x \right|_{p}^{\alpha _1}} \right|} \\ &=C^p\prod_{j=1}^m{\left\| f_j \right\| _{H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )}}. \end{align}\] For \(x\ne 0\), where we used the condition \(\alpha =\alpha _1+\cdots +\alpha _m\). This finishes the proof of Theorem 3. ◻

3.2 Sharp bound for \(\boldsymbol{p}\)-adic Hardy operator↩︎

Proof of Corollary 1:. Next, we refer to the methods in [24] to solve it. If we take the kernel \[K\left( y_1,...,y_m \right) =\chi _{\{ \left| \left( y_1,...,y_m \right) \right|_p\leqslant 1 \}}\left( y_1,...,y_m \right)\] in Theorems 3, by a change of variables, it is easy to verify that \(T^p=T_{1}^{p}\), and then \(T_{1}^{p}\) can be denoted by \[T_{1}^{p}=\int_{\left| \left( y_1,...,y_m \right) \right|_p\leqslant 1}{f_1( \left| x \right|_{p}^{-1}y_1 ) \cdots f_m( \left| x \right|_{p}^{-1}y_m ) dy_1\cdots dy_m},\] respectively, then all things reduce to calculating \[C_{1}^{p}=\int_{\left| \left( y_1,...,y_m \right) \right|_p\leqslant 1}{\prod_{j=1}^m{\left| y_j \right|_{p}^{-\alpha_j}}dy_1\cdots dy_m}.\] To calculate this integral, we divide the integral into \(m\) parts. Let \[\begin{align} &D_1=\{ \left( y_1,...,y_m \right) \in \mathbb{Q} _{p}^{n}\cdots \mathbb{Q} _{p}^{n}:\left| y_1 \right|_p\leqslant 1,\left| y_k \right|_p\leqslant \left| y_1 \right|_p,1<k\leqslant m \}, \\ &D_i=\{ \left( y_1,...,y_m \right) \in \mathbb{Q} _{p}^{n}\cdots \mathbb{Q} _{p}^{n}:\left| y_i \right|_p\leqslant 1,\left| y_j \right|_p<\left| y_i \right|_p,\left| y_k \right|_p\leqslant \left| y_i \right|_p,1\leqslant j<i<k\leqslant m \}, \\ &D_m=\{ \left( y_1,...,y_m \right) \in \mathbb{Q} _{p}^{n}\cdots \mathbb{Q} _{p}^{n}:\left| y_m \right|_p\leqslant 1,\left| y_j \right|_p<\left| y_m \right|_p,1\leqslant j<m \}. \end{align}\] It is clear that \[\bigcup_{j=1}^m{D_j=\{ \left( y_1,...,y_m \right) \in \mathbb{Q} _{p}^{n}\cdots \mathbb{Q} _{p}^{n}:\left| \left( y_1,...,y_m \right) \right|_p\leqslant 1 \}}\] and \(D_i\cap D_j=\varnothing \left( i\ne j \right)\). Let \[I_j:=\int_{D_j}{\prod_{k=1}^m{\left| y_k \right|_{p}^{-\alpha_k}}dy_1\cdots dy_m}.