Rough fractional integral and its multilinear commutators on $p$-adic generalized Morrey spaces

1.   Introduction

In the past few decades, $p$-adic analysis has gathered a lot of attention by its applications in many aspects of mathematical physics, such as quantum mechanics, the probability theory and the dynamical systems ^[1]. Significantly, the geometry of the field of $p$-adic numbers is surprisingly unlike the geometry of the real numbers field $\mathbb R$, in particular the Archimedean axiom is not true in the field of $p$-adic numbers^[2]. Therefore, $p$-adic analysis has also gained impeccable attraction in harmonic analysis ^[3,4,5,6,7].

In $p$-adic harmonic analysis, fractional calculus is a key area because of its heap of applications in engineering science and technology, see for instance ^[8,9]. Also, fractional integral operators (Riesz potentials) are significant in the mathematical analysis as they construct and formulate inequalities which have several applications in scientific areas that can be found in the existing literature ^[10,11]. The boundedness criteria of fractional integral operators on different functional spaces is a key area not only in harmonic analysis but also in partial differential equations, differentiation theory and potential theory ^[12,13]. In this connection, the fractional integral operator in $p$-adic analysis is defined by

Here, $\mathbb Q_p^n$ consists of all points ${\bf x} = (x_1, \cdots, x_n)$ for $n\in\mathbb{N}$, where $x_j\in \mathbb Q_p(j = 1, \cdots, n)$ and $\mathbb Q_p$ is the field of $p$-adic numbers.

When $n = 1$, Haran^[3,4] not only obtained the explicit formula of the fractional integral operator $T^p_\beta$ on $\mathbb Q_p$ but also developed the analytical potential theory on $\mathbb Q_p^n$. Taibleson ^[2] gave the fundamental analytic properties of $T^p_\beta$ on local fields, including $\mathbb Q_p^n$, as well as the classical Hardy-Littlewood-Sobolev theorem (also see ^[5]). Moreover, Volosivets ^[6,7] showed that $T^p_\beta$ is bounded on radial Morrey spaces. In 2015, Wu and Fu ^[14] established Hardy-Littlewood-Sobolev inequalities on $p$-adic central Morrey spaces and $\lambda$-central BMO estimates for commutators of $T^p_\beta$. In 2018, Mo et al. ^[15] showed the boundedness of $T^p_\beta$ on $p$-adic generalized Morrey spaces, as well as the boundedness of multilinear commutators generated by $T^p_\beta$ and generalized Campanato functions. In 2022, both Shi et al. ^[16] and Sarfraz et al. ^[17] studied the boundedness of $T^p_\beta$ and its commutators on Morrey-Herz spaces. At the same time, Sarfraz and Jarad ^[18] considered the roughness of the operator $T^p_\beta$, they introduced rough fractional integral operator $T_{\beta, \Omega}^p$. In the form, $T_{\beta, \Omega}^p$ has the following integral expression

for suitable measurable mappings $f: \mathbb Q_p^n\rightarrow \mathbb R$ and $\Omega:S_0({\bf 0})\rightarrow \mathbb R$. When $f\in L^q(\mathbb Q_p^n)$, $1\leq q < \infty$, by the same way in the book ^[2], Sarfraz and Jarad ^[18] showed the boundedness of $T_{\beta, \Omega}^p$ on Lebesgue spaces (see Lemma 2.2 in Section 2). Furthermore, they obtained the boundedness of $T_{\beta, \Omega}^p$ on $p$-adic central Morrey spaces, as well as the $\lambda$-central BMO estimates for commutator $T_{\beta, \Omega}^{p, b}$ defined by

In ^[19], Sarfraz and Aslam showed the boundedness of $T_{\beta, \Omega}^p$ and $T_{\beta, \Omega}^{p, b}$ on $p$-adic Herz spaces.

We observe above works, the boundedness of $T_{\beta, \Omega}^p$ on generalized Morrey spaces are remains open. Therefore, in this paper, we are going to devote to the boundedness of $T_{\beta, \Omega}^p$ on $p$-adic generalized Morrey spaces. Moreover, let ${\bf{b}}$ $= (b_1, b_2, \cdots, b_m)$ with $b_i\in L_{{\mathrm{loc}}}(\mathbb Q_p^n)$ for $1\leq i\leq m$, $m\in\mathbb{N}$, we will consider multilinear commutator defined by

and investigate the boundedness of $T_{\beta, \Omega}^{p, {\bf b}}$ on $p$-adic generalized Morrey spaces with symbols in Campanato spaces. It should be emphasized that our results are new and cover some existing results of $T^p_\beta$ and $T_{\beta, \Omega}^p$.

