Boundedness of Commutators Generated by the Rough Fractional Maximal Operator on Variable Exponent Central Morrey Spaces

Abstract

In this paper, using harmonic analysis tools−including spherical harmonic decomposition of kernels, sharp maximal function estimates, and variable exponent space theory—we investigate the boundedness of the commutator $[b,{\mathcal{M}}_{\Omega ,\beta }]$ on variable exponent central Morrey spaces, under suitable regularity conditions on the variable exponents. Here, $\Omega \in L^l\left({\mathbf{S}}^{n-1}\right)$ ($l\ge 1$) denotes a zero-degree homogeneous function on the unit sphere ${\mathbf{S}}^{n-1}$, $\beta$ satisfies $0\le \beta <n$, and $b\in CBMO\left({\mathbf{R}}^n\right)$.

1. Introduction

This paper investigates the boundedness of the commutator of the rough fractional maximal operator on variable exponent central Morrey spaces. Variable exponent Morrey spaces, first introduced in [1], are a natural extension of classical Morrey spaces, offering a flexible framework for characterizing problems with non-standard growth behaviour—such as those arising in electrorheological fluids and image processing [2,3,4]—via spatially dependent integrability exponents. With the advancement of the theory of these spaces, the boundedness of operators defined on them has become one of the central topics in harmonic analysis.

Due to fundamental differences between variable exponent spaces and their classical counterparts, however, many established results cannot be directly extended, which calls for the construction of suitably adapted theoretical tools. In recent years, considerable progress has been achieved in studying operator boundedness on variable exponent central Morrey spaces [5,6,7]. Despite the aforementioned advances, most studies to date have centered on the analysis of individual operators, with relatively little attention devoted to their commutators. This is particularly true for commutators of fractional maximal operators with rough kernels, whose boundedness remains an outstanding and significant open problem in the field.

Against this backdrop, Wang, Liu, Zhou, and Teng introduced variable exponent central bounded mean oscillation (CBMO) spaces in 2017 and established several boundedness results for associated operators in [6]. Subsequently, in 2019, Fu, Lu, Wang, and Wang constructed homogeneous variable exponent central Morrey spaces and studied the boundedness of singular integral operators with rough kernels together with their commutators in [8]. Building on this foundation, Husain et al. in [9] focused on fractional Hardy operators and their commutators, systematically deriving norm estimates and proving their boundedness on variable–exponent central Morrey spaces. Meanwhile, Ma in [10] investigated more general multilinear Calderón-Zygmund operators with Dini-type kernels and their commutators, verifying the boundedness of these multilinear operators and the commutators generated by variable-exponent $\lambda$-central BMO functions on the same spaces. Together, these studies provide more refined analytical tools for problems in partial differential equations and expand the applicability of operator theory in variable-exponent spaces.

Building on these earlier works, the present paper aims to systematically investigate the boundedness of the commutator formed by the rough fractional maximal operator and a CBMO function on variable exponent central Morrey spaces. Using extrapolation techniques (detailed in Section 3), we establish corresponding results within the variable exponent framework.

Our main contribution is the first systematic characterization of the boundedness of commutators of rough-kernel fractional maximal operators on variable exponent central Morrey spaces, filling the gap in the intersection of rough kernel operators, commutators, and variable exponent central spaces. Unlike previous works that only focus on single operators or commutators with smooth kernels, our results not only extend classic conclusions in harmonic analysis but also provide new tools and ideas for operator theory on variable exponent spaces through the innovative use of extrapolation techniques. This work deepens the fundamental understanding of commutators and rough kernel operators, and offers solid theoretical support for applications like partial differential equations and image processing that rely on variable exponent space modeling, making it both academically valuable and practically meaningful.

The remainder of the paper is organized as follows. Section 2 introduces necessary definitions and lemmas and states the main theorems. Section 3 completes the proof of the main results by employing extrapolation methods in the setting of variable exponent spaces. Section 4 demonstrates the practical value of the theoretical results in relevant real−world problems.

2. Preliminaries

As a fundamental class of variable exponent function spaces, variable exponent Lebesgue spaces extend beyond classical Lebesgue theory to offer a natural framework for problems exhibiting non-standard growth. Below we introduce their basic definition and essential notation, which underpin the analysis that follows.

Let $E\subset {\mathbb{R}}^n$ be an open set, and let $q(·):E\to [1,\infty )$ be a measurable function. We define the variable exponent Lebesgue space $L^{q(·)}\left(E\right)$ as the set of measurable functions f on E for which there exists $\eta >0$ such that

$${\int }_E{\left({\displaystyle \frac{\left|f\right(x\left)\right|}{\eta }}\right)}^{q\left(x\right)}dx<\infty .$$

(1)

Endowed with the Luxemburg–Nakano norm:

$${\parallel f\parallel }_{L^{q(·)}\left(E\right)}=\inf \left\{\eta >0:{\int }_E{\left({\displaystyle \frac{\left|f\right(x\left)\right|}{\eta }}\right)}^{q\left(x\right)}dx\le 1\right\},$$

(2)

$L^{q(·)}\left(E\right)$ forms a Banach function space. These spaces are termed variable $L^q$ spaces, as they generalize classical $L^q$ spaces: if $q\left(x\right)=q$(constant), $L^{q(·)}\left(E\right)$ is isometrically isomorphic to $L^q\left(E\right)$.

The local variable $L^q$ space $L_{loc}^{q(·)}\left(E\right)$ is defined as

$$L_{loc}^{q(·)}\left(E\right):=\left\{f:f\in L^{q(·)}\left(F\right)forallcompactsubsetsF\subset E\right\}.$$

(3)

Define $\mathcal{P}\left(E\right)$ as the collection of measurable functions $q(·):E⟶[1,\infty )$ satisfying

$$q^{-}=ess\inf \{q\left(x\right):x\in E\}>1,q^{+}=ess\sup \{q\left(x\right):x\in E\}<\infty .$$

For $q(·)\in \mathcal{P}\left(E\right)$, its conjugate exponent $q^{\prime }(·)$ is determined by the identify $1/q\left(x\right)+1/q^{\prime }\left(x\right)=1$ for all $x\in E$.

A key condition for the boundedness of classical operators in the variable exponent setting is the log-Hölder continuity of the exponent. An exponent $q(·)\in \mathcal{P}\left(E\right)$ is said to belong to the class $\mathcal{B}\left(E\right)$ if the Hardy–Littlewood maximal operator M is bounded on $L^{q(·)}\left(E\right)$. The following lemma provides a sufficient condition for this property.