\] Then \[C_{1}^{p}=\sum_{j=1}^m{I_j:}=\sum_{j=1}^m{\int_{D_j}{\prod_{k=1}^m{\left| y_k \right|_{p}^{-\alpha_k}}dy_1\cdots dy_m}}.\] Now let us calculate \(I_j\), \(j=1,2,...,m\). Since \(\alpha_j<n\), then \(\alpha<mn\), so we have \[\begin{align} I_1&=\int_{D_1}{\prod_{k=1}^m{\left| y_k \right|_{p}^{-\alpha _k}dy_1}\cdots dy_m}\\ &=\int_{\left| y_1 \right|_p\leqslant 1}{\left( \int_{\left| y_2 \right|_p\leqslant \left| y_1 \right|_p}{\cdots \int_{\left| y_m \right|_p\leqslant \left| y_1 \right|_p}{\prod_{k=1}^m{\left| y_k \right|_{p}^{-\alpha _k}}dy_m}}\cdots dy_2 \right) dy_1}\\ &=\int_{\left| y_1 \right|_p\leqslant 1}{\left| y_1 \right|_{p}^{-\alpha _1}\left( \prod_{k=2}^m{\int_{\left| y_k \right|_p\leqslant \left| y_1 \right|_p}{\left| y_k \right|_{p}^{-\alpha _k}dy_k}} \right) dy_1}=\int_{\left| y_1 \right|_p\leqslant 1}{\left| y_1 \right|_{p}^{-\alpha _1}\prod_{k=2}^m{\left( \sum_{i=-\infty}^{\log _p\left| y_1 \right|_p}{\int_{S_i}{\left| y_k \right|_{p}^{-\alpha _k}dy_k}} \right) dy_1}}\\ &=\int_{\left| y_1 \right|_p\leqslant 1}{\left| y_1 \right|_{p}^{-\alpha _1}\prod_{k=2}^m{\left( \sum_{i=-\infty}^{\log _p\left| y_1 \right|_p}{p^{-i\alpha _k}}\times \int_{S_i}{dy_k} \right) dy_1}}=\int_{\left| y_1 \right|_p\leqslant 1}{\left| y_1 \right|_{p}^{-\alpha _1}\prod_{k=2}^m{\left( \left( 1-p^{-n} \right) \sum_{i=-\infty}^{\log _p\left| y_1 \right|_p}{p^{i\left( n-\alpha _k \right)}} \right) dy_1}}\\ &=\left( 1-p^{-n} \right) ^{m-1}\int_{\left| y_1 \right|_p\leqslant 1}{\left| y_1 \right|_{p}^{-\alpha _1}\prod_{k=2}^m{\left( \frac{p^{\left( n-\alpha _k \right) \log _p\left| y_1 \right|_p}}{1-p^{\alpha _k-n}} \right) dy_1}}=\frac{\left( 1-p^{-n} \right) ^{m-1}}{\prod\nolimits_{k=2}^m{\left( 1-p^{\alpha _k-n} \right)}}\int_{\left| y_1 \right|_p\leqslant 1}{\left| y_1 \right|_{p}^{-\alpha +(m-1)n}dy_1}\\ &=\frac{\left( 1-p^{-n} \right) ^{m-1}}{\prod\nolimits_{k=2}^m{\left( 1-p^{\alpha _k-n} \right)}}\sum_{i=-\infty}^0{\left( p^{-i\alpha +\left( m-1 \right) n}\int_{S_i}{dy_1} \right)}=\frac{\left( 1-p^{-n} \right) ^m}{\left( 1-p^{\alpha -mn} \right) \prod\nolimits_{k=2}^m{\left( 1-p^{\alpha _k-n} \right)}}.\\ \end{align}\]

