On the Weighted Boundedness of Fractional Rough Hardy Operators and Their Associated Commutators on Central Morrey Spaces

Abstract

This paper investigates the boundedness of the fractional rough Hardy operator and its adjoint counterpart on weighted central Morrey spaces with variable exponents by combining the analytical framework of weighted variable exponent spaces with the structural properties of rough kernels, through which we establish a series of corresponding weighted norm estimates. Furthermore, by imposing the membership of the generating symbol in the variable exponent $\lambda$-central BMO space, we prove the boundedness of the associated commutator operators, thereby extending the existing boundedness theory for fractional Hardy-type operators and providing a unified framework for the analysis of rough operators and their commutators within weighted variable exponent function spaces.

1. Introduction

The Hardy operator occupies a central and indispensable position in the realm of mathematical analysis and has been the subject of extensive investigation in contemporary literature [1,2]. In the year 1920, G. H. Hardy [3] introduced an operator defined for a locally integrable function F on $(0,\infty )$ by

$$\begin{array}{c}\hfill HF\left(\xi \right)={\displaystyle \frac{1}{\xi }}{\displaystyle {\int }_0^{\xi }}F\left(\eta \right)d\eta ,\xi >0,\end{array}$$

(1)

and established a sharp inequality associated with this operator. Subsequently, William G. Faris (see [4]) extended this construction by formulating the $\mathit{n}$-dimensional analogue of (1). Later, Loukas Grafakos and Michael Christ [5] determined the exact value of the operator norm of the n-dimensional Hardy operator on Lebesgue spaces. Moreover, the classical Hardy integral inequality has attracted considerable attention within the mathematical analysis community [1]. Its alternative proofs, numerous variants, diverse applications, and far-reaching generalizations have been extensively explored in the literature [1,6]. Several contributions addressing such inequalities and their related analytical frameworks can be found in [5,7,8]. In [9], the $\mathit{n}$-dimensional fractional Hardy operator was introduced in the following form:

$$H_{\mathsf{\alpha }}F\left(\xi \right)={\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\int }_{|\eta |\le |\xi |}F\left(\eta \right)d\eta ,{\displaystyle H_{\mathsf{\alpha }}^{*}}F\left(\xi \right)={\int }_{|\eta |>|\xi |}{\displaystyle \frac{F\left(\eta \right)}{{|\eta |}^{\mathit{n}-\mathsf{\alpha }}}}d\eta ,\xi \in {\mathbb{R}}^{\mathit{n}}\setminus \left\{0\right\},$$

(2)

where $|\xi |=\sqrt{{\displaystyle {\sum }_{k=1}^{\mathit{n}}}{\displaystyle {\xi }_k^2}}$ denotes the standard Euclidean norm in ${\mathbb{R}}^{\mathit{n}}$. Furthermore, the commutators associated with these operators are defined by

$$[\mathit{b},H]F=\mathit{b}HF-H\left(\mathit{b}F\right),[\mathit{b},H^{*}]F=\mathit{b}{H}^{*}F-H^{*}\left(\mathit{b}F\right),$$

(3)

where the function $\mathit{b}$ is assumed to be locally integrable. These commutator operators are essential in analyzing more subtle boundedness behavior and regularity effects across different function spaces. The boundedness properties of the commutators associated with the $\mathit{n}$-dimensional fractional Hardy operator were established by Shanzhen Fu, Zhaohui Liu, and Dachun Wang in [9]. Subsequently, Guanghong Ren and Shu Tao [10] introduced the $\mathit{n}$-dimensional fractional rough Hardy operator together with its adjoint operator, which are formulated as follows:

$$H_{\Omega ,\mathsf{\alpha }}F\left(\xi \right)={\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\int }_{|\eta |\le |\xi |}\Omega (\xi -\eta )F\left(\eta \right)d\eta ,{\displaystyle H_{\Omega ,\mathsf{\alpha }}^{*}}F\left(\xi \right)={\int }_{|\eta |>|\xi |}\Omega (\xi -\eta ){\displaystyle \frac{F\left(\eta \right)}{{|\eta |}^{\mathit{n}-\mathsf{\alpha }}}}d\eta ,$$

where $\Omega \in L^s\left(S^{\mathit{n}-1}\right),$$1<s\le \infty ,$ is assumed to be homogeneous of degree zero and is identified with its homogeneous extension to ${\mathbb{R}}^{\mathit{n}}\setminus 0$, given by $[\Omega \left(\xi \right)=\Omega \left({\displaystyle \frac{\xi }{|\xi |}}\right),\xi \in {\mathbb{R}}^{\mathit{n}}\setminus 0]$. Furthermore, the commutators corresponding to these rough Hardy operators are defined by

$${\displaystyle H_{\Omega ,\mathsf{\alpha }}^{\mathit{b}}}F\left(\xi \right)={\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\int }_{|\eta |\le |\xi |}(\mathit{b}\left(\xi \right)-\mathit{b}\left(\eta \right))\Omega (\xi -\eta )F\left(\eta \right)d\eta ,$$

$${\displaystyle H_{\Omega ,\mathsf{\alpha }}^{*,\mathit{b}}}F\left(\xi \right)={\int }_{|\eta |>|\xi |}(\mathit{b}\left(\xi \right)-\mathit{b}\left(\eta \right))\Omega (\xi -\eta ){\displaystyle \frac{F\left(\eta \right)}{{|\eta |}^{\mathit{n}-\mathsf{\alpha }}}}d\eta .$$

These operators were later utilized by Wei Wang, Zhen Zhen, and Dachun Wang in [11], where they established important estimates for commutators in the setting of Herz spaces.