Lemma 1

([11]). Given an open set $E\subset {\mathbf{R}}^n$. If $q(·)\in \mathcal{P}\left(E\right)$ and satisfies

$$\begin{array}{c}\hfill |q\left(x\right)-q\left(z\right)|\le {\displaystyle \frac{C}{-log\left(\right|x-z\left|\right)}},|x-z|\le 1/2,\end{array}$$

(4)

and

$$\begin{array}{c}\hfill |q\left(x\right)-q\left(z\right)|\le {\displaystyle \frac{C}{-log\left(\right|x|+e)}},\left|z\right|\ge \left|x\right|,\end{array}$$

(5)

then $q(·)\in \mathcal{B}\left(E\right),$ that is, the Hardy-Littlewood maximal operator M bounded on $L^{q(·)}\left(E\right)$.

Lemma 1 establishes the log–Hölder continuity condition for variable exponents. This is a basic smoothness requirement. It guarantees that the Hardy–Littlewood maximal operator M is bounded on the variable exponent space $L^{q(·)}\left(E\right)$. This result is essential for verifying the well-posedness of the variable exponent framework. It also directly supports the development of core functional analysis tools tailored to this framework.

Building on the theoretical foundation laid by Lemma 1, we now present the generalized Hölder inequality.

Lemma 2

([12]). Given an open set $E\subset {\mathbf{R}}^n$ and let $q(·)\in \mathcal{P}\left(E\right)$. If $f\in L^{q(·)}\left(E\right)$ and $g\in L^{q^{\prime }(·)}\left(E\right),$ then $fg$ is integrable on E and

$$\begin{array}{c}\hfill {\int }_E\left|f\left(x\right)g\left(x\right)\right|dx\le r_q{\parallel f\parallel }_{L^{q(·)}\left(E\right)}{\parallel g\parallel }_{L^{q^{\prime }(·)}\left(E\right)},\end{array}$$

(6)

where

$$r_q=1+1/q^{-}-1q^{+}.$$

This inequality is referred to as the generalized Hölder inequality adapted to variable $L^q$ spaces.

Remark 1.

The log–Hölder continuity condition in Lemma 1 is a regularity property of the exponent q. It provides support for operator boundedness. The generalized Hölder inequality in Lemma 2 is an independent integral estimation tool. It is used to control the integral of the product of functions in variable exponent spaces. It has no relation to the smoothness of the exponent.

For Morrey-type estimates, it is often essential to compare the norms of characteristic functions supported on cubes or balls. The following two lemmas provide such local norm comparisons, which play a crucial role in handling the scaling properties of central Morrey spaces.

Lemma 3

([13]). Let $q(·)\in \mathcal{B}\left({\mathbf{R}}^n\right)$. Then there exists a positive constant C such that for all balls B in ${\mathbf{R}}^n$ and all measurable subsets $S\subset B,$

$$\begin{array}{c}\hfill {\displaystyle \frac{\parallel {\chi }_B{\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}}{\parallel {\chi }_S{\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}}}\le C{\displaystyle \frac{\left|B\right|}{\left|S\right|}},{\displaystyle \frac{\parallel {\chi }_S{\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}}{\parallel {\chi }_B{\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}}}\le C{\left({\displaystyle \frac{\left|S\right|}{\left|B\right|}}\right)}^{{\delta }_1},{\displaystyle \frac{\parallel {\chi }_S{\parallel }_{L^{q^{\prime }(·)}\left({\mathbf{R}}^n\right)}}{\parallel {\chi }_B{\parallel }_{L^{q^{\prime }(·)}\left({\mathbf{R}}^n\right)}}}\le C{\left({\displaystyle \frac{\left|S\right|}{\left|B\right|}}\right)}^{{\delta }_2},\end{array}$$

(7)

where $0<{\delta }_1,{\delta }_2<1$ depend on $q(·)$.

Lemma 3 establishes quantitative norm ratios between the characteristic functions of a ball and its subsets. These ratios are critical for managing local scaling behavior in variable exponent spaces. From these results, we directly infer the uniform boundedness of the product of the characteristic function norms of the same ball. In what follows, we present the formal statement of this boundedness result.

Lemma 4

([13]). Suppose $q(·)\in \mathcal{B}\left({\mathbf{R}}^n\right)$. Then there exists a positive constant C such that for all balls B in ${\mathbf{R}}^n$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{B}}\parallel {\chi }_B{\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}{\parallel {\chi }_B\parallel }_{L^{q^{\prime }(·)}\left({\mathbf{R}}^n\right)}\le C.\end{array}$$

(8)

We have established core norm estimates for characteristic functions in variable exponent spaces via Lemmas 3 and 4. Building on this foundation, we next introduce the core function spaces that support the analysis of operator boundedness.These spaces are the λ−central BMO space and the central Morrey space. They are variable exponent generalizations of the corresponding classical spaces.

These spaces extend their classical counterparts by allowing the integrability exponent to depend on the spatial variable. They provide a natural framework for capturing functions with inhomogeneous local behavior. We now present their precise definitions:

Definition 1

([8]). Let $q(·)\in \mathcal{P}\left({\mathbf{R}}^n\right)$ and $\lambda <{\displaystyle \frac{1}{n}}$. The λ−central BMO space with variable exponent $CBM{O}^{q(·),\lambda }\left({\mathbf{R}}^n\right)$ is defined by

$$\begin{array}{c}\hfill CBM{O}^{q(·)\lambda }\left({\mathbf{R}}^n\right)=\{f\in L_{loc}^{q(·)}\left({\mathbf{R}}^n\right){:\parallel f\parallel }_{CBM{O}^{q(·),\lambda }\left({\mathbf{R}}^n\right)}<\infty \},\end{array}$$

(9)

where

$$\begin{array}{c}\hfill {\parallel f\parallel }_{CBM{O}^{q(·),\lambda }\left({\mathbf{R}}^n\right)}=\underset{R>0}{sup}{\displaystyle \frac{\parallel (f-f_{B(0,R)}){\chi }_{B(0,R)}{\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}}{{\left|B(0,R)\right|}^{\lambda }{\parallel {\chi }_{B(0,R)}\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}}}.\end{array}$$

(10)

Definition 2

([8]). Let $q(·)\in \mathcal{P}\left({\mathbf{R}}^n\right)$ and $\lambda \in R$. The central Morrey space with variable exponent ${\dot{B}}^{q(·),\lambda }\left({\mathbf{R}}^n\right)$ is defined by

$$\begin{array}{c}\hfill {\dot{B}}^{q(·),\lambda }\left({\mathbf{R}}^n\right)=\{f\in L_{loc}^{q(·)}\left({\mathbf{R}}^n\right){:\parallel f\parallel }_{{\dot{B}}^{q(·),\lambda }\left({\mathbf{R}}^n\right)}<\infty \},\end{array}$$

(11)

where

$$\begin{array}{c}\hfill {\parallel f\parallel }_{{\dot{B}}^{q(·),\lambda }\left({\mathbf{R}}^n\right)}=\underset{R>0}{sup}{\displaystyle \frac{\parallel f{\chi }_{B(0,R)}{\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}}{{\left|B(0,R)\right|}^{\lambda }{\parallel {\chi }_{B(0,R)}\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}}}.f_{B(0,R)}={\displaystyle \frac{1}{\left|B\right(0,R\left)\right|}}{\int }_{B(0,R)}f\left(y\right)dy\end{array}$$

(12)

These two spaces provide the essential functional framework for our investigation. we first need to introduce several key lemmas as technical support, these lemmas will characterize the integral behavior of the rough kernel and the embedding properties of variable-exponent spaces, respectively, providing core tools for the subsequent derivation of the operator boundedness.

