Sharp estimates for the ${p}$-adic ${m}$-linear ${n}$-dimensional Hardy and Hilbert operators on ${p}$-adic weighted Morrey space

1.   Introduction

In recent years, researchers on mathematical physics have paid increasing attention to $p$-adic fields, because $p$-adic numbers are widely used in theoretical and mathematical physics ^[1,2,22]. For this reason, the harmonic analysis of $p$-adic fields has attracted significant attention ^[16,17,18].

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 rational numbers $\mathbb{Q}$ with respect to the non-Archimedean $p$-adic norm $| \cdot |_p$. This norm is defined as follows: $| 0 |_p = 0$. If any non-zero rational number $x$ is represented as $x = p^{\gamma}\frac{m}{n}$, where $m$ and $n$ are integers that 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:

and

It follows from the second property that when $| x |_p\ne | y |_p$, then $| x+y |_p = \max \{ | x |_p, | y |_p \}$. From the standard $p$-adic analysis ^[22], we see that any non-zero $p$-adic number $x\in \mathbb{Q} _p$ can be uniquely represented in the canonical series

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 $|a_jp^j|_p = p^{-\gamma}$. The space $\mathbb{Q} _{p}^{n}$ consists of points $x = \left(x_1, x_2, ..., x_n \right)$, where $x_j\in \mathbb{Q} _p$, $j = 1, 2, ..., n$. The $p$-adic norm on $\mathbb{Q} _{p}^{n}$ is

Denoted by $B_{\gamma} = \{ x\in \mathbb{Q} _{p}^{n}:| x-a |_p\leqslant p^{\gamma} \}$, the ball with center at $x_j\in \mathbb{Q} _p$ and radius $p^{\gamma}$, $\gamma \in \mathbb{Z}$. It is clear that $S_{\gamma}(a) = B_{\gamma}(a) \backslash B_{\gamma -1}(a)$ and

We set $B_{\gamma}(0) = B_{\gamma}$ and $S_{\gamma}(0) = S_{\gamma}$.

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 $dx$ on $\mathbb{Q} _{p}^{n}$ (up to positive constant multiple) that is translation invariant. We normalize the measure $dx$ so that

where $|E|_H$ denotes the Haar measure of a measurable subset $E$ of $\mathbb{Q} _{p}^{n}$. From this integral theory, it is easy to obtain that $| B_{\gamma}(a)|_H = p^{\gamma n}$ and $|S_{\gamma}(a) |_H = p^{\gamma n}(1-p^{-n})$ for any $a\in \mathbb{Q} _{p}^{n}$. For a more complete introduction to $p$-adic fields, see ^[19].

In recent years, $p$-adic analysis has received a lot of attention due to its application in mathematical physics (cf. ^[1,2]). There are numerous papers on $p$-adic analysis, such as ^[11,12] about Riesz potentials, and ^[3] about $p$-adic pseudo-differential equations. The harmonic analysis on $p$-adic fields has been drawing more and more attention (cf. ^[14,15,20] and references therein).

The $p$-adic $m$-linear $n$-dimensional Hardy operator ^[5] is defined by

where $x\in \mathbb{Q}_p ^n\backslash \left\{ 0 \right\}$. Batbold and Sawano ^[5] obtained that the norm of $T_1^p$ on $p$-adic Lebesgue spaces and $p$-adic Morrey spaces is

and

The $p$-adic $m$-linear $n$-dimensional Hilbert operator ^[5] is defined by

where $x\in \mathbb{Q}_p ^n\backslash \left\{ 0 \right\}$. The $p$-adic $m$-linear $n$-dimensional Hausdorff operator ^[21] is defined by

where $x\in \mathbb{Q}_p ^n\backslash \left\{ 0 \right\}$, and $\Phi$ is a nonnegative function on $\mathbb{Q}_p ^n$. Chen et al. ^[6] obtained the norm of the multilinear Hausdorff operator on a $p$-adic function space.