Similar, for \(i=2,...,m-1\), we have \[\begin{align} I_i&=\int_{D_i}{\prod_{k=1}^m{\left| y_k \right|_{p}^{-\alpha _k}}dy_1}\cdots dy_m\\ &=\int_{\left| y_i \right|_p\leqslant 1}{\left| y_i \right|_{p}^{-\alpha _i}\left( \prod_{j=1}^{i-1}{\int_{\left| y_j \right|_p<\left| y_i \right|_p}{\left| y_j \right|_{p}^{-\alpha _j}dy_j}} \right) \left( \prod_{k=i+1}^m{\int_{\left| y_k \right|_p\leqslant \left| y_i \right|_p}{\left| y_k \right|_{p}^{-\alpha _k}dy_k}} \right) dy_i}\\ &=\int_{\left| y_i \right|_p\leqslant 1}{\left| y_i \right|_{p}^{-\alpha _i}\left( \prod_{j=1}^{i-1}{\sum_{u=-\infty}^{\log _p\left| y_i \right|_p-1}{\int_{S_u}{\left| y_j \right|_{p}^{-\alpha _j}}dy_j}} \right) \left( \prod_{k=i+1}^m{\sum_{v=-\infty}^{\log _p\left| y_i \right|_p}{\int_{S_v}{\left| y_k \right|_{p}^{-\alpha _k}}dy_k}} \right) dy_i}\\ &=\left( 1-p^{-n} \right) ^{m-1}\int_{\left| y_i \right|_p\leqslant 1}{\left| y_i \right|_{p}^{-\alpha _i}\left( \prod_{j=1}^{i-1}{\sum_{u=-\infty}^{\log _p\left| y_i \right|_p-1}{p^{u\left( -\alpha _j+n \right)}}} \right) \left( \prod_{k=i+1}^m{\sum_{v=-\infty}^{\log _p\left| y_i \right|_p}{p^{v(-\alpha _k+n)}}} \right) dy_i}\\ &=\left( 1-p^{-n} \right) ^{m-1}\int_{\left| y_i \right|_p\leqslant 1}{\left| y_i \right|_{p}^{-\alpha _i}\left( \prod_{j=1}^{i-1}{\frac{p^{(\alpha _j-n)}p^{\log _p\left| y_i \right|_p(-\alpha _j+n)}}{1-p^{\alpha _j-n}}} \right) \left( \prod_{k=i+1}^m{\frac{p^{\log _p\left| y_i \right|_p(-\alpha _k+n)}}{1-p^{\alpha _k-n}}} \right) dy_i}\\ &=\frac{\left( 1-p^{-n} \right) ^{m-1}\prod_{j=1}^{i-1}{p^{\alpha _j-n}}}{\prod_{1\leqslant k\leqslant m,k\ne i}{(1-p^{\alpha _k-n})}}\int_{\left| y_i \right|\leqslant 1}{\left| y_i \right|_{p}^{-\alpha +\left( m-1 \right) n}dy_i}\\ &=\frac{\left( 1-p^{-n} \right) ^m\prod_{j=1}^{i-1}{p^{\alpha _j-n}}}{(1-p^{\alpha -mn})\prod_{1\leqslant k\leqslant m,k\ne i}{(1-p^{\alpha _k-n})}}.\\ \end{align}\]