Remark 2.

Throughout this paper, for any measurable set $B\subset {\mathbb{R}}^n$ with $\left|B\right|>0$, the integral average of a locally integrable function f over B is denoted $f_B={\displaystyle \frac{1}{\left|B\right|}}{\int }_{B}f\left(y\right)dy$. For the ball $B(0,R)=\{x\in {\mathbb{R}}^n:|x|<R\}$, we have $f_{B(0,R)}={\displaystyle \frac{1}{{\omega }_n{R}^n}}{\int }_{B(0,R)}f\left(y\right)dy$, where ${\omega }_n$ is the volume of the unit ball in ${\mathbb{R}}^n$.

The indicator function of a measurable set $E\subset {\mathbb{R}}^n$ is ${\chi }_E\left(x\right)=1$ if $x\in E$ and 0 otherwise; for $B(0,R)$, this becomes ${\chi }_{B(0,R)}\left(x\right)=1$ if $\left|x\right|<R$ and 0 otherwise.

These tools are essential for analyzing local function behavior in BMO (bounded mean oscillation), Morrey and variable exponent spaces.

Fractional maximal operators are core research objects in harmonic analysis and closely related to partial differential equations (PDEs). Based on the variable exponent central Morrey spaces and central BMO spaces introduced earlier, this paper carries out relevant research. These spaces support the analysis of the boundedness of rough kernel operators and their commutators-key to characterizing the nonlinear effects of operators and the regularity of functions. We will present the definition of the core research object of this paper, namely the commutators of rough fractional maximal operators, and analyze their boundedness in the variable exponent framework.

Let ${\mathbf{S}}^{n-1}$ stand for the unit sphere within ${\mathbf{R}}^n$, and let $\Omega \in L^l\left({\mathbf{S}}^{n-1}\right)$($l\ge 1$) be a homogeneous function of degree zero-meaning $\Omega \left(\lambda x\right)=\Omega \left(x\right)$ holds for all $\lambda >0$ and $x\in {\mathbf{R}}^n$. For any $0\le \beta <n$, the fractional maximal operator with a rough kernel is formulated as

$$M_{\Omega ,\beta }f\left(x\right)=\underset{Q}{\sup }{\displaystyle \frac{1}{{\left|Q\right|}^{1-\frac{\beta }{n}}}}{\int }_Q|\Omega (x-z)|\left|f\left(z\right)\right|dz,$$

(13)

where Q denotes a cube in ${\mathbf{R}}^n$.

Correspondingly, for a locally integrable function b, the commutator induced by b and $M_{\Omega ,\beta }$ is defined as

$$\begin{array}{cc}\hfill [b,M_{\Omega ,\beta }]f\left(x\right)& =M_{\Omega ,\beta }\left(bf\right)\left(x\right)-b\left(x\right)M_{\Omega ,\beta }f\left(x\right)\hfill \\ \hfill & =\underset{Q}{\sup }{\displaystyle \frac{1}{{\left|Q\right|}^{1-\frac{\beta }{n}}}}{\int }_Q|\Omega (x-z)|\left|f\left(z\right)\right|[b\left(x\right)-b\left(z\right)]dz.\hfill \end{array}$$

(14)

Building on the spatial framework, preliminary lemmas and operator definitions established earlier, the main result of this paper can be stated as the following theorem. This theorem establishes the boundedness conditions for the commutator $[b,M_{\Omega ,\beta }]$ between variable exponent central Morrey spaces.

Theorem 1.

Suppose that $0\le \beta <n$, and let $\Omega \in L^l\left({\mathbf{S}}^{n-1}\right)$ with $l>{\displaystyle \frac{n}{n-\beta }}$, Let $q_1(·)$, $q_2(·)$, $p(·)\in \mathcal{P}\left({\mathbf{R}}^n\right)$ satisfies conditions (4) and (5) given in Lemma 1, where $q_1(·)<{\displaystyle \frac{n}{\beta }}$, $q_1^{\prime }(·)<q_2(·)$, and ${\displaystyle \frac{1}{p(·)}}={\displaystyle \frac{1}{q_1(·)}}+{\displaystyle \frac{1}{q_2(·)}}-{\displaystyle \frac{\beta }{n}}$. Let $0\le {\lambda }_2<{\displaystyle \frac{1}{n}}$, and suppose ${\lambda }_1$ satisfies one of the following two conditions:

(1) If ${\displaystyle \frac{1}{q_2(·)}}+{\displaystyle \frac{1}{l}}={\displaystyle \frac{1}{q_1^{\prime }(·)}}$, then ${\lambda }_1<-{\lambda }_2-{\displaystyle \frac{\beta }{n}}$;

(2) If ${\displaystyle \frac{1}{q_1(·)}}={\displaystyle \frac{\beta }{n}}+{\displaystyle \frac{1}{l}}$, then ${\lambda }_1<-{\lambda }_2-{\displaystyle \frac{\beta }{n}}-{\displaystyle \frac{1}{l}}$,

Define $\lambda ={\lambda }_1+{\lambda }_2+{\displaystyle \frac{\beta }{n}}$. If $b\in CBM{O}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)$, then the commutator $[b,M_{\Omega ,\beta }]$ is bounded from ${\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)$ to ${\dot{\mathcal{B}}}^{p(·),\lambda }\left({\mathbf{R}}^n\right)$, and the following inequality holds:

$$\begin{array}{c}\hfill \parallel [b,M_{\Omega ,\beta }]{f\parallel }_{{\dot{\mathcal{B}}}^{p(·),\lambda }\left({\mathbf{R}}^n\right)}\le {C\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{CBM{O}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}.\end{array}$$

(15)

To establish the conclusion of Theorem 1, we require the following preliminary lemmas. They serve as key technical foundations for the proof.

The next two lemmas provide such mixed-norm estimates, which are indispensable when dealing with multi-parameter situations arising in commutator estimates.