Computation of the operator norm of integral operators is a challenging work in harmonic analysis. In 2017, Batbold and Sawano ^[4] studied one-dimensional $m$-linear Hilbert-type operators, incuding the sharp bounds for the Hardy-Littlewood-Pólya operator on weighted Morrey spaces. He et al. ^[13] extended the results in ^[4] and obtained the sharp bound for the generalized Hardy-Littlewood-Pólya operator on weighted central and noncentral homogenous Morrey spaces. In 2011, Wu and Fu ^[23] obtained the sharp estimate of the $m$-linear $p$-adic Hardy operator on Lebesgue spaces with power weight.

Inspired by the above, we study a more general operator, which includes the $p$-adic Hardy and Hilbert operator as a special case. We consider their operator norm on two power weighted $p$-adic Morrey space and its central version. Finally, we also find the sharp bound for the Hausdorff operator on power weighted central and noncentral Morrey spaces, which generalizes the previous results. Fu and Wu et al.^[8,9,10] conducted many related research, which is the basis for our research.

We present the definition of the weighted $p$-adic Morrey space $B^{q, \lambda}(\mathbb{Q} _{p}^{n}, \omega _1, \omega _2)$ on $\mathbb{Q} _{p}^{n}$, where $\omega _1$, $\omega _2$: $\mathbb{Q} _{p}^{n}\rightarrow \left(0, \infty \right)$ are positive measurable functions.

Definition 1.1. Let $1 < q < \infty$, $\frac{1}{q} < \lambda < 0$, then the weighted $p$-adic Morrey space $L^{q, \lambda}(\mathbb{Q} _{p}^{n}, \omega _1, \omega _2)$ is the set of all $f\in L_{loc}^{q}(\mathbb{Q} _{p}^{n})$ for which the norm

Definition 1.2. Let $1 < q < \infty$, $\frac{1}{q} < \lambda < 0$, $B_{\gamma}$ replace with $B_{\gamma}\left(0 \right)$ in the above definition, then the weighted $p$-adic Morrey space $B^{q, \lambda}(\mathbb{Q} _{p}^{n}, \omega _1, \omega _2)$ is the set of all $f\in L_{loc}^{q}(\mathbb{Q} _{p}^{n})$ for which the norm

2.   Proofs of the main results

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 radial kernel such that $K\left(y_1, ..., y_m \right) = K(\left| y_1 \right|_{p}^{-1}, ..., \left| y_m \right|_{p}^{-1})$, then satisfies that

where $\lambda _j$, $\beta _j$, $q_j, q$, $\alpha$ are pre-defined indicator and some fixed indicators, $j = 1, 2, ..., m$. The $p$-adic $m$-linear $n$-dimensional integral operator with a kernel is defined by

where $x\in \mathbb{R} ^n\backslash \left\{ 0 \right\}$ and $f_j$ is a measurable radial function on $\mathbb{Q} _{p}^{n}$ with $j = 1, 2, ..., m$. Note that $T^p$ is in fact an integral operator with a homogeneous radial $K$ of degree $-mn$.

In this paper, we will obatin the sharp bound of the $p$-adic $m$-linear $n$-dimensional integral operator with a kernel on $p$-adic weighted Morrey space. Finally, by taking a particular kernel $K$ in operator $C^p$ defined by (2.1), then we obtain the sharp bounds of the $p$-adic Hardy and Hilbert operators.

Lemma 2.1. Let $\alpha \in \mathbb{R}$, $1 < q < q_j < \infty$, $\frac{1}{q} = \frac{1}{q_1}+\cdots +\frac{1}{q_m}$, $\lambda = \lambda _1+\cdots +\lambda _m$, $\beta = \beta _1+\cdots +\beta _m$, $-\frac{1}{q_j} < \lambda _j < 0$, $-\frac{1}{q} < \lambda < 0$, $y_j\in \mathbb{Q} _{p}^{n}$, if $f_j\in L^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{q_j\beta _j/q})$ or $B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{q_j\beta _j/q})$, where $j = 1, 2, ..., m$, then we have

and

For convenience of this paper, we define the dilation index in (2.3) and (2.4) by