It is noteworthy that function spaces endowed with variable exponents occupy a fundamental position in both modern harmonic analysis and various branches of applied mathematics. The foundational framework of variable exponent Lebesgue spaces was first initiated by Wladyslaw Orlicz in [12]. Subsequently, the broader framework of Musielak–Orlicz spaces was introduced and rigorously developed in [13].

Thereafter, the structural and functional properties of Sobolev and Lebesgue spaces with variable integrability exponents were extensively investigated; comprehensive treatments of these developments can be found in [14,15,16]. In the ensuing period, substantial attention was devoted to the systematic study of variable exponent Lebesgue spaces, together with the analysis of boundedness phenomena for a wide spectrum of operators. In particular, the boundedness of the maximal operator on variable Lebesgue spaces has been thoroughly investigated in various studies (see, for example, refs. [14,16,17]).

Central bounded mean oscillation spaces, $\mathsf{\lambda }$-central Morrey spaces, and other closely related function spaces possess significant analytical importance due to their remarkable applicability in the investigation of operator estimates, particularly in conjunction with singular integral operators. These developments have been extensively explored in several contributions, including [18,19,20,21,22,23].

The theoretical origins of Morrey-type spaces can be traced back to the pioneering work of Charles B. Morrey [24], who introduced such spaces in the context of studying the regularity properties of solutions to partial differential equations. Subsequently, in [18], the authors introduced the notions of $\mathsf{\lambda }$-central Morrey spaces together with central bounded mean oscillation $\left(BMO\right)$ spaces, which constitute a natural generalization based on the concept of bounded central mean oscillation. These function spaces have demonstrated substantial effectiveness in the analysis of boundedness properties for a wide class of operators; representative results in this direction may be found in [25,26,27].

Moreover, owing to their extensive applicability in diverse scientific and applied disciplines—such as image processing [28], electrorheological fluid models [29], and the theory of partial differential equations [30]—function spaces with variable exponents have attracted considerable scholarly attention. Following the seminal contribution of Ondrej Kovacik [31], the theoretical development of variable exponent spaces experienced substantial progress. In particular, the concept of non-homogeneous variable exponent central Morrey spaces was formulated by Yoshihiro Mizuta [32].

More recently, Dachun Wang and collaborators introduced the notion of variable exponent central BMO spaces and established boundedness results for several operators in [33]. These findings were subsequently generalized by Zunwei Fu [34], who extended the analysis to the setting of variable exponent $\mathsf{\lambda }$-central Morrey spaces and the corresponding central BMO spaces.

In [35,36,37,38,39,40,41,42], several authors established significant results concerning the boundedness properties of various integral operators acting on function spaces with variable exponents. Furthermore, a number of contributions have also addressed the boundedness behavior of multilinear integral operators together with their corresponding commutators; see, for instance, refs. [43,44,45,46].

Motivated by the developments presented in [34,47,48] on singular integral operators with rough kernels on central Morrey spaces with variable exponents, the present study is devoted to the investigation of the boundedness properties of fractional Hardy operators, as well as the boundedness of their associated commutators, within the framework of variable exponent $\mathsf{\lambda }$-central Morrey spaces.

For the convenience of the reader, the analytical framework of the present manuscript is briefly outlined as follows. In Section 2, several fundamental definitions, auxiliary lemmas, and preliminary propositions associated with variable exponent Lebesgue spaces are recalled. In Section 3, the boundedness properties of fractional rough Hardy operators on weighted central Morrey spaces with variable exponents are rigorously established. Subsequently, Section 4 is devoted to the investigation of the corresponding commutator estimates generated by rough Hardy operators and $\mathsf{\lambda }$-central BMO functions within the setting of weighted central Morrey spaces endowed with variable exponents.

Moreover, in this article, we use $|\mathcal{B}|$ to denote the Lebesgue measure of a measurable set $\mathcal{B}\subset {\mathbb{R}}^{\mathit{n}}$, and ${\mathcal{X}}_{\mathcal{B}}$ to represent its characteristic function. The notation $G\approx H,$ will be employed to indicate the existence of positive constants $c_1,c_2>0$ such that

$$c_{1}G\le H\le c_{2}G.$$

Furthermore, the ball centered at the origin with radius $2^{\mathit{n}}$ will be denoted by

$${\mathcal{B}}_n=\mathcal{B}(0,2^{\mathit{n}})=\{\xi \in {\mathbb{R}}^{\mathit{n}}:|\xi |\le 2^{\mathit{n}}\},$$

and the corresponding characteristic function will be represented by ${\mathcal{X}}_{\mathit{n}}={\mathcal{X}}_{A_{\mathit{n}}}$ for $\mathit{n}\in \mathbb{Z}.$

2. Function Spaces Along Variable Exponent

Primarily, we delineate some foundational definitions and notational conventions pertaining to Lebesgue spaces endowed with variable exponents. Let $\mathcal{B}\subseteq {\mathbb{R}}^{\mathit{n}}$ denote an open subset, and let $l(·):\mathcal{B}\to [1,\infty )$ be a measurable function. We denote by $l^{\prime }(·)$ its conjugate exponent, defined pointwise via the relation

$${\displaystyle \frac{1}{l^{\prime }(·)}}+{\displaystyle \frac{1}{l(·)}}=1.$$

The class $\mathcal{P}\left(\mathcal{B}\right)$ is defined as the class of measurable functions $l(·)$ satisfying the prescribed lower and upper bounds, with $l^{\prime }(·)$ subsequently defined as its pointwise conjugate exponent. In particular

$$1<l^{-}\le l^{+}<\infty ,$$

where

$$1<l^{-}=essinf\{l\left(\xi \right):\xi \in \mathcal{B}\}$$

$$l^{+}=esssup\{l\left(\xi \right):\xi \in \mathcal{B}\}<\infty .$$

The variable exponent Lebesgue space $L^{l(·)}$ consists of all measurable functions $F:\mathcal{B}\to \mathbb{R}$ for which there exists a $\rho >0$ such that

$${\int }_{\mathcal{B}}{\left({\displaystyle \frac{|F(\xi )|}{\rho }}\right)}^{l\left(\xi \right)}d\xi <\infty .$$