Lemma 5

([14]). Define a variable exponent $\tilde{p}(·)$ by ${\displaystyle \frac{1}{q\left(x\right)}}={\displaystyle \frac{1}{\tilde{p}\left(x\right)}}+{\displaystyle \frac{1}{p}}(x\in {\mathbf{R}}^n).$ Then we have

$$\begin{array}{c}\hfill {\parallel fg\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}\le {C\parallel f\parallel }_{L^{\tilde{p}(·)}\left({\mathbf{R}}^n\right)}{\parallel g\parallel }_{L^p\left({\mathbf{R}}^n\right)}\end{array}$$

(16)

for all measurable functions f and g.

A more symmetric version of the preceding result, which is often more convenient in applications, is the following.

Lemma 6

([15]). Let $q(·),p(·),l(·)\in \mathcal{P}\left({\mathbf{R}}^n\right)$ be such that

$${\displaystyle \frac{1}{l\left(x\right)}}={\displaystyle \frac{1}{q\left(x\right)}}+{\displaystyle \frac{1}{p\left(x\right)}}$$

for almost every $x\in {\mathbf{R}}^n$. Then

$$\begin{array}{c}\hfill {\parallel fg\parallel }_{L^{l(·)}\left({\mathbf{R}}^n\right)}\le {2\parallel f\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}{\parallel g\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\end{array}$$

(17)

for all $f\in L^{q(·)}\left({\mathbf{R}}^n\right)$ and $g\in L^{p(·)}\left({\mathbf{R}}^n\right)$.

To effectively apply the inequalities presented above, one often needs explicit asymptotic estimates for the norms of characteristic functions on cubes. The next lemma provides a scaling relation that yields such valuable estimates.

Lemma 7

([15]). Let $q(·)\in \mathcal{P}\left({\mathbf{R}}^n\right)$ satisfy conditions (4) and (5) in Lemma 1. Then

$$\begin{array}{c}\hfill {\parallel {\chi }_Q\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}\approx \left\{\begin{array}{cc}{\left|Q\right|}^{{\displaystyle \frac{1}{q\left(x\right)}}}\hfill & if\left|Q\right|\le 2^{n}andx\in Q\hfill \\ {\left|Q\right|}^{{\displaystyle \frac{1}{q(\infty )}}}\hfill & if\left|Q\right|\ge 1\hfill \end{array}\right.,\end{array}$$

(18)

for every cube (or ball) $Q\subset {\mathbf{R}}^n$, where $q(\infty )={lim}_{x\to \infty }q\left(x\right)$.

A pivotal technique in dealing with integral operators consists in exchanging the order of integration and norm evaluation. The following lemma provides a Minkowski−type inequality adapted to variable exponent spaces, which plays an essential role in our main estimates.

Lemma 8

([16]). Given a measurable set E and $q(·)\in \mathcal{P}\left(E\right)$, let $f:E\times E\to \mathbf{R}$ be a measurable function (with respect to the product measure) such that for almost every $z\in E$, $f(·,z)\in L^{q(·)}\left(E\right)$. Then

$$\begin{array}{c}\hfill {∥{\int }_{E}f(·,z)dz∥}_{L^{q(·)}\left(E\right)}\le C{\int }_E{\parallel f(·,z)\parallel }_{L^{q(·)}\left(E\right)}dz.\end{array}$$

(19)

The lemmas presented in this section collectively form a coherent toolkit for addressing commutator problems in variable-exponent Morrey spaces. They are interconnected and mutually reinforcing, establishing a unified and rigorous analytical framework. This framework will provide a methodological foundation for subsequently proving the boundedness of the commutator $[b,M_{\Omega ,\beta }]$ on the corresponding variable–exponent central Morrey spaces.

The proof of this theorem is based on the lemmas established in Section 3 and will be accomplished by appropriately decomposing the function and controlling the norms of the local terms. Below we give the detailed proof.

3. Proof of Theorem

In order to prove Theorem 1 concerning the boundedness of the commutator of the fractional maximal operator with rough kernels on variable exponent central Morrey spaces, we require the following lemmas.

Lemma 9

([17]). Let Ω be an open set in ${\mathbb{R}}^n$, and $\mathcal{F}$ be a collection of pairs of non-negative functions. Suppose that $0<q_0\le p_0<\infty$, and for any $w\in A_1$, there holds

$${\left({\int }_{{\mathbb{R}}^n}f{\left(x\right)}^{p_0}w\left(x\right)dx\right)}^{1/p_0}\le C_0{\left({\int }_{{\mathbb{R}}^n}g{\left(x\right)}^{q_0}w{\left(x\right)}^{q_0/p_0}dx\right)}^{1/q_0},(f,g)\in \mathcal{F}.$$

We denote $q^{-}={\inf }_{x\in {\mathbb{R}}^n}q\left(x\right)$ and $q^{+}={\sup }_{x\in {\mathbb{R}}^n}q\left(x\right)$. If $q(·)\in \mathcal{P}\left({\mathbb{R}}^n\right)$ and ${\displaystyle p_0<q^{-}\le q^{+}<\frac{q_0{p}_0}{p_0-q_0}}$, we define the function $p(·)$ by

$${\displaystyle \frac{1}{q\left(x\right)}}-{\displaystyle \frac{1}{p\left(x\right)}}={\displaystyle \frac{1}{q_0}}-{\displaystyle \frac{1}{p_0}},x\in {\mathbb{R}}^n.$$

If $q(·)\in LH\left({\mathbb{R}}^n\right)$ and the function pair $(f,g)\in \mathcal{F}$ satisfies $f\in L^{p(·)}\left({\mathbb{R}}^n\right)$, then

$${\parallel f\parallel }_{L^{p(·)}\left({\mathbb{R}}^n\right)}\le C{\parallel g\parallel }_{L^{q(·)}\left({\mathbb{R}}^n\right)},$$

(20)

where the constant C is independent of the function pair $(f,g)$.

The general extrapolation principle established in Lemma 9 enables the transfer of weighted norm inequalities into the variable exponent framework. To apply it, a suitable initial weighted estimate is required. The following lemma provides precisely such an estimate for the rough fractional maximal operator, which will serve as the concrete input $(g=M_{\Omega ,\beta }f,f=f)$ in the extrapolation scheme.

Lemma 10

([18]). Let $0<\beta <n$, $1\le l^{\prime }<q<{\displaystyle \frac{n}{\beta }}$ and ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{q}}-{\displaystyle \frac{\beta }{n}}$. If $\Omega \in L^{l^{\prime }}\left({\mathbf{S}}^{n-1}\right)$ and $w^{l^{\prime }}\in A\left({\displaystyle \frac{q}{l^{\prime }}},{\displaystyle \frac{p}{l^{\prime }}}\right)$, then there exists a constant C independent of f such that

$$\begin{array}{c}\hfill {\left({\int }_{{\mathbf{R}}^n}{\left|M_{\Omega ,\beta }f\left(x\right)w\left(x\right)\right|}^{p}dx\right)}^{1/p}\le C{\left({\int }_{{\mathbf{R}}^n}{\left|f\left(x\right)w\left(x\right)\right|}^{q}dx\right)}^{1/q}.\end{array}$$

(21)

Based on Lemmas 9 and 10, we can obtain the following lemma and its proof.