Lemma 2.2. Assume that real parameters $q$, $q_j$, $\lambda$, $\lambda_j$, $\beta$, $\beta_j$ with $j = 1, 2, ..., m$ are the same as in Lemma 2.1, $\alpha +n > 0$. Then we have $\left| x \right|_{p}^{d(\lambda _j, q_j, \alpha, \frac{q_j\beta _j}{q})}\in B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$. Moreover,

Lemma 2.3. Assume that real parameters $q$, $q_j$, $\lambda$, $\lambda_j$, $\beta$, $\beta_j$ with $j = 1, 2, ..., m$ are the same as in Lemma 2.1, $\alpha +n > 0$. Then we have $\left| x \right|_{p}^{d(\lambda _j, q_j, \alpha, \frac{q_j\beta _j}{q})}\in L^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$. Moreover,

Theorem 2.1. Let $\alpha \in \mathbb{R}$, $1 < q < q_j < \infty$, $\frac{1}{q} = \frac{1}{q_1}+\cdots +\frac{1}{q_m}$, $\lambda = \lambda _1+\cdots +\lambda _m$, $\beta = \beta _1+\cdots +\beta _m$, $-\frac{1}{q_j} < \lambda _j < 0$, $-\frac{1}{q} < \lambda < 0$ with $j = 1, 2, ..., m$, $f_j$ be a radial function in $B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$. We obtain

where $C^p$ is the constant defined by (2.1). Moreover, if $\alpha +n > 0$ and $q\lambda = q_1\lambda _1 = \cdots = q_m\lambda _m$, then $C^p$ is the sharp constant in (2.7), then

Theorem 2.2. Assume that the real parameters $q$, $q_j$, $\lambda$, $\lambda_j$, $\beta$, $\beta_j$ with $j = 1, 2, ..., m$ are the same as in Theorem 2.1, $f_j$ be a radial function in $L^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$. We get

where $C^p$ is the constant defined by (2.1). Moreover, if $\alpha +n > 0$ and $q\lambda = q_1\lambda _1 = \cdots = q_m\lambda _m$, then $C^p$ is the sharp constant in (2.8), and we have

Corollary 2.1. Assume that the real parameters $q$, $q_j$, $\lambda$, $\lambda_j$, $\beta$, $\beta_j$ with $j = 1, 2, ..., m$ are the same as in Theorem 2.1, $f_j$ be a radial function in $B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$. Assume also that $d(\lambda _j, q_j, \alpha, \frac{q_j\beta _j}{q}) +n > 0$. Then $T_{1}^{p}$ is bounded from $\prod_{j = 1}^m{B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})}$ to $B^{q, \lambda}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\beta})$. Furthermore, if $\alpha+n > 0$, then

The weighted Morrey space $L^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$ is similar.

Corollary 2.2. Assume that the real parameters $q$, $q_j$, $\lambda$, $\lambda_j$, $\beta$, $\beta_j$ with $j = 1, 2, ..., m$ are the same as in Theorem 2.1, $f_j$ be a radial function in $B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$. Assume also that $d(\lambda _j, q_j, \alpha, \frac{q_j\beta _j}{q}) +n >$$0$ and $d(\lambda, q, \alpha, \beta) < 0$. Then $T_{2}^{p}$ is bounded from $\prod_{j = 1}^m{B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})}$ to $B^{q, \lambda}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\beta})$. Furthermore, if $\alpha+n > 0$, then

The weighted Morrey space $L^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$ is similar.

2.1.   Sharp bound for the $p$-adic integral operator with kernel

Proof of Lemma 2.1. We only prove the scaling in $L^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$, as the other case is similar. By computing, we obtain

This finishes the proof of Lemma 2.1. □

Next, we will give the proofs of Lemmas 2.2 and 2.3.