Endowed with the Luxemburg norm

$${||F||}_{L^{l(·)}\left(\mathcal{B}\right)}=inf\left\{\rho >0:{\int }_{\mathcal{B}}{\left({\displaystyle \frac{|F(\xi )|}{\rho }}\right)}^{l\left(\xi \right)}d\xi \le 1\right\}.$$

The corresponding local space ${\displaystyle L_{loc}^{l(·)}}(\partial )$ is defined as

$${\displaystyle L_{loc}^{l(·)}}(\partial )=\left\{F:F\in L^{l(·)}\left(E\right)\forall compactsubset\mathcal{B}\subset \partial \right\}.$$

Let $\mathbb{M}$ denote the Hardy–Littlewood maximal operator acting on functions $F\in {\displaystyle L_{loc}^1}\left({\mathbb{R}}^{\mathit{n}}\right)$, specified by

$$\mathbb{M}F=\underset{r>0}{sup}{\displaystyle \frac{1}{|\mathcal{B}(\xi ,r)|}}{\int }_{\mathcal{B}(\xi ,r)}|F\left(\zeta \right)|,d\zeta ,\xi \in {\mathbb{R}}^{\mathit{n}},$$

where $\mathcal{B}(\xi ,r)=\zeta \in {\mathbb{R}}^{\mathit{n}}:|\xi -\zeta |<r.$

We denote by $\mathit{B}$ the subclass of $\mathcal{P}\left({\mathbb{R}}^{\mathit{n}}\right)$ consisting of those variable exponent functions $l(·)$ for which the operator $\mathbb{M}$ is bounded on $L^{l(·)}$. Subsequently, we recall several pertinent properties of variable exponent functions affiliated with the class $\mathit{B}\left(\mathcal{B}\right).$ The following inequalities have been established by Neugebauer, Cruz-Uribe and Fiorenza [49], as well as by Nakvinda [50], and are summarized in the ensuing proposition.

Proposition 1

([50]). Let $\mathcal{B}$ be an open subset of $R^{\mathit{n}}$, and suppose that $q(·)\in \mathcal{P}\left(\mathcal{B}\right)$ is a variable exponent function satisfying the following regularity conditions. Specifically, assume that there exists a positive constant C, independent of ξ and η, such that

$$|q\left(\eta \right)-q\left(\xi \right)|\le {\displaystyle \frac{-C}{ln(|\eta -\xi |)}},{\displaystyle \frac{1}{2}}\ge |\eta -\xi |,$$

(4)

$$|q\left(\eta \right)-q\left(\xi \right)|\le {\displaystyle \frac{C}{ln(|\xi |+e)}},|\xi |\le |\eta |.$$

(5)

Under the aforementioned logarithmic continuity conditions, it follows that the exponent function $q(·)$ belongs to the class $\mathit{B}\left(\mathcal{B}\right).$ In this context, the symbol C represents a positive constant that is independent of the variables $\eta$ and $\xi$.

Definition 1.

A weight function ω is said to belong to the $A_1$ class of Muckenhoupt weights provided that it satisfies the fundamental domination condition $\mathbb{M}\omega \left(\xi \right)\lesssim \omega \left(\xi \right)$ for almost every $\xi \in {\mathbb{R}}^{\mathit{n}}.$

Definition 2.

Let $l(·)\in \mathcal{P}\left({\mathbb{R}}^{\mathit{n}}\right).$ A weight function ω is said to belong to the class $A_{l(·)}$ provided that it satisfies the following boundedness condition:

$$\underset{\mathcal{B}}{sup}{\displaystyle \frac{1}{|\mathcal{B}|}}\parallel {\omega }^{{\displaystyle \frac{1}{l(·)}}}{\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l(·)}}{\parallel {\omega }^{-{\displaystyle \frac{1}{l(·)}}}{\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{l^{\prime }(·)}}<\infty .$$

Furthermore, a weight ω is said to belong to the class $\widetilde{A_{l(·)}}$ if the subsequent condition is fulfilled:

$$\underset{\mathcal{B}}{sup}{\displaystyle \frac{1}{{|\mathcal{B}|}^{l_{\mathcal{B}}}}}\parallel \omega {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^1}{\parallel {\omega }^{-1}{\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{{\displaystyle \frac{l^{\prime }(·)}{l(·)}}}}<\infty .$$

where $l_{\mathcal{B}}={\left({\displaystyle \frac{1}{|\mathcal{B}|}}{\int }_{\mathcal{B}}{\displaystyle \frac{1}{l\left(\xi \right)}}d\xi \right)}^{-1}.$

Based on the above definition, Hitoshi Izuki and Takuma Noi [51] proved a significant monotonicity property for these classes of variable exponent weights, which is restated in the following proposition.

Definition 3.