Lemma 11.

Let $p(·),q(·)\in \mathcal{P}\left({\mathbf{R}}^n\right)$, $0<\beta <n$, $1<p^{-}\le p^{+}<{\displaystyle \frac{n}{\beta }}$ and ${\displaystyle \frac{1}{q\left(x\right)}}={\displaystyle \frac{1}{p\left(x\right)}}-{\displaystyle \frac{\beta }{n}}$. If $p(·)\in LH\left({\mathbf{R}}^n\right)$, $\Omega \in L^l\left({\mathbf{S}}^{n-1}\right)$ and $1\le l^{\prime }<p^{-}$, then

$$\begin{array}{c}\hfill \parallel M_{\Omega ,\beta }{f\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}\le C{\parallel f\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}.\end{array}$$

(22)

Proof of Lemma 11. 

Choose $0<q_0\le p_0<\infty$, such that $l^{\prime }<q_0<q^{-}$ and $1/p_0=1/q_0-\beta /n$. For any weight function $W\left(x\right)=w{\left(x\right)}^{p_0}\in A_1$ and any cube Q, we have

$${\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}w{\left(x\right)}^{p_0}dx\le C\underset{x\in Q}{\inf }w{\left(x\right)}^{p_0}$$

and

$${\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}w{\left(x\right)}^{-l^{\prime }{\left({\displaystyle \frac{q_0}{l^{\prime }}}\right)}^{\prime }}dx\right)}^{{\displaystyle \frac{1}{{\left(\frac{q_0}{l^{\prime }}\right)}^{\prime }}}}\le \underset{x\in Q}{\sup }w{\left(x\right)}^{-l^{\prime }}={\left(\underset{x\in Q}{\inf }w\left(x\right)\right)}^{-l^{\prime }}.$$

Thus, we obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}w{\left(x\right)}^{l^{\prime }\left({\displaystyle \frac{p_0}{l^{\prime }}}\right)}dx\right)}^{{\displaystyle \frac{l^{\prime }}{p_0}}}{\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}w{\left(x\right)}^{-l^{\prime }{\left({\displaystyle \frac{q_0}{l^{\prime }}}\right)}^{\prime }}dx\right)}^{{\displaystyle \frac{1}{{\left(\frac{q_0}{l^{\prime }}\right)}^{\prime }}}}& \le C^{{\displaystyle \frac{l^{\prime }}{p_0}}}{\left(\underset{Q}{\inf }w\right)}^{l^{\prime }}·{\left(\underset{Q}{\inf }w\right)}^{-l^{\prime }}\le C.\hfill \end{array}$$

From this, $w^{l^{\prime }}\in A\left({\displaystyle \frac{q_0}{l^{\prime }}},{\displaystyle \frac{p_0}{l^{\prime }}}\right)$. According to Lemma 10, we have

$${\left({\int }_{{\mathbb{R}}^n}{\left|M_{\Omega ,\beta }f\left(x\right)\right|}^{p_0}w{\left(x\right)}^{p_0}dx\right)}^{1/p_0}\le C{\left({\int }_{{\mathbb{R}}^n}{\left|f\left(x\right)\right|}^{q_0}w{\left(x\right)}^{q_0}dx\right)}^{1/q_0}.$$

Finally, we choose variable exponent functions $q(·)$ and $p(·)$ such that $q_0<q^{-}\le q^{+}<q_0{p}_0/(p_0-q_0)$, $q(·)\in LH\left({\mathbf{R}}^n\right)$, and

$${\displaystyle \frac{1}{q\left(x\right)}}-{\displaystyle \frac{1}{p\left(x\right)}}={\displaystyle \frac{1}{q_0}}-{\displaystyle \frac{1}{p_0}}.$$

By Lemma 9, we have

$$\begin{array}{c}\hfill \parallel M_{\Omega ,\beta }{f\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\le C{\parallel f\parallel }_{L^{q(·)}\left({\mathbf{R}}^n\right)}.\end{array}$$

(23)

Thus, the proof of Lemma 11 is concluded. □

Based on the boundedness result of the operator established in Lemma 11, we now proceed to prove the main theorem of this paper (Theorem 1). The following proof will employ this lemma together with norm estimation techniques in variable exponent spaces.

Proof of Theorem 1. 

We begin by assuming that $f\in {\dot{B}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)$ and fixing a ball $B=B(0,r)$. Our goal is to establish the norm inequality

$$\begin{array}{c}\hfill {\parallel [b,M_{\Omega ,\beta }]f{\chi }_B\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\le {C\left|B\right|}^{\lambda }{\parallel f\parallel }_{{\dot{B}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}{\parallel {\chi }_B\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)},\end{array}$$

(24)

where C is a constant independent of $\mathbf{R}$.

To this end, we first split the function f into the near part $f{\chi }_{2B}$ and the far part $f{\chi }_{{\left(2B\right)}^c}$. By means of the Minkowski inequality, the commutator can be correspondingly decomposed into four terms, denoted by $J_1,J_2,J_3,J_4$. In the sequel, we estimate each term separately, using a combination of integral estimates, variable–exponent norm inequalities, and properties of central BMO and central Morrey spaces.

For any $x\in {\mathbf{R}}^n$, given any ball $B=B(x,r)$, where $r=\sqrt{n}l\left(Q\right)$ and $l\left(Q\right)$ denotes the side length of the cube Q, decompose $f=f{\chi }_{2B}+f{\chi }_{{\left(2B\right)}^c}$. Then using the Minkowski inequality, we obtain

$$\begin{array}{cc}\hfill \parallel [b,M_{\Omega ,\beta }]f{\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}& \le \parallel (b-b_B)\left(M_{\Omega ,\beta }\left(f{\chi }_{2B}\right)\right){\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & +\parallel (b-b_B)\left(M_{\Omega ,\beta }\left(f{\chi }_{{\left(2B\right)}^c}\right)\right){\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & +\parallel M_{\Omega ,\beta }(b-b_B)\left(f{\chi }_{2B}\right){\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & +\parallel M_{\Omega ,\beta }(b-b_B)\left(f{\chi }_{{\left(2B\right)}^c}\right){\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & :=J_1+J_2+J_3+J_4.\hfill \end{array}$$

(25)