The case of \(i=m\) similar to the previous step, we show that \[\begin{align} I_m=\int_{\left| y_m \right|_p\leqslant 1}{\left| y_m \right|^{-\alpha _m}\left( \prod_{j=1}^{m-1}{\int_{\left| y_j \right|<\left| y_m \right|_p}{\left| y_j \right|^{-\alpha _j}dy_i}} \right) dy_m}=\frac{\left( 1-p^{-n} \right) ^m\prod_{j=1}^{m-1}{p^{\alpha _j-n}}}{\left( 1-p^{\alpha -mn} \right) \prod_{k=1}^{m-1}{(1-p^{\alpha _k-n})}}. \end{align}\] Now, we will calculate their sum, we set \[A_m=\frac{\left( 1-p^{-n} \right) ^m}{\left( 1-p^{\alpha -mn} \right) \prod\nolimits_{k=1}^m{\left( 1-p^{\alpha _k-n} \right)}}\,\,\,\text{and} \,\,\, d_k=\sum_{i=1}^k{\alpha _i}.\] Notice that \(d_m=\alpha\), then \[\begin{align} C_{1}^{p} &=I_1+\sum_{i=2}^{m-1}{I_i+I_m} \\ &=\frac{\left( 1-p^{-n} \right) ^m}{\left( 1-p^{\alpha -mn} \right) \prod\nolimits_{k=2}^m{\left( 1-p^{\alpha _k-n} \right)}}+\sum_{i=2}^{m-1}{\frac{\left( 1-p^{-n} \right) ^m\prod_{j=1}^{i-1}{p^{\alpha _j-n}}}{(1-p^{\alpha -mn})\prod_{1\leqslant k\leqslant m,k\ne i}{(1-p^{\alpha _k-n})}}} \\ &\quad+\frac{\left( 1-p^{-n} \right) ^m\prod_{j=1}^{m-1}{p^{\alpha _j-n}}}{\left( 1-p^{\alpha -mn} \right) \prod_{k=1}^{m-1}{(1-p^{\alpha _k-n})}}\\ &=A_m\left( (1-p^{d_1-n})+p^{\alpha _1-n}(1-p^{\alpha _2-n})+\prod_{j=1}^2{p^{\alpha _j-n}}(1-p^{\alpha _3-n})+\cdots +\prod_{j=1}^{m-1}{p^{\alpha _j-n}}(1-p^{\alpha _m-n}) \right) \\ &=A_m\left( (1-p^{d_1-n})+(p^{d_1-n}-p^{d_2-2n})+(p^{d_2-n}-p^{d_3-3n})+\cdots +(p^{d_{m-1}-(m-1)n}-p^{d_m-mn}) \right) \\ &=A_m( 1-p^{d_m-mn} ) =\frac{( 1-p^{-n} ) ^m}{\prod_{j=1}^m{(1-p^{\alpha _j-n})}}. \end{align}\] This finishes the proof of Corollary 1. ◻