Let $\mathsf{\alpha }\in (0,\mathit{n})$ and suppose that $l_2(·),l_1(·)\in \mathcal{P}\left({\mathbb{R}}^{\mathit{n}}\right)$ are variable exponent functions satisfying the structural relation

$${\displaystyle \frac{1}{l_1(·)}}={\displaystyle \frac{1}{l_2(·)}}+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}$$

implies $l_1(·)<l_2(·)$ for almost every $\xi .$ Thus, $l_2(·)$ is the larger exponent, while $l_1(·)$ is the smaller exponent, which is consistent with the usual fractional-integral mapping

$$L^{l_1(·)}\to L^{l_2(·)}.$$

A weight ω is said to belong to the class $A(l_2(·),l_1(·))$ if it fulfills the following integral inequality for every ball $\mathcal{B}\subset {\mathbb{R}}^{\mathit{n}}$

$$\parallel {\omega }^{-1}{\mathcal{X}}_{\mathcal{B}}{\displaystyle {\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}}^{{\displaystyle \frac{1}{1-\frac{\mathsf{\alpha }}{\mathit{n}}}}}}\parallel \omega {\mathcal{X}}_{\mathcal{B}}{\displaystyle {\parallel }_{L^{l_2(·)}}^{{\displaystyle \frac{1}{1-\frac{\mathsf{\alpha }}{\mathit{n}}}}}}\le |\mathcal{B}|.$$

Definition 4.

Let $l(·)\in \mathcal{P}$ and let $\mathsf{\lambda }\in \mathbb{R}.$ The weighted Morrey space with variable exponent, denoted by ${\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left({\omega }^{l(·)}\right)$, is defined as the collection of all measurable functions F belonging locally to the weighted variable exponent Lebesgue space ${\displaystyle L_{loc}^{l(·)}}\left(\omega \right)$ for which the associated Morrey-type norm is finite. More precisely,

$${\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)=\{F\in {\displaystyle L_{loc}^{l(·)}}{\left(\omega \right):\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)}<\infty \},$$

where

$${\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)}=\underset{R>0}{sup}{\displaystyle \frac{\parallel F{\mathcal{X}}_{\mathcal{B}(0,R)}{\parallel }_{L^{l(·)}\left(\omega \right)}}{{|\mathcal{B}(0,R)|}^{\mathsf{\lambda }}{\parallel {\mathcal{X}}_{\mathcal{B}(0,R)}\parallel }_{L^{l(·)}\left(\omega \right)}}}.$$

Definition 5.

Let $l(·)\in \mathcal{P}$ and suppose that $\mathsf{\lambda }<{\displaystyle \frac{1}{\mathit{n}}}$. The weighted $\mathsf{\lambda }$-central BMO space with variable exponent, denoted by $CBM{O}^{l(·),\mathsf{\lambda }}\left({\omega }^{l(·)}\right)$, is defined as the class of all measurable functions F which belong locally to the weighted variable exponent Lebesgue space ${\displaystyle L_{loc}^{l(·)}}\left(\omega \right)$ and whose associated central BMO-type norm is finite. More precisely,

$$CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)=\{F\in {\displaystyle L_{loc}^{l(·)}}{\left(\omega \right):\parallel F\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}<\infty \},$$

where

$${\parallel F\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}=\underset{R>0}{sup}{\displaystyle \frac{\parallel (F-F_{\mathcal{B}(0,R))}{\mathcal{X}}_{\mathcal{B}(0,R)}{\parallel }_{L^{l(·)}\left(\omega \right)}}{{|\mathcal{B}(0,R)|}^{\mathsf{\lambda }}{\parallel {\mathcal{X}}_{\mathcal{B}(0,R)}\parallel }_{L^{l(·)}\left(\omega \right)}}}.$$

Lemma 1

([51]). Let $l_1(·),l_2(·)\in \mathcal{P}\left({\mathbb{R}}^{\mathit{n}}\right)\bigcap LH\left({\mathbb{R}}^{\mathit{n}}\right)$ be variable exponent functions satisfying the pointwise inequality $l_1(·)<l_2(·)$. Then the corresponding classes of Muckenhoupt weights obey the nested inclusion property

$$A_1\subset A_{l_1(·)}\subset A_{l_2(·)}.$$

Lemma 2

([51]). Let $0<\alpha <\mathit{n}$ and suppose that the variable exponents $l_2(·),l_1(·)\in \mathcal{P}\left({\mathbb{R}}^{\mathit{n}}\right)$ satisfy the functional relation ${\displaystyle \frac{1}{l_1(·)}}={\displaystyle \frac{1}{l_2(·)}}+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}.$ Then a weight function ω belongs to the generalized Muckenhoupt class $A(l_2(·),l_1(·))$ if and only if the transformed weight ${\omega }^{l_2(·)}$ lies in the variable exponent Muckenhoupt class $A_{1+{\displaystyle \frac{l_2(·)}{{\displaystyle l_1^{\prime }}(·)}}}.$

Lemma 3

([31], Generalized Hölder’s inequality). Let $l(·),l_1(·),l_2(·)\in \mathcal{P}\left({\mathbb{R}}^n\right)$ and let ω be a positive measurable weight on ${\mathbb{R}}^n$.