Proof of Lemma 2.2. For the case when $\alpha+n > 0$, let $f_j(x) = \left| x \right|_{p}^{d(\lambda _j, q_j, \alpha, \frac{q_j\beta _j}{q})}$ for all $x\in \mathbb{Q} _{p}^{n}\backslash \left\{ 0 \right\}$ and $f_j\left(0 \right) : = 0\left(j = 1, 2, ..., m \right)$. Then for any $B_{\gamma} = B\left(0, p^{\gamma} \right)$, we need to show that $f_j\in B_{\gamma}^{q_{j, }\lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$. By computing, we have

Since $\alpha +n > 0$ and $1+q_j\lambda _j > 0$, we have

Notice that $\left(\alpha +n \right) \gamma (-\lambda _j-\frac{1}{q_j}) +\frac{1}{q_j}\gamma(q_j\lambda _j+1) \left(\alpha +n \right) = 0$, so we obtain

Using (2.11), we have that $f_j\in B^{q_{j, }\lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$.

This finishes the proof of Lemma 2.2. □

Proof of Lemma 2.3. For the case when $\alpha+n > 0$, let $f_j\left(x \right) = \left| x \right|_{p}^{d(\lambda _j, q_j, \alpha, \frac{q_j\beta _j}{q})}$ for all $x\in \mathbb{Q} _{p}^{n}\backslash \left\{ 0 \right\}$ and $f_j\left(0 \right) : = 0\left(j = 1, 2, ..., m \right)$. Then for any $B_{\gamma}\left(a \right) = B\left(a, p^{\gamma} \right)$, we need to show that $f_j\in L_{\gamma}^{q_{j, }\lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$. We will consider the following two cases:

(ⅰ) If $\left| a \right|_p > p^{\gamma}$ and $x\in B_{\gamma}\left(a \right)$, since $\left| x-a \right|_p\leqslant p^{\gamma}$, it follows that $\left| x \right|_p\leqslant \max \{ \left| x-a \right|_p, \left| a \right|_p \} = \left| a \right|_p > p^{\gamma}$. Thus, we have

(ⅱ) If $\left| a \right|_p\leqslant p^{\gamma}$ and $x\in B_{\gamma}\left(a \right)$, since $\left| x-a \right|_p\leqslant p^{\gamma}$ and $\left| a \right|_p\leqslant p^{\gamma}$, then $\left| x \right|_p = \max \{ \left| x-a \right|_p, \left| a \right|_p \} \leqslant p^{\gamma}$, so we have $x\in B_{\gamma}$. For $x\in B_{\gamma}$, then $\left| x \right|_p\leqslant p^{\gamma}$, we have $\left| x-a \right|_p\leqslant p^{\gamma}$ and $x\in B_{\gamma}\left(a \right)$. So we have $B_{\gamma}\left(a \right) = B_{\gamma}$, thus

By the argument in the proof of Lemma 2.2, we have that for $\left| a \right|_p\leqslant p^{\gamma}$, $x\in B_{\gamma}\left(a \right)$,

In conclusion, we can see that $f_j\in L^{q_{j, }\lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$.

This finishes the proof of Lemma 2.3. □

Proof of Theorem 2.1. First, we claim that the operator $T^p(f_1, .., f_m)(x)$ and its restriction to the functions $g(x) = g(|x|_{p}^{-1})$ have the same operator norm. In fact, set

Obviously, $g_j$ satisfies $g_j\left(x \right) = g_j(\left| x \right|_{p}^{-1})$, $T^p\left(g_1, ..., g_m \right) \left(x \right)$ is equal to

In the fourth to fifth lines, we let $z_j = | y_j |_{p}^{-1}\xi _j$. From the fifth to sixth lines, we perform an integral permutation. In the sixth to seventh lines, we set $y_j = | z_j |_{p}^{-1}t_j$. On the other hand, using the Holder's inequality, we conclude