Our paper is organized as follows. In Section 2, we present some notations and preliminaries. In Section 3, we present our main results. In Section 4, we will give the proof of main results. Throughout this paper, $q' = q/(q-1)$ for $1 < q < \infty$ and $q' = \infty$ when $q = 1$, the letter $C$ will be used to denote various constants, .

2.   Preliminaries

We begin this section with recalling some preliminaries of $p$-adic analysis pertaining to our work. For a prime number $p$, let $\mathbb Q_p$ be the field of $p$-adic numbers defined as the completion of the field of rational numbers $\mathbb Q$ with respect to non-Archimedean $p$-adic normal $|\cdot|_p$. This normal $|\cdot|_p$ is defined as follows: if $x = 0$, $|0|_p = 0$; if $x\neq0$ is an arbitrary rational number with the unique representation $x = p^\gamma m/n$, where $m$, $n$ are not divisible by $p$, $\gamma = \gamma(x)\in \mathbb Z$, then $|x|_p = p^{-\gamma}$. It's not hard to see that the norm satisfies the following properties:

(i)$|x|_p\geq0$, $\forall x\in \mathbb Q_p$ and $|x|_p = 0\Leftrightarrow x = 0$;

(ii) $|xy|_p = |x|_p|y|_p$, $\forall x, y\in \mathbb Q_p$;

(iii) $|x+y|_p\leq\max(|x|_p, |y|_p)$, $\forall x, y\in \mathbb Q_p$ and when $|x|_p\neq|y|_p$, we have $|x+y|_p = \max(|x|_p, |y|_p)$.

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

where $\gamma = \gamma(x)\in \mathbb Z$, $x_k\in\left\{0, 1, \cdots, p-1\right\}$, $x_0\neq0$, $k = 0, 1, \cdots$. The series (2.1) converges in the $p$-adic norm because $|x_kp^k|_p = p^{-k}$.

The $p$-adic norm of $\mathbb Q_p^n = \mathbb Q\times \mathbb Q\times\cdots\times \mathbb Q$ is defined by

Denote by

the ball of radius $p^\gamma$ with center at ${\bf a}\in \mathbb Q_p^n$ and write $\mathbb{B} = \left\lbrace B_\gamma({\bf a}): {\bf a}\in \mathbb Q_p^n, \gamma\in \mathbb Z\right\rbrace$. If let

the sphere of radius $p^\gamma$ with center at ${\bf a}\in \mathbb Q_p^n$, it is easy to see that

Since the space $\mathbb Q_p^n$ is a locally compact commutative group under addition, there exists the Haar measure $dx$ on the additive group of $\mathbb Q_p^n$ normalized by $\int_{B_0}dx = \left|B_0\right| = 1$, where $B_0: = B_0({\bf 0})$ and $\left|E\right|$ denotes the Haar measure of a measurable set $E\subset \mathbb Q_p^n$. Then by a simple calculation, the Haar measures of any balls and spheres can be obtained. Especially, we frequently use

For a more complete introduction to the $p$-adic analysis, we refer the readers to ^[2] and the references therein.

Now, let us give the definitions of generalized Morrey spaces and generalized Campanato spaces on the $p$-adic number field as follows.

Definition 2.1. ^[15] Let $1\leq q < \infty$, $B_\gamma({\bf a})$ be a ball in $\mathbb Q_p^n$ and $\omega({\bf x})$ be a non-negative measurable function in $\mathbb Q_p^n$. A function $f\in L_{{\mathrm{loc}}}^q(\mathbb Q_p^n)$ is said to belong to the generalized Morrey space $L^{q, \omega}(\mathbb Q_p^n)$, if

where $\omega(B_\gamma({\bf a})) = \int_{B_\gamma({\bf a})}\omega({\bf x})d{\bf x}$.