(i) 

If $G\in L^{l(·)}({\mathbb{R}}^n,\omega )$ and $H\in L^{l^{\prime }(·)}({\mathbb{R}}^n,\omega )$, then

$${\int }_{{\mathbb{R}}^n}|G\left(\xi \right)H\left(\xi \right)|d\xi \le r_l{\parallel G\parallel }_{L^{l(·)}({\mathbb{R}}^n,\omega )}{\parallel H\parallel }_{L^{l^{\prime }(·)}({\mathbb{R}}^n,\omega )},$$

where

$$r_l=1+{\displaystyle \frac{1}{l^{-}}}-{\displaystyle \frac{1}{l^{+}}}.$$

(ii) 

Let $l(·),l_1(·),l_2(·)\in \mathcal{P}\left({\mathbb{R}}^n\right)$ satisfy

$${\displaystyle \frac{1}{l(·)}}={\displaystyle \frac{1}{l_1(·)}}+{\displaystyle \frac{1}{l_2(·)}}.$$

Let ${\omega }_1$ and ${\omega }_2$ be two positive measurable weights and consider

$$\omega ={\omega }_1{\omega }_2.$$

If

$$G\in L^{l_1(·)}({\mathbb{R}}^n,{\omega }_1)\mathit{and}H\in L^{l_2(·)}({\mathbb{R}}^n,{\omega }_2),$$

then

$${\parallel GH\parallel }_{L^{l(·)}({\mathbb{R}}^n,\omega )}\le r_{l,l_1}{\parallel G\parallel }_{L^{l_1(·)}({\mathbb{R}}^n,{\omega }_1)}{\parallel H\parallel }_{L^{l_2(·)}({\mathbb{R}}^n,{\omega }_2)},$$

where

$$r_{l,l_1}={\left(1+{\displaystyle \frac{1}{{\left(l_1\right)}^{-}}}-{\displaystyle \frac{1}{{\left(l_1\right)}^{+}}}\right)}^{1/l^{-}}.$$

In fact, given that $\omega ={\omega }_1{\omega }_2$, we have

$${\parallel GH\parallel }_{L^{l(·)}({\mathbb{R}}^n,\omega )}={\parallel \left(G{\omega }_1\right)\left(H{\omega }_2\right)\parallel }_{L^{l(·)}\left({\mathbb{R}}^n\right)},$$

The result is a consequence of the generalized Hölder inequality in variable exponent Lebesgue spaces.

Lemma 4

([52]). Let ω be a weight function defined on ${\mathbb{R}}^{\mathit{n}}$. Suppose that there exists a real number $l\in [1,\infty )$ such that $\omega \in A_l$. Then, for every measurable set $\mathcal{I}$ satisfying $\mathcal{I}\subseteq \mathcal{B}$, the following weighted measure estimates hold:

$$i.{\displaystyle \frac{\omega \left(\mathcal{B}\right)}{\omega \left(\mathcal{I}\right)}}\le C{\left({\displaystyle \frac{|\mathcal{B}|}{|\mathcal{I}|}}\right)}^l$$

$$ii.{\displaystyle \frac{\omega \left(\mathcal{I}\right)}{\omega \left(\mathcal{B}\right)}}\le C{\left({\displaystyle \frac{|\mathcal{I}|}{|\mathcal{B}|}}\right)}^{\partial },$$

where $C>0$ denotes a positive constant independent of the particular choice of $\mathcal{I}$ and $\mathcal{B}$, and the parameter ∂ satisfies $0<\partial <1$.

Lemma 5

([52]). Let us consider $l_1(·)$ and $l_2(·)\in \mathcal{P}\cap LH({\mathbb{R}}^{\mathit{n})}$. If ${\omega }^{l_2(·)}\in A_{l_2(·)}$ and ${\omega }^{l_1(·)}\in A_{l_1(·)}$, then this implies that ${\omega }^{-{\displaystyle l_2^{\prime }}(·)}\in A_{{\displaystyle l_2^{\prime }}(·)}$ and ${\omega }^{-{\displaystyle l_1^{\prime }}(·)}\in A_{{\displaystyle l_1^{\prime }}(·)}$, respectively. Consequently, the Hardy–Littlewood maximal operator $\mathbb{M}$ is bounded on $L^{{\displaystyle l_2^{\prime }}(·)}\left({\omega }^{-{\displaystyle l_2^{\prime }}(·)}\right)$. Therefore, there exist constants ${\partial }_1,{\partial }_2\in (0,1)$ such that for any measurable sets $\mathcal{I}\subset \mathcal{B}$ the following holds

$${\displaystyle \frac{\parallel {\mathcal{X}}_{\mathcal{I}}{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}}{\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}}}={\displaystyle \frac{\parallel {\mathcal{X}}_{\mathcal{I}}{\parallel }_{{\left(L^{{\displaystyle l_2^{\prime }}(·)}{\omega }^{-{\displaystyle l_2^{\prime }}(·)}\right)}^{\prime }}}{\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{{\left(L^{{\displaystyle l_2^{\prime }}(·)}{w}^{-{\displaystyle l_2^{\prime }}(·)}\right)}^{\prime }}}}\lesssim {\left({\displaystyle \frac{|\mathcal{I}|}{|\mathcal{B}|}}\right)}^{{\partial }_1},$$

(6)

$${\displaystyle \frac{{\parallel {\mathcal{X}}_{\mathcal{I}}\parallel }_{{\left(L^{l_1(·)}{\omega }^{l_1(·)}\right)}^{\prime }}}{{\parallel {\mathcal{X}}_{\mathcal{B}}\parallel }_{{\left(L^{l_1(·)}{\omega }^{l_1(·)}\right)}^{\prime }}}}\lesssim {\left({\displaystyle \frac{|\mathcal{I}|}{|\mathcal{B}|}}\right)}^{{\partial }_2}.$$

(7)

Lemma 6

([53]). Let ζ be a Banach function space. Then, for every ball $\mathcal{B}\subset {\mathbb{R}}^{\mathit{n}}$, the following inequality holds:

$$1\le {\displaystyle \frac{1}{|\mathcal{B}|}}\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{\zeta }{\parallel {\mathcal{X}}_{\mathcal{B}}\parallel }_{{\zeta }^{\prime }}.$$

In other words, the normalized product of the norms of the characteristic function ${\mathcal{X}}_{\mathcal{B}}$ in the space ζ and its associate space ${\zeta }^{\prime }$ is bounded below by unity for every ball $\mathcal{B}$.