From the third to fourth lines, we apply Hölder's inequality. In the fifth to sixth lines, we let $z_j = \left| x \right|_{p}^{-1}\xi _j$. From the sixth to seventh lines, we perform an integral permutation. In the seventh to eighth lines, we set $x = | z_j |_{p}^{-1}t_j$. Therefore, we have

which implies that the operators $T^p$ and their restriction to the function $g$ satisfying $g_j\left(x \right) = g_j(\left| x \right|_{p}^{-1})$ have the same operator norm in $B^{q, \lambda}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\beta})$. So without loss of generality, we assume that $f_j\in B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$ with $j = 1, 2, ..., m$ satisfies that $f_j\left(x \right) = f_j(\left| x \right|_{p}^{-1})$ in the rest of the proof.

The following sequence is obtained by Minkowski's inequality and Holder's inequality; notice that $\frac{1}{q} = \frac{1}{q_j}+\cdots +\frac{1}{q_m}$, $\lambda = \lambda _1+\cdots +\lambda _m$, $\beta = \beta _1+\cdots +\beta _m$, then $\frac{q}{q_j}+\cdots +\frac{q}{q_m} = 1$ and $f(\left| x \right|_{p}^{-1}y_j) = f(x| y_j |_{p}^{-1})$, thus we have

Using Lemma 2.1, we can deduce that

Now, we will show that the operator norm of $T^p(f_1, ..., f_m)(x)$ is equal to $C^p$. Taking

Since $\alpha+n > 0$, using Lemma 2.2, then $f_j\in B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$, we calculate that

Since $\lambda = \lambda _1+\cdots +\lambda _m$, $\frac{1}{q} = \frac{1}{q_j}+\cdots +\frac{1}{q_m}$, $q\lambda = q_1\lambda _1 = \cdots = q_m\lambda _m$, we have

Notice that $(nq_j\lambda _j+\alpha (q_j\lambda _j+1) +n) \frac{1}{q_j}+\left(n+\alpha \right) (-\lambda _j-\frac{1}{q_j}) = 0$, we obtain

In the first to second lines, we let $x_j = p^{-\gamma}y_j$. From the second to third lines, we let $y_j = z_jp^{\gamma _j}$. Through the above steps, we have completed the proof of Theorem 2.1. It is

□

Proof of Theorem 2.2. The previous step is similar to the proof of Theorem 2.1, using Lemma 2.3, then $f_j\in L^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$, finaly we obtain that

Notice that $(nq_j\lambda _j+\alpha (q_j\lambda _j+1) +n) \frac{1}{q_j}+\left(n+\alpha \right) (-\lambda _j-\frac{1}{q_j}) = 0$, we obtain

In the first to second lines, we let $x_j = p^{-\gamma}y_j$. From the third to fourth lines, we let $y_j = z_jp^{\gamma _j}$. Through the above steps, we have completed the proof of Theorem 2.2. It is

□

2.2.   Sharp bound for $p$-adic Hardy operator

Proof of Corollary 2.1. Next, we will use the methods in ^[23]. If we take the kernel

in Theorems 2.1 and 2.2, by a change of variables, it is easy to verify that $T^p(f_1, ..., f_m)(x) = T_{1}^{p}(f_1, ..., f_m)(x)$, then $T_{1}^{p}(f_1, ..., f_m)(x)$ can be denoted by

respectively, then it is all reduced to calculating

To calculate this integral, we divide the integral into $m$ parts. Let

It is clear that

and $D_i\cap D_j = \varnothing \left(i\ne j \right)$. Let

Then

Now, let us calculate $I_j$, $j = 1, 2, ..., m$. Since $d(\lambda _j, q_j, \alpha, \frac{q_j\beta _j}{q}) +n > 0$, then $d\left(\lambda, q, \alpha, \beta \right) +mn > 0$, so we have