Notice that if let $\omega(B_\gamma({\bf a})) = |B_\gamma({\bf a})|^{\lambda}$, then $L^{q, \omega}(\mathbb Q_p^n)$ is the classical Morrey spaces $M_p^\lambda(\mathbb Q_p^n)$. Moreover, if let $\lambda\in \mathbb R$ and $B_\gamma({\bf a}) = B_\gamma({\bf 0})$, then $L^{q, \omega}(\mathbb Q_p^n)$ is the central Morrey spaces $\dot{B}^{q, \lambda}(\mathbb Q_p^n)$ (see ^[14,18]) defined by

Definition 2.2. ^[15] Let $1\leq q < \infty$, $B_\gamma({\bf a})$ be a ball in $\mathbb Q_p^n$ and $\omega({\bf{x}})$ be a non-negative measurable function in $\mathbb Q_p^n$. A function $f\in L_{{\mathrm{loc}}}^q(\mathbb Q_p^n)$ is said to belong to the generalized Campanato space $\mathcal{L}^{q, \omega}(\mathbb Q_p^n)$, if

where$f_{B_\gamma({\bf a})} = \frac{1}{\left|B_\gamma({\bf a})\right|_{H}}\int_{B_\gamma({\bf a})}f({\bf{x}})d{\bf{x}}$.

We invoke the following result.

Lemma 2.1. ^[15,20] Let $1\leq q < \infty$ and $\omega$ be a non-negative measurable function. Suppose that $b\in\mathcal{L}^{q, \omega}(\mathbb Q_p^n)$, then

for any $j, k\in \mathbb Z$ and any fixed ${\bf a}\in \mathbb Q_p^n$. Thus, for $j > k$, we have

In addition, for $\lambda < 1/n$, if let $B_\gamma({\bf a}) = |B_\gamma({\bf a})|^{\lambda}$ in Definition 2.2, then $\mathcal{L}^{q, \omega}(\mathbb Q_p^n) = {\mathrm{BMO}}^{q, \lambda}(\mathbb Q_p^n)$. Moreover, let $B_\gamma({\bf a}) = B_\gamma({\bf 0})$, then $\mathcal{L}^{q, \omega}(\mathbb Q_p^n)$ is the $\lambda$-central BMO space ${\mathrm{CBMO}}^{q, \lambda}(\mathbb Q_p^n)$ (see ^[14,18]) defined by

Furthermore, when $\lambda = 0$, the particular case of ${\mathrm{CBMO}}^{q, \lambda}(\mathbb Q_p^n)$ is ${\mathrm{CBMO}}^{q}(\mathbb Q_p^n)$ defined in ^[21].

Now, we present two desired lemmas which will be used in the proof of our main results.

Lemma 2.2. ^[18] Let $0 < \beta < n$, $1\leq q < r < \infty$, $1/r = 1/q-\beta/n$, $\Omega\in L^{q'}(S_0({\bf 0}))$ and $f\in L^q(\mathbb Q_p^n)$.

(i) If $q > 1$, then

(ii) If $q = 1$, for any $\sigma > 0$, then

Lemma 2.3. Let $0 < \beta < n$, $1\leq q < r < \infty$, $1/r = 1/q-\beta/n$, $\Omega\in L^{q'}(S_0({\bf 0}))$, $f\in L^{q, \nu}(\mathbb Q_p^n)$ and $\nu$ is a non-negative measurable functionin $\mathbb Q_p^n$. For any $B_\gamma({\bf a})\in\mathbb{B}$, then

Proof. For any ${\bf x}\in B_\gamma({\bf a})$, we have $|{\bf x}-{\bf a}|_p\leq p^{\gamma}$. For any ${\bf y}$ satisfying $p^{k-1} < |{\bf y}-{\bf a}|_p\leq p^{k}$ for some $k\geq \gamma+1$, the property (ii) of $|\cdot|_p$ shows that $p^{k-1} < |{\bf y}-{\bf a}|_p\leq \max(|{\bf x}-{\bf y}|_p, |{\bf x}-{\bf a}|_p)$, the inequality $|{\bf x}-{\bf a}|_p\leq p^\gamma\leq p^{k-1}$ guarantees that $|{\bf x}-{\bf y}|_p > p^{k-1}$. Consequently, by Hölder's inequality, we have

Let nonzero ${\bf{y}}\in \mathbb Q_p^n$ has a form ${\bf{y}} = (y_1, \cdots, y_n)$, applying (2.1), we proceed as

Then there exists $i_0\in\{1, \cdots, n\}$ such that $|y_{i_0}|_p = p^{-\gamma_{i_0}}\geq p^{-\gamma_i} = |y_i|_p$, whenever $y_i\neq0$. Using (2.2), we obtain $|{\bf{y}}|_p = p^{-\gamma_{i_0}}$. It follows that

Thus, for every nonzero ${\bf{y}}\in \mathbb Q_p^n$, the vector $|{\bf{y}}|_p{\bf{y}}$ belongs to sphere $S_0(\bf{0}) = \left\{{\bf{y}}\in \mathbb Q_p^n:|{\bf{y}}|_p = 1\right\}$. Notice that $\Omega\in L^{q'}(S_0({\bf 0}))$, then

Hence, we obtain

Lemma 2.3 is proved. □

3.   Main results

Before giving the main results in this paper, according to the idea of the article ^[22], we will first state how to define the action of $T_{\beta, \Omega}^p$ on generalized Morrey spaces.