Lemma 7

([52]). Suppose that the Hardy–Littlewood maximal operator $\mathbb{M}$ is weakly bounded on a Banach function space ζ, that is,

$$\parallel {\mathcal{X}}_{\{\mathbb{M}F>\mathsf{\lambda }\}}{\parallel }_{\zeta }\le C{\mathsf{\lambda }}^{-1}{\parallel F\parallel }_{\zeta },$$

(8)

for every $\mathsf{\lambda }>0$ and for all $F\in \zeta$. Then it follows that

$$sup{\displaystyle \frac{1}{|\mathcal{B}|}}\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{\zeta }{\parallel {\mathcal{X}}_{\mathcal{B}}\parallel }_{{\zeta }^{\prime }}<\infty ,$$

(9)

where the supremum is taken over all balls $\mathcal{B}\subset {\mathbb{R}}^{\mathit{n}}$.

We now turn to the definition of weighted Banach function spaces and present several of their basic properties. Let $\omega \left(\xi \right)$ be a measurable function satisfying $0<\omega \left(\xi \right)<\infty$, such that $\omega \left(\xi \right)\in {\zeta }_{\mathrm{loc}}$ and ${\omega }^{-1}\left(\xi \right)\in {\displaystyle {\zeta }_{\mathrm{loc}}^{\prime }}$. Here, ${\zeta }_{\mathrm{loc}}\left({\mathbb{R}}^{\mathit{n}}\right)$ denotes the collection of all measurable functions F for which $F{\mathcal{X}}_{\mathcal{B}}\in \zeta$ for every compact set $\mathcal{B}$ with finite Lebesgue measure $|\mathcal{B}|<\infty$.

Under these assumptions, the weighted Banach function space is defined by

$$\zeta ({\mathbb{R}}^{\mathit{n}},\omega )=\{F\in \mathbb{M}:F\omega \in \zeta \}.$$

In this setting, the following lemma holds.

Lemma 8

([54]).

1. 

The weighted space $\zeta ({\mathbb{R}}^{\mathit{n}},\omega )$ constitutes a Banach function space when equipped with the norm

$${\parallel F\parallel }_{\zeta ({\mathbb{R}}^{\mathit{n}},\omega )}={\parallel F\omega \parallel }_{\zeta }.$$

2. 

Furthermore, the associate (Kothe dual) space corresponding to the weighted Banach function space $\zeta ({\mathbb{R}}^{\mathit{n}},\omega )$ itself inherits the structure of a Banach function space, thereby preserving completeness and the normed lattice properties intrinsic to $\zeta ({\mathbb{R}}^{\mathit{n}},\omega )$.

Remark 1.

Let $l(·)\in \mathcal{P}\left({\mathbb{R}}^{\mathit{n}}\right)$. By invoking the definition of the weighted Banach function space $\zeta ({\mathbb{R}}^{\mathit{n}},\omega )$, together with the associated weighted variable exponent Lebesgue spaces $L^{l(·)}\left({\omega }^{l(·)}\right)$ and $L^{l^{\prime }(·)}\left({\omega }^{-l^{\prime }(·)}\right)$, we deduce the following observations:

1. 

If we set $\zeta =L^{l(·)}\left({\mathbb{R}}^{\mathit{n}}\right)$ and $W=\omega ,$ it follows immediately that $L^{l(·)}({\mathbb{R}}^{\mathit{n}},\omega )=L^{l(·)}\left({\omega }^{l(·)}\right).$

2. 

For $\zeta =L^{l^{\prime }(·)}\left({\mathbb{R}}^{\mathit{n}}\right)$ and $W={\omega }^{-1}$. It then follows that $L^{l^{\prime }(·)}({\mathbb{R}}^{\mathit{n}},{\omega }^{-1})=L^{l^{\prime }(·)}\left({\omega }^{-l^{\prime }(·)}\right).$

Consequently, by invoking Lemma 8, one deduces the duality relation

$${\left(L^{l(·)}\left({\omega }^{l(·)}\right)\right)}^{\prime }={\left(L^{l(·)}({\mathbb{R}}^{\mathit{n}},{\omega }^{-1})\right)}^{\prime }=L^{l^{\prime }(·)}\left({\omega }^{-l^{\prime }(·)}\right).$$

Furthermore, let us define the exponent function $l_1(·)$ via the relation ${\displaystyle \frac{1}{l_1(·)}}={\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}+{\displaystyle \frac{1}{l_2(·)}}$. Suppose that $l_1(·)\in \mathcal{P}\cap LH\left({\mathbb{R}}^{\mathit{n}}\right)$ and that $0<\mathsf{\alpha }<{\displaystyle \frac{\mathit{n}}{l_1}}$. By invoking the monotonicity property of the weight classes, it then follows that ${\omega }^{l_1(·)}\in A_1\subset A_{1+{\displaystyle \frac{l_2(·)}{{\displaystyle l_1^{\prime }}(·)}}}$. So, by Lemma 8 we obtain $\omega \in A(l_2(·),l_1(·)).$

${\omega }^{l(·)}\in A_1$ is used when estimates in ${\displaystyle L_{\omega }^{l(·)}}\left({\mathbb{R}}^{\mathit{n}}\right)$ are required, whereas ${\omega }^{l_1(·)}\in A_1$ is used in the estimates involving the weighted space ${\displaystyle L_{\omega }^{l_1(·)}}\left({\mathbb{R}}^{\mathit{n}}\right)$.