3.3 Sharp bound for \(\boldsymbol{p}\)-adic Hilbert operator↩︎

Proof of Corollary 2:. Next, we refer to the methods in [25] to solve it. If we take the kernel \[K\left( y_1,...,y_m \right) =\frac{1}{( 1+\left| y_1 \right|_{p}^{n}+\cdots +\left| y_m \right|_{p}^{n} ) ^m}\] in Theorem 3, by a change of variables, we have \(T^p=T_{2}^{p}\), and then \(T_{2}^{p}\) can be denoted by \[T_{2}^{p}=\int_{\mathbb{Q}_p ^{nm}}{\frac{1}{( 1+\left| y_1 \right|_{p}^{n}+\cdots +\left| y_m \right|_{p}^{n} ) ^m}f_1( \left| x \right|_{p}^{-1}y_1 ) \cdots f_m( \left| x \right|_{p}^{-1}y_m ) dy_1\cdots dy_m},\] respectively, then all things reduce to calculating \[C_{2}^{p}=\int_{\mathbb{Q} _{p}^{nm}}{\frac{1}{(1+\left| y_1 \right|_{p}^{n}+\cdots +\left| y_m \right|_{p}^{n})^m}\prod_{j=1}^m{\left| y_j \right|_{p}^{-\alpha _j}dy_1}\cdots dy_m}.\] After a series of simple operations, we have \[\begin{align} C_{2}^{p} &=\int_{\mathbb{Q} _{p}^{n}}{\cdots \int_{\mathbb{Q} _{p}^{n}}{\sum_{k_m=-\infty}^{+\infty}{\int_{S_{k_m}}{\frac{1}{( 1+\left| y_1 \right|_{p}^{n}+\cdots +\left| y_m \right|_{p}^{n}) ^m}}}}}\prod_{j=1}^m{\left| y_j \right|_{p}^{-\alpha _j}}dy_m\cdots dy_1 \\ &=\sum_{k_1=-\infty}^{+\infty}{\sum_{k_2=-\infty}^{+\infty}{\cdots}\sum_{k_m=-\infty}^{+\infty}{\int_{S_{k_1}}{\cdots \int_{S_{k_m}}{\frac{1}{( 1+\left| y_1 \right|_{p}^{n}+\cdots +\left| y_m \right|_{p}^{n}) ^m}\prod_{j=1}^m{\left| y_j \right|_{p}^{-\alpha _j}}}}}}dy_m\cdots dy_1 \\ &=\sum_{k_1=-\infty}^{+\infty}{\sum_{k_2=-\infty}^{+\infty}{\cdots}\sum_{k_m=-\infty}^{+\infty}{\frac{1}{(1+p^{k_1n}+\cdots +p^{k_mn})^m}}}\prod_{j=1}^m{p^{-k_j\alpha _j}}\int_{S_{k_1}}{\cdots \int_{S_{k_m}}{dy_m\cdots dy_1}} \\ &=\sum_{k_1=-\infty}^{+\infty}{\sum_{k_2=-\infty}^{+\infty}{\cdots}\sum_{k_m=-\infty}^{+\infty}{\frac{1}{(1+p^{k_1n}+\cdots +p^{k_mn})^m}}}\prod_{j=1}^m{p^{-k_j\alpha _j}}\prod_{j=1}^m({p^{k_jn}( 1-p^{-n} ))} \\ &=\left( 1-p^{-n} \right) ^m\sum_{k_1=-\infty}^{+\infty}{\sum_{k_2=-\infty}^{+\infty}{\cdots}\sum_{k_m=-\infty}^{+\infty}{\frac{1}{(1+p^{k_1n}+\cdots +p^{k_mn})^m}}}\prod_{j=1}^m{p^{k_j(-\alpha _j+n)}}. \end{align}\] What we want to prove is that the sum of this series is bounded. Because it is a challenging problem to calculate the sum of this series, so we can indirectly prove that this series sum is bounded by an inequality. Clearly, \[[ \max ( 1,\left| y_1 \right|_{p}^{n},\cdots ,\left| y_m \right|_{p}^{n}) ] ^m=\underset{1\leqslant j\leqslant m}{\max}\{ 1,\left| y_j \right|_{p}^{mn} \} \leqslant ( 1+\left| y_1 \right|_{p}^{n}+\cdots +\left| y_m \right|_{p}^{n}) ^m.\] Then we have \[C_{2}^{p}\leqslant D^p=:\int_{\mathbb{Q} _{p}^{nm}}{\frac{1}{[\max\mathrm{(}1,\left| y_1 \right|_{p}^{n},...,\left| y_m \right|_{p}^{n})]^m}\prod_{j=1}^m{\left| y_j \right|_{p}^{-\alpha _j}}dy_1\cdots dy_m}.\] Thus if we prove that \(D^p\) is bounded, it means that \(C_2^p\) is bounded. Next, we refer to the methods in [25] to calculate \(D^p\).