Similarly, for $i = 2, ..., m-1$, we have

The case of $i = m$ is similar to the previous step, and we have

Now, we will calculate their sum, let

Notice that $-d_m = -d\left(\lambda, q, \alpha, \beta \right)$, then

This finishes the proof of Corollary 2.1. □

2.3.   Sharp bound for $p$-adic Hilbert operator

Proof of Corollary 2.2. If we take the kernel

in Theorems 2.1 and 2.2, by a change of variables, it is easy to verify that $T^p(f_1, ..., f_m)(x) = T_{2}^{p}(f_1, ..., f_m)(x)$, then $T_{2}^{p}(f_1, ..., f_m)(x)$ can be denoted by

respectively, then it is all reduced to calculating

After a series of simple operations, we have

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, we can indirectly prove that this series sum is bounded by an inequality. Clearly,

Then we have

So if we prove that $D^p$ is bounded, it means that $C_2^p$ is bounded. Next, we refer to the methods in ^[7] to calculate $D^p$.

To calculate this integral, we divide the integral into $m$ parts. Let

Its clear that

and $E_i\cap E_j = \varnothing \left(i\ne j \right)$. Let

then we have

Now, let us calculate $J_j$, $j = 1, 2, ..., m$. Since $d(\lambda _j, q_j, \alpha, \frac{q_j\beta _j}{q}) +n > 0$, we have

For $j = 1$, since $d\left(\lambda, q, \alpha, \beta \right) < 0$, $d(\lambda _1, q_1, \alpha, \frac{q_1\beta _1}{q}) +n > 0$, then we have

where

Similarly for $i = 2, ..., m-1$, we have

When $i = m$, similarly to the previous step, we show that

Now, we will calculate their sum, let

Notice that $-d_m = -d\left(\lambda, q, \alpha, \beta \right)$, then

In conclusion, we prove that $D^p$ is bounded, which also means that $C_2^p$ is bounded, that is

Our results also show that $D^p$ is Hilbert's bound. This finishes the proof of Corollary 2.2. □

3.   Further results: sharp estimate for the Hausdorff operator

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 Morrey spaces.

Corollary 3.1. Assume that the real parameters $q$, $q_j$, $\lambda$, $\lambda_j$, $\beta$, and $\beta_j$ with $j = 1, 2, ..., m$ are the same as in Theorem 2.1, and a nonnegative function $\Phi$ on $\mathbb{R} ^n$ satisfies

then $T_{\Phi}^{p}$ is bounded from $\prod_{j = 1}^m{B^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})}$ to $B^{q, \lambda}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\beta}).$ Furthermore, if $\alpha+n > 0$, then

The weighted Morrey space $L^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$ is similar.

Proof of Corollary 3.1. By a change of variables, the $p$-adic $m$-linear $n$-dimensional Hausdorff operator becomes

According to the proof of Theorem 2.1 and Lemma 2.2, notice that there is no need to let $f_j$ be a radial function, and we have

The weighted Morrey space $L^{q_j, \lambda _j}(\mathbb{Q} _{p}^{n}, \left| x \right|_{p}^{\alpha}, \left| x \right|_{p}^{\frac{q_j\beta _j}{q}})$ is similar, so we omit the details. This finishes the proof of Corollary 3.1. □

4.   Conclusions

First, the $m$-linear $n$-dimensional integral operator with a kernel has a sharp estimate. The sharp estimate on central and noncentral $p$-adic Morrey spaces with power weighted is given by

where the kernel $K(y_1, .., y_m)$ satisfies $C^p < \infty$.

Second, as an application, we derive the sharp bounds for the $m$-linear $n$-dimensional Hardy operator and Hilbert operators on weighted Morrey spaces, that is

and

Finally, based on the previous result, we also find the estimate for the Hausdorff operator on weighted Morrey spaces:

where the nonnegative function $\Phi$ on $\mathbb{R} ^n$ satisfies $C_{\Phi}^{p} < \infty$.

Author contributions

Tingting Xu: Conceptualization, methodology; Zaiyong Feng: Writing-original draft; Tianyang He: Writing-review and editing, validation, methodology; Xiaona Fan: Theoretical derivation, proof verification. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by Foundation of the Natural Science Foundation of China under Grant No. 12271262. The funders had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.