It should be emphasized that these two conditions are, in general, distinct and neither one is assumed to imply the other. Moreover, the conditions ${\omega }^{l(·)}\in A_1$ and ${\omega }^{l_1(·)}\in A_1$ should not be confused with the generalized variable exponent Muckenhoupt conditions $\omega \in A_{l(·)}$ and $\omega \in A(l_1(·),l_2(·))$. Each weight condition is used only at the stage of the proof where the corresponding weighted norm or maximal-operator estimate is invoked. Consequently, no additional weight assumption is required when it is not used in the corresponding argument.

Lemma 9

([51]). Let $l_1(·)\in \mathcal{P}\cap LH\left({\mathbb{R}}^{\mathit{n}}\right)$, and let α be a positive real number satisfying $0<\alpha <{\displaystyle \frac{\mathit{n}}{l_1{(·)}^{+}}}$. Define $l_2(·)$ via the relation ${\displaystyle \frac{1}{l_2{(}^{.})}}={\displaystyle \frac{1}{l_1{(}^{.})}}-{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}$. If $\omega \in A(l_2{(}^{.}),l_1{(}^{.})),$ then $I_{\mathsf{\alpha }}$ is bounded from $L^{l_1{(}^{.})}\left({\omega }^{l_1{(}^{.})}\right)$ to $L^{l_2{(}^{.})}\left({\omega }^{l_2{(}^{.})}\right).$

Lemma 10

([55]). Suppose that the Hardy–Littlewood maximal operator is bounded on the Banach function space ζ. Then, for any measurable subset $\mathcal{I}\subset \mathcal{B}$, the following estimate holds:

$${\displaystyle \frac{\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{\zeta }}{\parallel {\mathcal{X}}_{\mathcal{I}}{\parallel }_{\zeta }}}\lesssim {\displaystyle \frac{|\mathcal{B}|}{|\mathcal{I}|}}.$$

Furthermore, by exploiting the boundedness properties of the fractional integral operator $I_{\mathsf{\alpha }}$, we establish the boundedness of the fractional rough Hardy operator. The fractional integral operator is defined by

$$I_{\mathsf{\alpha }}\left(F\right)\left(\xi \right)={\int }_{{\mathbb{R}}^{\mathit{n}}}{\displaystyle \frac{F\left(\eta \right)}{{|\xi -\eta |}^{\mathit{n}-\mathsf{\alpha }}}}d\eta ,$$

where $0<\mathsf{\alpha }<\mathit{n}$. This operator will play a central role in establishing the boundedness properties of the fractional rough Hardy operator under study.

Lemma 11

([51]). Let $l_1(·)\in \mathcal{P}\cap LH\left({\mathbb{R}}^{\mathit{n}}\right)$, and let $0<\mathsf{\alpha }<{\displaystyle \frac{\mathit{n}}{l_1{(}^{.})+}}$, with

$${\displaystyle \frac{1}{l_2{(}^{.})}}={\displaystyle \frac{1}{l_1{(}^{.})}}-{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}.$$

If $\omega \in A(l_2{(}^{.}),l_1{(}^{.})),$ then the operator $I_{\mathsf{\alpha }}$ is bounded from $L^{l_1{(}^{.})}\left({\omega }^{l_1{(}^{.})}\right)$ to $L^{l_2{(}^{.})}\left({\omega }^{l_2{(}^{.})}\right).$

Lemma 12.

Let $s>1$, and let $l_1(·),l_2(·)\in \mathcal{P}\left({\mathbb{R}}^{\mathit{n}}\right)$ satisfy

$$1<{\displaystyle l_1^{-}}\le {\displaystyle l_1^{+}}<\infty ,1<{\displaystyle l_2^{-}}\le {\displaystyle l_2^{+}}<\infty ,$$

and

$${\displaystyle \frac{1}{l_2\left(\xi \right)}}={\displaystyle \frac{1}{{\displaystyle l_1^{\prime }}\left(\xi \right)}}-{\displaystyle \frac{1}{s}}\mathit{for}\mathit{a}.\mathit{e}.\xi \in {\mathbb{R}}^{\mathit{n}}.$$

Let us assume that there exists a weight ω, for which the assumptions of the weighted variable-exponent characteristic-function estimate are fulfilled. For each measurable set $B\subset {\mathbb{R}}^{\mathit{n}}$,

$$0<\omega \left(B\right)<\infty ,$$

we have

$$\parallel {\chi }_B{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le C\omega {\left(B\right)}^{-{\displaystyle \frac{1}{s}}}{\parallel {\chi }_B\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}.$$

In this case, the value of $C>0$ does not depend on B.

Proof of Lemma 12. 

Based on the assumed relation between the exponents,

$${\displaystyle \frac{1}{{\displaystyle l_1^{\prime }}\left(\xi \right)}}={\displaystyle \frac{1}{l_2\left(\xi \right)}}+{\displaystyle \frac{1}{s}}.$$

So, the exponents $l_2(·)$, s and ${\displaystyle l_1^{\prime }}(·)$ satisfy the necessary fractional Hölder relation. If we use the weighted characteristic-function estimate associated with this exponent relation, then we have

$${\displaystyle \frac{\parallel {\chi }_B{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}}{\parallel {\chi }_B{\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}}}\le C\omega {\left(B\right)}^{-{\displaystyle \frac{1}{s}}}.$$

Let us multiply both sides by $\parallel {\chi }_B{\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}$ to yield

$$\parallel {\chi }_B{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le C\omega {\left(B\right)}^{-{\displaystyle \frac{1}{s}}}{\parallel {\chi }_B\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}.$$

It is proved by this result.