To calculate this integral, divide the integral into \(m\) parts. Let \[\begin{align} &E_0=\{ \left( y_1,...,y_m \right) \in \mathbb{Q} _{p}^{n}\times \cdots \times \mathbb{Q} _{p}^{n}:\left| y_k \right|_p\leqslant 1,1\leqslant k\leqslant m\} ; \\ &E_1=\{ \left( y_1,...,y_m \right) \in \mathbb{Q} _{p}^{n}\times \cdots \times \mathbb{Q} _{p}^{n}:\left| y_1 \right|_p>1,\left| y_k \right|_p\leqslant \left| y_1 \right|_p,1<k\leqslant m \} ; \\ &E_i=\{ \left( y_1,...,y_m \right) \in \mathbb{Q} _{p}^{n}\times \cdots \times \mathbb{Q} _{p}^{n}:\left| y_i \right|_p>1,\left| y_j \right|_p<\left| y_i \right|_p,\left| y_k \right|_p\leqslant \left| y_i \right|_p,1\leqslant j<i<k\leqslant m \} ; \\ &E_m=\{ \left( y_1,...,y_m \right) \in \mathbb{Q} _{p}^{n}\times \cdots \times \mathbb{Q} _{p}^{n}:\left| y_m \right|_p>1,\left| y_j \right|_p<\left| y_m \right|_p,1\leqslant j\leqslant m \}. \end{align}\] Its clear that \[\bigcup_{j=0}^m{E_j=}\mathbb{Q} _{p}^{n}\times \cdots \times \mathbb{Q} _{p}^{n}\] and \(E_i\cap E_j=\varnothing \left( i\ne j \right)\). We let \[J_j:=\int_{E_j}{\frac{1}{[ \max ( 1,\left| y_1 \right|_{p}^{n},...,\left| y_m \right|_{p}^{n} )] ^m}\prod_{k=1}^m{\left| y_k \right|_{p}^{-\alpha_k}}}dy_1\cdots dy_m.\] Then \[D^{p}=\sum_{j=1}^m{J_j:}=\sum_{j=1}^m{\int_{E_j}{\frac{1}{[ \max ( 1,\left| y_1 \right|_{p}^{n},...,\left| y_m \right|_{p}^{n} ) ] ^m}\prod_{k=1}^m{\left| y_k \right|_{p}^{-\alpha_k}}dy_1\cdots dy_m}}.\] Now let us calculate \(I_j\), \(j=1,2,...,m\). Since \(\alpha_j<n\), so we have \[\begin{align} J_0&=\prod_{k=1}^m{\int_{\left| y_k \right|_p\leqslant 1}{\left| y_k \right|_{p}^{-\alpha _k}}dy_k=\prod_{k=1}^m{\left( \sum_{i=-\infty}^0{\int_{S_i}{\left| y_k \right|_{p}^{-\alpha _k}}dy_k} \right)}} \\ &=\prod_{k=1}^m{\left( \sum_{i=-\infty}^0{p^{-i\alpha _k}p^{in}(1}-p^{-n}) \right)}=\frac{\left( 1-p^{-n} \right) ^m}{\prod_{k=1}^m{(1-p^{\alpha _k-n})}}. \end{align}\] Similar for \(i=1\), since \(\alpha>0\) and \(\alpha_j<n\), we have \[\begin{align} J_1&=\int_{E_1}{\frac{1}{[\max\mathrm{(}1,\left| y_1 \right|_{p}^{n},...,\left| y_m \right|_{p}^{n})]^m}\prod_{k=1}^m{\left| y_k \right|_{p}^{-\alpha_k}}}dy_1\cdots dy_m \\ &=\int_{\left| y_1 \right|_p>1}{\left| y_1 \right|_{p}^{-\alpha _1-mn}}\prod_{k=2}^m{\left( \int_{\left| y_k \right|_p\leqslant \left| y_1 \right|_p}{\left| y_k \right|_{p}^{-\alpha _k}}dy_k \right) dy_1}=\frac{\left( 1-p^{-n} \right) ^{m-1}}{\prod_{k=2}^m{(1-p^{\alpha _k-n})}}\int_{\left| y_1 \right|_p>1}{\left| y_1 \right|_{p}^{-\alpha -n}}dy_1 \\ &=\frac{\left( 1-p^{-n} \right) ^{m-1}}{\prod_{k=2}^m{(1-p^{\alpha _k-n})}}\left( \int_{\left| y_1 \right|_p<\infty}{\left| y_1 \right|_{p}^{-\alpha -n}dy_1}-\int_{\left| y_1 \right|_p\leqslant 1}{\left| y_1 \right|_{p}^{-\alpha -n}dy_1} \right) \\ &=\frac{\left( 1-p^{-n} \right) ^{m-1}}{\prod_{k=2}^m{(1-p^{\alpha _k-n})}}\left( \sum_{i=-\infty}^{+\infty}{\int_{S_i}{\left| y_1 \right|_{p}^{-\alpha -n}}dy_1-\sum_{j=-\infty}^0{\int_{S_j}{\left| y_1 \right|_{p}^{-\alpha -n}}dy_1}} \right) \\ &=\frac{\left( 1-p^{-n} \right) ^{m-1}}{\prod_{k=2}^m{(1-p^{\alpha _k-n})}}\sum_{i=1}^{+\infty}{\int_{S_j}{\left| y_1 \right|_{p}^{-\alpha -n}}dy_1}=\frac{\left( 1-p^{-n} \right) ^mp^{-\alpha}}{\left( 1-p^{-\alpha} \right) \prod_{k=2}^m{(1-p^{\alpha _k-n})}}. \end{align}\] Similar for \(i=2,...,m-1\), we have \[\begin{align} J_i&=\int_{\left| y_i \right|_p>1}{\left| y_i \right|_{p}^{-\alpha _i-mn}}\left( \prod_{j=1}^{i-1}{\int_{\left| y_j \right|_p<\left| y_i \right|_p}{\left| y_j \right|_{p}^{-\alpha _j}}dy_j} \right) \left( \prod_{k=i+1}^m{\int_{\left| y_k \right|_p\leqslant \left| y_i \right|_p}{\left| y_k \right|_{p}^{-\alpha _k}}dy_k} \right)dy_i \\ &=\frac{\left( 1-p^{-n} \right) ^{m-1}\prod_{j=1}^{i-1}{p^{\alpha _j-n}}}{\prod_{1\leqslant k\leqslant m,k\ne i}{(1-p^{\alpha _k-n})}}\int_{\left| y_i \right|_p>1}{\left| y_i \right|_{p}^{-\alpha -n}}dy_i \\ &=\frac{p^{-\alpha}\left( 1-p^{-n} \right) ^m\prod_{j=1}^{i-1}{p^{\alpha _j-n}}}{(1-p^{-\alpha})\prod_{1\leqslant k\leqslant m,k\ne i}{(1-p^{\alpha _k-n})}}. \end{align}\] The case of \(i=m\) similar to the previous step, we show that \[\begin{align} J_m&=\int_{\left| y_m \right|_p}{\left| y_m \right|^{-\alpha _m-mn}\left( \prod_{j=1}^{i-1}{\int_{\left| y_j \right|_p<\left| y_i \right|_p}{\left| y_i \right|_{p}^{-\alpha _j}}dy_j} \right)}\left( \prod_{k=i+1}^m{\int_{\left| y_k \right|_p\leqslant \left| y_i \right|_p}{\left| y_k \right|_{p}^{-\alpha _k}}dy_k} \right)dy_m \\ &=\frac{p^{-\alpha}\left( 1-p^{-n} \right) ^m\prod_{j=1}^{m-1}{p^{\alpha _j-n}}}{\left( 1-p^{-\alpha} \right) \prod_{k=1}^{m-1}{(1-p^{\alpha _k-n})}}. \end{align}\] Now, we will calculate their sum, let \[B_m=\frac{p^{-\alpha}\left( 1-p^{-n} \right) ^m}{\left( 1-p^{-\alpha} \right) \prod_{k=1}^m{(1-p^{\alpha _k-n})}}\,\,\,\text{and} \,\,\, d_k=\sum_{i=1}^k{\alpha _i}.\] Notice that \(d_m=\alpha\), then \[\begin{align} D^{p}&=J_0+J_1+\sum_{i=2}^{m-1}{J_i}+J_m \\ &=\frac{\left( 1-p^{-n} \right) ^m}{\prod_{k=1}^m{(1-p^{\alpha _k-n})}}+\frac{\left( 1-p^{-n} \right) ^mp^{-\alpha}}{\left( 1-p^{-\alpha} \right) \prod_{k=2}^m{(1-p^{\alpha _k-n})}} \\ &\quad+\prod_{i=2}^{m-1}{\frac{p^{-\alpha}\left( 1-p^{-n} \right) ^m\prod_{j=1}^{i-1}{p^{\alpha _j-n}}}{(1-p^{-\alpha})\prod_{1\leqslant k\leqslant m,k\ne i}{(1-p^{\alpha _k-n})}}+\frac{p^{-\alpha}\left( 1-p^{-n} \right) ^m\prod_{j=1}^{m-1}{p^{\alpha _j-n}}}{\left( 1-p^{-\alpha} \right) \prod_{k=1}^{m-1}{(1-p^{\alpha _k-n})}}}. \\ &=B_m\left( \frac{1-p^{-\alpha}}{p^{-\alpha}}+(1-p^{d_1-n})+(p^{d_1-n}-p^{d_2-2n})+\cdots +(p^{d_{m-1}-(m-1)n}-p^{d_m-mn}) \right) \\ &=B_m\left( \frac{1-p^{-d_m}}{p^{-d_m}}+1-p^{d_m-mn} \right) =B_mp^{d_m}\left( 1-p^{-mn} \right) \\ &=\frac{\left( 1-p^{-n} \right) ^m\left( 1-p^{-mn} \right)}{\left( 1-p^{-\alpha} \right) \prod\nolimits_{k=1}^m{\left( 1-p^{\alpha _k-n} \right)}}<\infty. \end{align}\] In conclusion, we prove that \(D^p\) is bounded, it also means that \(C_2^p\) is bounded, that is \[(1-p^{-n})^m\sum_{k_1=-\infty}^{+\infty}{\sum_{k_2=-\infty}^{+\infty}{\cdots}\sum_{k_m=-\infty}^{+\infty}{\frac{1}{(1+p^{k_1n}+\cdots +p^{k_mn})^m}}}\prod_{j=1}^m{p^{k_j(-\alpha _j+n)}}<\infty.\] Our results also show that \(D^p\) is Hilbert’s upper bound. This finishes the proof of Corollary 2. ◻