Definition 3.1. Let $0 < \beta < n$, $1\leq q < \infty$, $\Omega\in L^{q'}(S_0({\bf 0}))$, $T_{\beta, \Omega}^p$ be a fractional integral operator defined by (1.1), $\nu$ be a non-negative measurable function such that

For any $f\in L^{q, \nu}(\mathbb Q_p^n)$ and any fixed $B_\gamma({\bf a})\in\mathbb{B}$, define

Remark 3.1. For any $B_\gamma({\bf a})\in\mathbb{B}$ and $f\in L^{q, \nu}(\mathbb Q_p^n)$, write $f = f\chi_{B_\gamma({\bf a})}+f\chi_{B^c_\gamma({\bf a})}$. We can see that the definition of $L^{q, \nu}(\mathbb Q_p^n)$ assures that $f\chi_{B_\gamma({\bf a})}\in L^q(\mathbb Q_p^n)$, so Lemma 2.3 guarantees that $T_{\beta, \Omega}^p(f\chi_{B_\gamma({\bf a})})$ is well defined. Besides that, if $\nu$ satisfies (3.1), Lemma 2.3 implies

for ${\bf x}\in B_\gamma({\bf a})$. That is, $T_{\beta, \Omega}^p(f\chi_{B^c_\gamma({\bf a})})$ is well defined when $\nu$ satisfies (3.1). Consequently, the linearity of $T_{\beta, \Omega}^p$ on $L^q(\mathbb Q_p^n)$ yields

for $f\in L^{q, \nu}(\mathbb Q_p^n)$ and ${\bf x}\in B_\gamma({\bf a})\in\mathbb{B}$.

Now we give the boundedness result of $T_{\beta, \Omega}^p$ on generalized Morrey spaces in the following.

Theorem 3.1. Let $0 < \beta < n$, $1\leq q < r < \infty$, $1/r = 1/q-\beta/n$, $\Omega\in L^{q'}(S_0({\bf 0}))$ and $f\in L^{q, \nu}(\mathbb Q_p^n)$. Suppose that $\omega$ and $\nu$ are non-negative measurable functions such that

(i) If $q > 1$, then

(ii) If $q = 1$, for any $\sigma > 0$, $\gamma\in \mathbb Z$ and ${\bf a}\in \mathbb Q_p^n$, then

Remark 3.2. Notice that $\nu$ satisfy (3.1) if $\omega$ and $\nu$ satisfy (3.2). That is, (3.2) assures that $T_{\beta, \Omega}^p$ is well defined on $L^{q, \nu}(\mathbb Q_p^n)$.

Significantly, our results not only extend Theorem 1 in ^[18] to generalized Morrey spaces, but also extend Theorem 3.1 in ^[15] to rough $p$-adic fractional integral operator. At the same time, for $\lambda < -\beta/n$ and $\mu = \lambda+\beta/n$, if we take $\omega(B_\gamma({\bf a})) = |B_\gamma({\bf a})|_H^{\mu}$, $\nu(B_\gamma({\bf a})) = |B_\gamma({\bf a})|_H^{\lambda}$ for any fixed $B_\gamma({\bf a})$, it is easy to check that $\omega$ and $\nu$ satisfy (3.2). Hence, by Theorem 3.1, we can obtain the following corollary.

Corollary 3.1. Let $0 < \beta < n$, $1\leq q < r < \infty$, $1/r = 1/q-\beta/n$, $\lambda+\beta/n < 0$, $\mu = \lambda+\beta/n$, $\Omega\in L^{q'}(S_0({\bf 0}))$ and $f\in M_q^{\lambda}(\mathbb Q_p^n)$.

(i)If $q > 1$, then

(ii)If $q = 1$, for any $\sigma > 0$, $\gamma\in \mathbb Z$ and ${\bf a}\in \mathbb Q_p^n$, then

Our second main result is the boundedness of multilinear commutators generated by rough $p$-adic fractional integral operator and $p$-adic generalized Campanato functions.