We begin by estimating $J_1$, which contains the near part of f. Let $1/q_1(·)-1/s(·)=\beta /n$, so that $1/p(·)=1/q_2(·)+1/s(·)$. Given $1<q_1(·)<n/\beta$, application of Lemma 3, Lemmas 6, 7, and 11, we obtain

$$\begin{array}{cc}\hfill J_1& \le \parallel (b-b_B){\chi }_B{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}{\parallel M_{\Omega ,\beta }\left(f{\chi }_{2B}\right)\parallel }_{L^{s(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le \parallel (b-b_B){\chi }_B{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\parallel f{\chi }_{2B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C\parallel b\parallel }_{{\mathrm{CMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}{\left|B\right|}^{{\lambda }_2}\parallel {\chi }_B{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}{\parallel f{\chi }_{2B}\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}{\left|B\right|}^{{\lambda }_2}\parallel {\chi }_B{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}{\left|2B\right|}^{{\lambda }_1}{\parallel {\chi }_{2B}\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C\left|B\right|}^{\lambda }{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}{\parallel {\chi }_B\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)},\hfill \end{array}$$

(26)

where

$$\begin{array}{c}\hfill \parallel {\chi }_B{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}{\parallel {\chi }_B\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\approx {\left|B\right|}^{1/q_1(·)}·{\left|B\right|}^{1/q_2(·)}\hfill \\ \hfill ={\left|B\right|}^{\beta /n}·{\left|B\right|}^{1/p(·)}\hfill \\ \hfill \hspace{1em}\hspace{1em}\approx {\left|B\right|}^{\beta /n}{\parallel {\chi }_B\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

(27)

This completes the estimate for term $J_1$, thereby establishing the necessary bound for this part of the proof.

Next, we proceed to estimate the term $J_3$. To this end, we define the exponent function $t(·)$ by the relation

$${\displaystyle \frac{1}{t(·)}}={\displaystyle \frac{1}{q_1(·)}}+{\displaystyle \frac{1}{q_2(·)}}.$$

This directly implies its connection with $p(·)$:

$${\displaystyle \frac{1}{t(·)}}-{\displaystyle \frac{1}{p(·)}}={\displaystyle \frac{\beta }{n}}.$$

Given the condition $1<q_1(·)<n/\beta$ and $q_1^{\prime }(·)<q_2(·)<\infty$, it follows that $1<t(·)<n/\beta$. Now, by combining Lemmas 4, 6, 7, and 11, we obtain the following estimate:

$$\begin{array}{cc}\hfill J_3& \le C\parallel (b-b_B)f{\chi }_{2B}{\parallel }_{L^{t(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\parallel (b-b_B){\chi }_{2B}{\parallel }_{L^{q_2(·)}({\mathbf{R}}^n)}{\parallel f{\chi }_{2B}\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\left[\parallel (b-b_{2B}){\chi }_{2B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}+|b_{2B}-b_B|\parallel {\chi }_{2B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\right]{\parallel f{\chi }_{2B}\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\parallel (b-b_{2B}){\chi }_{2B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}{\parallel f{\chi }_{2B}\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C\left|2B\right|}^{{\lambda }_2}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}{\parallel f{\chi }_{2B}\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}{\left|2B\right|}^{{\lambda }_1+{\lambda }_2}{\parallel {\chi }_{2B}\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C\left|B\right|}^{\lambda }{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

(28)

where

$$\begin{array}{cc}\hfill |b_{2B}-b_B|& \le {\displaystyle \frac{1}{\left|B\right|}}\parallel (b-b_{2B}){\chi }_{2B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2B}{\parallel }_{L^{q_2^{\prime }(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\parallel (b-b_{2B}){\chi }_{2B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}{\displaystyle \frac{1}{{\parallel {\chi }_{2B}\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}}}.\hfill \end{array}$$

(29)

We now turn to estimating the terms $J_2$ and $J_4$. To this end, we distinguish two distinct cases based on the relations among the exponents and treat them separately.

(i) $1/q_2(·)=1/l+1/q_1^{\prime }(·)$.

For $x\in B$ and $z\in 2^{j+1}B$, the triangle inequality readily yields $x-z\in 2^{j+2}B$. Notice that the kernel $\Omega$ is homogeneous of degree zero and satisfies the integrability condition $\Omega \in L^l\left({\mathbf{S}}^{n-1}\right)$. Consequently, we obtain

$$\begin{array}{cc}\hfill \parallel \Omega (x-·){\chi }_{2^{j+1}B}(·){\parallel }_{L^l\left({\mathbf{R}}^n\right)}& ={\left({\int }_{2^{j+1}B}{|\Omega (x-z)|}^{l}dz\right)}^{1/l}\hfill \\ \hfill & \le {\left({\int }_{2^{j+2}B}{|\Omega \left(y\right)|}^{l}dy\right)}^{1/l}\hfill \\ \hfill & ={\left({\int }_0^{2^{j+2}R}{\int }_{{\mathbf{S}}^{n-1}}{|\Omega \left(y^{\prime }\right)|}^{l}d\sigma \left(y^{\prime }\right)r^{n-1}dr\right)}^{1/l}\hfill \\ \hfill & ={C\parallel \Omega \parallel }_{L^l\left({\mathbf{S}}^{n-1}\right)}{\left|2^{j}B\right|}^{1/l}.\hfill \end{array}$$

(30)

This estimate plays a central role in the subsequent term-by-term control of $J_2$ and $J_4$.

We first consider the index relation $1/q_2(·)=1/l+1/q_1^{\prime }(·)$, this relation directly implies $1/l<1/q_1^{\prime }(·)$. Combining this with the conjugate exponent property $1/q_1(·)+1/q_1^{\prime }(·)=1$, we deduce that $1-1/l+1/q_1(·)>0$. Now, take any $x\in B$ and fix z satisfying the distance condition $|x-x_Q|-\sqrt{n}l\left(Q\right)\le |x-z|$. Based on this, we will apply (30), Lemma 5 and the generalized Hölder inequality to proceed with the estimation.

$$\begin{array}{cc}\hfill |M_{\Omega ,\beta }\left(f{\chi }_{{\left(2B\right)}^c}\right)\left(x\right)|& ={\displaystyle \frac{1}{{|Q|}^{1-\frac{\beta }{n}}}}{\int }_{Q\bigcap B_{(x,2r)}^c}|\Omega (x-z)||f\left(z\right)|dz\hfill \\ \hfill & \le {\displaystyle \frac{1}{{|B|}^{1-\frac{\beta }{n}}}}\sum _{j=1}^{\infty }{\int }_{2^{j}r<|z|\le 2^{j+1}r}|\Omega (x-z)||f\left(z\right)|dz\hfill \\ \hfill & \le {\displaystyle \frac{1}{{|B|}^{1-\frac{\beta }{n}}}}\sum _{j=1}^{\infty }{\int }_{2^{j+1}B\setminus 2^{j}B}|\Omega (x-z)||f\left(z\right)|dz\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n}{\int }_{2^{j+1}B\setminus 2^{j}B}|\Omega (x-z)||f\left(z\right)|dz\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }{|2^{j}B|}^{-1+\beta /n}\parallel \Omega (x-·){\chi }_{2^{j+1}B}(·){\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbf{R}}^n\right)}\parallel f{\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }{|2^{j}B|}^{-1+\beta /n+1/l}{\parallel \Omega \parallel }_{L^l\left({\mathbf{S}}^{n-1}\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\parallel f{\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