4 Further Results↩︎

In this section, we will use the previous results to give the sharp bound for the \(p\)-adic \(m\)-linear \(n\)-dimensional Hausdorff operator on \(p\)-adic weighted space.

Corollary 3. Assume that the real paramenters \(\alpha\), \(\alpha_j\) with \(j=1,2,...,m\) are the same as Theorem 3, \(f_j\) be a measurable function in \(H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )\). A non-negative function \(\Phi\) on \(\mathbb{Q} _{p}^{n}\) satisfies

\[\begin{align} C_{\Phi}^{p}=\int_{\mathbb{Q} _{p}^{n}}{\cdots}\int_{\mathbb{Q} _{p}^{n}}{\frac{\Phi \left( y_1,...,y_m \right)}{\left| y_1 \right|_{p}^{n}\cdots \left| y_m \right|_{p}^{n}}\prod_{j=1}^m{\left| y_j \right|^{-\alpha _j}}}dy_1\cdots dy_m<\infty. \end{align}\] Then \[\begin{align} \left\| T_{\Phi}^{p} \right\| _{\prod\nolimits_{j=1}^m{H_{\alpha _j}^{\infty}( \mathbb{Q} _{p}^{n} )}\rightarrow H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )}=C_{\Phi}^{p}. \end{align}\]

Proof of Corollary 3:. By a change of variables, the \(p\)-adic \(m\)-linear \(n\)-dimensional Hausdorff operator become \[T_{\Phi}^{p}=\int_{\mathbb{Q} _{p}^{n}}{\cdots}\int_{\mathbb{Q} _{p}^{n}}{\frac{\Phi \left( y_1,...,y_m \right)}{\left| y_1 \right|_{p}^{n}\cdots \left| y_m \right|_{p}^{n}}}f_1( x\left| y_1 \right|_{p}^{-1} ) \cdots f_m( x\left| y_m \right|_{p}^{-1} ) dy_1\cdots dy_m.\] We can obtain \[\begin{align} \left\| T_{\Phi}^{p} \right\| _{\prod\nolimits_{j=1}^m{H_{\alpha _j}^{\infty}\left( \mathbb{Q} _{p}^{n} \right)}\rightarrow H_{\alpha}^{\infty}\left( \mathbb{Q} _{p}^{n} \right)}=\int_{\mathbb{Q} _{p}^{n}}{\cdots}\int_{\mathbb{Q} _{p}^{n}}{\frac{\Phi \left( y_1,...,y_m \right)}{\left| y_1 \right|_{p}^{n}\cdots \left| y_m \right|_{p}^{n}}\prod_{j=1}^m{\left| y_j \right|^{-\alpha _j}}}dy_1\cdots dy_m=C_{\Phi}^{p}. \end{align}\] This is similar to the proof of Theorem 3, so we omit the details. This finishes the proof of Corollary 3. ◻

Declarations↩︎

Acknowledgments↩︎

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Data availability↩︎

Not applicable for this article as it is pure mathematical research with no associated data.

Conflict of interest↩︎

The authors declare that they have no conflicts of interest/competing interests regarding this work.

Tianyang He
Research Center for Mathematics and Interdisciplinary Sciences, Frontiers Science Center for Nonlinear Expectations (Ministry of Education)
Shandong University
Qingdao, 266237
P. R. China E-mail address: hty13791059207@163.com

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Corresponding author, E-mail: hty13791059207@163.com. ORCID: 0009-0000-8811-3074↩︎

Sharp bounds for the \(\boldsymbol{p}\)-adic \(\boldsymbol{n}\)-dimensional fractional Hardy operator and a class of integral operators on \(\boldsymbol{p}\)-adic function spaces