Theorem 3.2. Let $m\in\mathbb{N}$, $0 < \beta < n$, $1 < q, q_1, \cdots, q_m < \infty$, $r > n/(n-\beta)$, $1/r = 1/q_1+\cdots+1/q_m+1/q-\beta/n$ and $\Omega\in L^{q'}(S_0({\bf 0}))$. Suppose that $\omega$, $\nu$ and $\nu_i$ $(i = 1, 2, \cdots, m)$ are non-negative measurable functions, and satisfy

and

for any $\gamma\in \mathbb Z$ and ${\bf a}\in \mathbb Q_p^n$. If $b_i\in \mathcal{L}^{q_i, \nu_i}(\mathbb Q_p^n)$, $f\in L^{q, \nu}(\mathbb Q_p^n)$, then

For $0\leq\lambda_1, \lambda_2, \cdots, \lambda_m < 1/n$, $\lambda+\Sigma\lambda_i+\beta/n < 0$ and $\mu = \lambda+\Sigma\lambda_i+\beta/n$, if we take $\omega(B_\gamma({\bf a})) = |B_\gamma({\bf a})|_H^{\mu}$, $\nu(B_\gamma({\bf a})) = |B_\gamma({\bf a})|_H^{\lambda}$ and $\nu_i(B_\gamma({\bf a})) = |B_\gamma({\bf a})|_H^{\lambda_i}$ for any $B_\gamma({\bf a})$, it is not difficult to check that $\omega$, $\nu$ and $\nu_i$ satisfy (3.3) and (3.4). Hence, Theorem 3.2 implies the following corollary.

Corollary 3.2. Let $m\in\mathbb{N}$, $0 < \beta < n$, $1 < q, q_1, \cdots, q_m < \infty$, $r > n/(n-\beta)$, $1/r = 1/q_1+\cdots+1/q_m+1/q-\beta/n$ and $\Omega\in L^{q'}(S_0({\bf 0}))$. Suppose that $0\leq\lambda_1, \lambda_2, \cdots, \lambda_m < 1/n$, $\lambda+\Sigma\lambda_i+\beta/n < 0$, $\mu = \lambda+\Sigma\lambda_i+\beta/n$. If $b_i\in {\mathrm{BMO}}^{q_i, \lambda_i}(\mathbb Q_p^n)$, $f\in M_q^{\lambda}(\mathbb Q_p^n)$, then

If we let $b_i\in {\mathrm{CBMO}}^{q_i, \lambda_i}(\mathbb Q_p^n)$, by Corollary 3.2, we will obtain the follwing boundedness of $T_{\beta, \Omega}^{p, {\bf b}}$ on central Morrey spaces.

Corollary 3.3. Let $m\in\mathbb{N}$, $0 < \beta < n$, $1 < q, q_1, \cdots, q_m < \infty$, $r > n/(n-\beta)$, $1/r = 1/q_1+\cdots+1/q_m+1/q-\beta/n$ and $\Omega\in L^{q'}(S_0({\bf 0}))$. Suppose that $0\leq\lambda_1, \lambda_2, \cdots, \lambda_m < 1/n$, $\lambda+\Sigma\lambda_i+\beta/n < 0$, $\mu = \lambda+\Sigma\lambda_i+\beta/n$. If $b_i\in {\mathrm{CBMO}}^{q_i, \lambda_i}(\mathbb Q_p^n)$, $f\in \dot{B}^{q, \lambda}(\mathbb Q_p^n)$, then

Here we point out that Corollary 3.3 extends Theorem 2 in ^[18] to the multilinear case.

4.   Proof of theorems

The proof of Theorem 3.1. As non-negative measurable functions $\omega$ and $\nu$ satisfy (3.2), $\nu$ fulfills (3.1), so Definition 3.1 assures that $T_{\beta, \Omega}^p$ is well defined on $L^{q, \nu}(\mathbb Q_p^n)$. For $f\in L^{q, \nu}(\mathbb Q_p^n)$ and any fixed $B_\gamma({\bf a})\in\mathbb{B}$, it follows that

Consequently, we only need to estimate $T_{\beta, \Omega}^p(f\chi_{B_\gamma({\bf a})})(\bf{x})$ and $T_{\beta, \Omega}^p(f\chi_{B^c_\gamma({\bf a})})(\bf{x})$ respectively.

(i) If $q > 1$, for fixed $B_\gamma({\bf a})$, we have

For $I$, by the $L^r$-boundedness of $T_{\beta, \Omega}^p$ (see Lemma 2.2) and (3.2), it follows that

For $II$, Lemma 2.3 yields that

Consequently, combining the estimation of $I$ and $II$, we have

which implies the desired inequality $\|T_{\beta, \Omega}^p f\|_{L^{r, \omega}(\mathbb Q_p^n)}\leq C \|f\|_{L^{q, \nu}(\mathbb Q_p^n)}$.