When $|2^{j+1}B|\le 2^n$ and $x\in 2^{j+1}B$, by Lemma 7 and $1/q_2(·)+1/l=1/q_1^{\prime }(·)$, we have

$$\begin{array}{c}\hfill \parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\approx |2^{j+1}{B|}^{{\displaystyle \frac{1}{q_2\left(x\right)}}}\approx \parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbf{R}}^n\right)}{|2^{j+1}B|}^{-{\displaystyle \frac{1}{l}}}.\end{array}$$

(31)

When $|2^{j+1}B|\ge 1$, we have

$$\begin{array}{c}\hfill \parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\approx |2^{j+1}{B|}^{{\displaystyle \frac{1}{q_2(\infty )}}}\approx \parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbf{R}}^n\right)}{|2^{j+1}B|}^{-{\displaystyle \frac{1}{l}}}.\end{array}$$

(32)

By (31) and (32), we obtain

$$\begin{array}{c}\hfill \parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\approx \parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbf{R}}^n\right)}{|2^{j+1}B|}^{-{\displaystyle \frac{1}{l}}}.\end{array}$$

(33)

Thus, it follows from Lemma 4 and ${\lambda }_1<-{\lambda }_2-\beta /n\le -\beta /n$ that

$$\begin{array}{cc}\hfill |M_{\Omega ,\beta }\left(f{\chi }_{{\left(2B\right)}^c}\right)\left(x\right)|& \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n+1/l+{\lambda }_1}{\parallel f\parallel }_{{\dot{B}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n+{\lambda }_1}{\parallel f\parallel }_{{\dot{B}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C|B|}^{\beta /n+{\lambda }_1}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

Since $1/p(·)-1/q_2(·)=1/s(·)$, application of Lemmas 6 and 7, we have

$$\begin{array}{cc}\hfill J_2& \le {C|B|}^{\beta /n+{\lambda }_1}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}\parallel (b-b_B){\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C|B|}^{\beta /n+{\lambda }_1+{\lambda }_2}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{s(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C|B|}^{\lambda }{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

(34)

On the other hand, the exponent equality $1/q_2(·)+1/l=1/q_1^{\prime }(·)$ directly yields the relation $1-1/l-1/q_1(·)-1/q_2(·)=0$. Now, for $x\in B$, we note that the parameter conditions ${\lambda }_2\ge 0$ and ${\lambda }_1<-{\lambda }_2-\beta /n$ hold. Combining this with the distance condition $|x-x_Q|-\sqrt{n}l\left(Q\right)\le |x-z|$, we comprehensively apply (30), Lemma 5, the Minkowski inequality, and the generalized Hölder inequality to systematically derive the following estimate:

$$\begin{array}{cc}\hfill |M_{\Omega ,\beta }\left((b-b_B)f{\chi }_{{\left(2B\right)}^c}\right)\left(x\right)|& \le {\displaystyle \frac{1}{{|B|}^{1-\frac{\beta }{n}}}}\sum _{j=1}^{\infty }{\int }_{2^{j}r<|z|\le 2^{j+1}r}|b\left(z\right)-b_B||\Omega (x-z)||f\left(z\right)|dz\hfill \\ \hfill & \le {\displaystyle \frac{1}{{|B|}^{1-\frac{\beta }{n}}}}\sum _{j=1}^{\infty }{\int }_{2^{j+1}B\setminus 2^{j}B}|b\left(z\right)-b_B||\Omega (x-z)||f\left(z\right)|dz\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n}{\int }_{2^{j+1}B\setminus 2^{j}B}|b\left(z\right)-b_B||\Omega (x-z)||f\left(z\right)|dz\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }{|2^{j}B|}^{-1+\beta /n}\parallel (b(·)-b_B)\Omega (x-·){\chi }_{2^{j+1}B}(·){\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbf{R}}^n\right)}\parallel f{\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n+1/l+{\lambda }_1}{\parallel \Omega \parallel }_{L^l\left({\mathbf{S}}^{n-1}\right)}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & ·\left[\parallel (b-b_{2^{j+1}B}){\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}+|b_{2^{j+1}B}-b_B|\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\right].\hfill \end{array}$$

For ${\lambda }_2\ge 0$,

$$\begin{array}{cc}\hfill |b_{2^{j+1}B}-b_B|& \le \sum _{i=0}^j|b_{2^{i+1}B}-b_{2^{i}B}|\hfill \\ \hfill & \le C\sum _{i=0}^j{\displaystyle \frac{1}{|2^{i}B|}}\parallel (b(·)-b_{2^{i+1}B}){\chi }_{2^{i+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{i+1}B}{\parallel }_{L^{q_2^{\prime }(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\sum _{i=0}^j{|2^{i+1}B|}^{{\lambda }_2}\parallel {\chi }_{2^{i+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}{\displaystyle \frac{1}{\parallel {\chi }_{2^{i+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}}}\hfill \\ \hfill & \le {C\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}(j+1){|2^{j+1}B|}^{{\lambda }_2}.\hfill \end{array}$$

(35)

Thus, by (37) and Lemma 7, we derive that

$$\begin{array}{cc}\hfill |M_{\Omega ,\beta }\left((b-b_B)f{\chi }_{{\left(2B\right)}^c}\right)\left(x\right)|& \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n+1/l+{\lambda }_1}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}j{|2^{j+1}B|}^{{\lambda }_2}\hfill \\ \hfill & ·{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }j|2^j{B|}^{\beta /n+{\lambda }_1+{\lambda }_2}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C|B|}^{\lambda }{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill J_4& \le {C|B|}^{\lambda }{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

(36)

(ii) $1/q_1(·)=\beta /n+1/l$.