(ii) If $q = 1$, for fixed $B_\gamma({\bf a})$ and $T_{\beta, \Omega}^p(f\chi_{B_\gamma({\bf a})})(\bf{x})$, by the weak $L^r$-boundedness of $T_{\beta, \Omega}^p$ (see Lemma 2.2) and (3.2), we get

For $T_{\beta, \Omega}^p(f\chi_{B^c_\gamma({\bf a})})(\bf{x})$, by Chebychev's inequality, Lemma 2.3 and (3.2), we have

Thus, we obtain

Ultimately,

for any $\sigma > 0$, $\gamma\in \mathbb Z$ and ${\bf a}\in \mathbb Q_p^n$. This completes the proof of Theorem 3.1.

The proof of Theorem 3.2. In order to simplify the proving process, for a positive integer $m$ and $0\leq i\leq m$, we denote by $C_i^m$ the family of all finite subsets $\theta = \{\theta_1, \theta_2, \cdots, \theta_i\}$ of $\{1, 2, \cdots, m\}$ of $i$ different elements, let $\theta^c = \{1, 2, \cdots, m\}\backslash\theta$ for any $\theta\in C_i^m$. For ${\bf{b}} = (b_1, b_2, \cdots, b_m)$ and $\theta = \{\theta_1, \theta_2, \cdots, \theta_i\}\in C_i^m$, set ${\bf{b}}_\theta = (b_{\theta_1}, b_{\theta_2}, \cdots, b_{\theta_i})$ and the product $b_\theta = b_{\theta_1}b_{\theta_2}\cdots b_{\theta_i}$. With this notation, we write

and

where $1/\bar{q} = 1/\bar{q}_1+\cdots+1/\bar{q}_i$ and $\bar{\nu}(B_\gamma({\bf a})) = \bar{\nu}_1(B_\gamma({\bf a}))\cdots\bar{\nu}_i(B_\gamma({\bf a}))$ for any ball $B_\gamma({\bf a})$. Especially, when $i = m$, $\theta = \{1, 2, \cdots, m\}$ and $\theta^c = \emptyset$, we have ${\bf b}_\theta = {\bf b}$, hence

When $i = 0$, $\theta = \emptyset$ and $\theta^c = \{1, 2, \cdots, m\}$, we have${\bf b}_{\theta^c} = {\bf b}$, hence

We write $b_i({\bf{x}})-b_i({\bf{y}}) = (b_i({\bf{x}})-{b_i}_{B_\gamma})+(b_i({\bf{y}})-{b_i}_{B_\gamma})$ for $i = 1, 2, \cdots, m$, then

For $f\in L^{q, \nu}(\mathbb Q_p^n)$, $q > 1$, let $f = f\chi_{B_\gamma({\bf a})}+f\chi_{B^c_\gamma({\bf a})} = :f_1+f_2$, then we get

To facilitate estimates $J_1$ and $J_2$, set

and

then $1/r = 1/s+1/t$ and $g > 1$.

First, using Minkowski's inequality, Hölder's inequality and $L^t$-boundedness of $T_{\beta, \Omega}^{p}$, we obtain

Then, it is not difficult for us to get

Second, we will turn to the estimation of $J_2$. Given ${\bf{x}}\in B_\gamma({\bf a})$, by Hölder's inequality, Minkowski's inequality and Lemma 2.1, we have

Then, it follows that

At last, combining the estimation of $J_1$ and $J_2$, we obtain

for any $B_\gamma({\bf a})\subset \mathbb Q_p^n$ and $f\in L^{q, \nu}(\mathbb Q_p^n)(q > 1)$, and the proof of Theorem 3.2 is finished.

5.   Conclusions

In this article, the boundedness of the rough $p$-adic fractional integral operator on $p$-adic generalized Morrey spaces is studied. In addition, the boundedness for multilinear commutators generated by rough $p$-adic fractional integral operator and $p$-adic generalized Campanato functions is also obtained. Moreover, the boundedness in classical Morrey is given as corollaries.

Acknowledgments

We would like to thank the editors and reviewers for their helpful suggestions.

This research is partially supported by Teacher Professional Development Program of Domestic University Visiting Scholar in Zhejiang Province under grant No. FX2022076 and National Natural Science Foundation of China under grant No. 12271483.

Conflict of interest

The authors declare that they have no conflict of interest.