Following the reasoning analogous to that used for (30), when $z\in 2^{j+1}B$, we derive the following result:

$$\begin{array}{c}\hfill \parallel \Omega (·-z){\chi }_B(·){\parallel }_{L^l\left({\mathbf{R}}^n\right)}\le {C\parallel \Omega \parallel }_{L^l\left({\mathbf{S}}^{n-1}\right)}{|2^{j}B|}^{1/l}.\end{array}$$

(37)

On one hand, the index relation $1/q_1(·)=\beta /n+1/l$ yields two consequences: $p(·)<l$ and $1/p(·)-1/q_1(·)-1/l=0$. By combining (37), Lemmas 5, 7 and 8, the generalized Hölder inequality, and the condition ${\lambda }_1<-{\lambda }_2-\beta /n-1/l\le -\beta /n-1/l$, we obtain the following result:

$$\begin{array}{cc}\hfill J_2& \le C\sum _{j=1}^{\infty }{\int }_{2^{j+1}B\setminus 2^{j}B}{∥{\displaystyle \frac{(b(·)-b_B)\Omega (·-z){\chi }_B(·)}{{|·-z|}^{n-\beta }}}∥}_{L^{p(·)}\left({\mathbf{R}}^n\right)}|f\left(z\right)|dz\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n}\parallel (b(·)-b_B)\Omega (·-z){\chi }_B(·){\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}{\int }_{2^{j+1}B}|f\left(z\right)|dz\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n+1/l}{|B|}^{{\lambda }_2}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}{\parallel \Omega \parallel }_{L^l\left({\mathbf{S}}^{n-1}\right)}{\parallel {\chi }_B\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & ·\parallel f{\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}{\parallel {\chi }_{2^{j+1}B}\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n+1/l}{|B|}^{{\lambda }_2}|2^{j+1}{B|}^{{\lambda }_1}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}{\parallel {\chi }_B\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & ·\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}{\parallel {\chi }_{2^{j+1}B}\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C|B|}^{\lambda }{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}{\parallel {\chi }_B\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

(38)

On the other hand, the condition $q_1^{\prime }(·)<q_2(·)<\infty$ implies $1-1/q_1(·)-1/q_2(·)>0$. Let $q_1^{\prime }(·)=1/q_2(·)+1/m$, By combining (35), (37), Lemmas 7, and 8, together with the Minkowski inequality, generalized Hölder inequality, and the assumption that ${\lambda }_1<-{\lambda }_2-\beta /n-1/l$, we derive that

$$\begin{array}{cc}\hfill J_4& \le C\sum _{j=1}^{\infty }{\int }_{2^{j+1}B\setminus 2^{j}B}{∥{\displaystyle \frac{\Omega (·-z){\chi }_B(·)}{{|·-z|}^{n-\beta }}}∥}_{L^{p(·)}\left({\mathbf{R}}^n\right)}|b\left(z\right)-b_B||f\left(z\right)|dz\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n}\parallel \Omega (·-z){\chi }_B(·){\parallel }_{L^l\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}{|B|}^{-1/l}·{\int }_{2^{j+1}B}|b\left(z\right)-b_B||f\left(z\right)|dz\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{-1+\beta /n}{|B|}^{-1/l}{\parallel \Omega \parallel }_{L^l\left({\mathbf{S}}^{n-1}\right)}{|2^{j}B|}^{1/l}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\parallel f{\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & ·\parallel (b-b_B){\chi }_{2^{j+1}B}{\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{{\lambda }_1-1+\beta /n+1/l+1/m}{|B|}^{-1/l}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & ·\parallel (b-b_B){\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{{\lambda }_1-1+\beta /n+1/l+1/m}{|B|}^{-1/l}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & ·\left[\parallel (b-b_{2^{j+1}B}){\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}+|b_{2^{j+1}B}-b_B|\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\right]\hfill \\ \hfill & \le C\sum _{j=1}^{\infty }|2^j{B|}^{{\lambda }_1-1+\beta /n+1/l+1/m}{|B|}^{-1/l}j|2^{j+1}{B|}^{{\lambda }_2}{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & ·\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_1(·)}\left({\mathbf{R}}^n\right)}\parallel {\chi }_{2^{j+1}B}{\parallel }_{L^{q_2(·)}\left({\mathbf{R}}^n\right)}\hfill \\ \hfill & \le {C|B|}^{\lambda }{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

(39)

By combining (34) and (38), (36) and (39) corresponding, respectively, to the two cases, we obtain the estimates for $J_2$ and $J_4$, as follows:

$$\begin{array}{cc}\hfill J_2& \le {C\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{|B|}^{\lambda }{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill J_4& \le {C\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbf{R}}^n\right)}{|B|}^{\lambda }{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}\left({\mathbf{R}}^n\right)}\parallel {\chi }_B{\parallel }_{L^{p(·)}\left({\mathbf{R}}^n\right)}.\hfill \end{array}$$

(41)

Having obtained uniform estimates for $J_1,J_2,J_3,J_4$ in both scenarios, we combine them to conclude that

$$\begin{array}{c}\hfill {∥\left[b,M_{\Omega ,\beta }\right]f{\chi }_B∥}_{L^{p(·)}\left({\mathbb{R}}^n\right)}\le {C|B|}^{\lambda }{\parallel f\parallel }_{{\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}}{\parallel b\parallel }_{{\mathrm{CBMO}}^{q_2(·),{\lambda }_2}}{\parallel {\chi }_B\parallel }_{L^{p(·)}}.\end{array}$$

(42)

Dividing both sides by ${|B|}^{\lambda }{\parallel {\chi }_B\parallel }_{L^{p(·)}}$ and taking the supremum over all balls B centered at the origin yields precisely the norm inequality (24). This establishes the boundedness of the commutator $\left[b,M_{\Omega ,\beta }\right]$ from ${\dot{\mathcal{B}}}^{q_1(·),{\lambda }_1}\left({\mathbb{R}}^n\right)$ to ${\dot{\mathcal{B}}}^{p(·),\lambda }\left({\mathbb{R}}^n\right)$, thereby completing the proof of Theorem 1. □

4. Application

The boundedness theory established in this paper for commutators of the rough fractional maximal operator on variable exponent central Morrey spaces provides an effective functional space framework and tools for analyzing partial differential equations with non-standard growth properties. Specifically, the results can be applied to the study of mathematical models involving inhomogeneous media or complex geometric structures.

For instance, in electrochemical fluid dynamics, the constitutive relations of the fluid often exhibit power-law growth behavior that depends on spatial location; in such cases, the regularity of solutions is naturally described within variable exponent Morrey spaces. The commutator estimates proved in this work furnish key inequalities that support a priori estimates when nonlinear terms are coupled with fractional diffusion terms in the corresponding equations, thereby facilitating proofs of the existence and uniqueness of weak solutions.

Furthermore, in diffusion-based denoising models for image processing, variable exponent spaces offer a flexible way to capture the strongly diffusion-behaving differences between image edges and texture regions. The operator boundedness obtained here provides a theoretical foundation for designing and analyzing related adaptive diffusion equations.

Hence, this paper not only extends classical operator estimates in harmonic analysis, but also supplies rigorous mathematical tools for modeling and analysis in relevant applied fields.