To the dyadic annulus.

Taking

$$B={\mathcal{B}}_m$$

and

$${\mathcal{X}}_m={\chi }_{{\mathcal{B}}_m},$$

we immediately obtain

$$\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}={\parallel {\chi }_{{\mathcal{B}}_m}\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}$$

and hence

$$\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le C\omega {\left({\mathcal{B}}_m\right)}^{-{\displaystyle \frac{1}{s}}}{\parallel {\mathcal{X}}_m\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}.$$

The constant C is independent of m and only depends on the structural parameters in the hypotheses of the characteristic-function estimate. □

3. Boundedness of Fractional Hardy Operators

Theorem 1.

Let $\Omega \in L^s\left(S^{\mathit{n}-1}\right)$ with ${\displaystyle \frac{\mathit{n}}{\mathit{n}-1}}<s.$ Suppose that the variable exponents $l_1(·),l(·),l_2(·)\in \mathcal{P}\left({\mathbb{R}}^{\mathit{n}}\right)$ satisfy the inequalities (4) and (5) as stipulated in Proposition 1, and define the variable exponent $l(·)$ through the relation

$${\displaystyle \frac{1}{l_1(·)}}={\displaystyle \frac{1}{l(·)}}+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}.$$

Let ${\mathsf{\lambda }}_1$ be such that, for

$${\displaystyle \frac{1}{l_2(·)}}={\displaystyle \frac{1}{{\displaystyle l_1^{\prime }}(·)}}-{\displaystyle \frac{1}{s}},$$

there exists a constant ${\mathsf{\lambda }}_1>-{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}$, and define $\mathsf{\lambda }={\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}.$ Assume further that ${\partial }_2-{\displaystyle \frac{1}{s}}+\mathsf{\lambda }+\partial >0,$ and that the weight functions ${\omega }^{l(·)},{\displaystyle {\omega }_1^{l_1(·)}}\in A_1.$ Under these assumptions, the fractional rough Hardy operator $H_{\mathsf{\alpha },\Omega }$ is bounded as a mapping

$$H_{\mathsf{\alpha },\Omega }:{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)\to {\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)$$

and the following norm inequality is satisfied:

$$||H_{\mathsf{\alpha },\Omega }{F||}_{{\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)}\le {C||F||}_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)},$$

where $C>0$ is a constant independent of F.

Proof of Theorem 1. 

$$\begin{array}{cc}& |H_{\mathsf{\alpha },\Omega }F\left(x\right)·{\mathcal{X}}_k\left(\xi \right)|\le {\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\int }_{{\mathcal{B}}_k}|F\left(t\right)||\Omega (\xi -\eta )|d\eta ·{\mathcal{X}}_k\left(\xi \right)\hfill \\ & \le C{2}^{-k(\mathit{n}-\mathsf{\alpha })}{\displaystyle \sum _{m=-\infty }^k}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta ){\mathcal{X}}_m\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}{\mathcal{X}}_k\left(\xi \right).\hfill \end{array}$$

By invoking the fundamental relation

$${\displaystyle \frac{1}{l_2(·)}}+{\displaystyle \frac{1}{s}}={\displaystyle \frac{1}{{\displaystyle l_1^{\prime }}(·)}},$$

one may further refine the above bound as

$$\begin{array}{c}\hfill |H_{\mathsf{\alpha },\Omega }F\left(x\right)·{\mathcal{X}}_k\left(\xi \right)|\le C{2}^{-k(\mathit{n}-\mathsf{\alpha })}{\displaystyle \sum _{m=-\infty }^k}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\parallel \Omega (\xi -\eta ){\mathcal{X}}_m{\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-2}}\right)}{\parallel {\mathcal{X}}_m\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}·{\mathcal{X}}_k\left(\xi \right).\end{array}$$

Consequently, one derives

$$\begin{array}{cc}& \parallel H_{\mathsf{\alpha },\Omega }F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\le C{2}^{-k(\mathit{n}-\mathsf{\alpha })}{\displaystyle \sum _{m=-\infty }^k}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\parallel \Omega (\xi -\eta ){\mathcal{X}}_m{\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-1}}\right)}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}{\parallel {\mathcal{X}}_k\parallel }_{L^{l(·)}\left(\omega \right)}.\hfill \end{array}$$

By Lemma 12, we have

$$\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le C,\omega {\left({\mathcal{B}}_m\right)}^{-{\displaystyle \frac{1}{s}}}{\parallel {\mathcal{X}}_m\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}.$$

Therefore, substituting this estimate into the preceding inequality yields

$$\begin{array}{cc}\hfill \parallel H_{\mathsf{\alpha },\Omega }F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}& \le C{2}^{-k(\mathit{n}-\mathsf{\alpha })}{\displaystyle \sum _{m=-\infty }^k}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta ){\mathcal{X}}_m\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-1}}\right)}\hfill \\ & \omega {\left({\mathcal{B}}_m\right)}^{-{\displaystyle \frac{1}{s}}}\parallel {\mathcal{X}}_m{\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}{\parallel {\mathcal{X}}_k\parallel }_{L^{l(·)}\left(\omega \right)}.\hfill \end{array}$$

(10)

Invoking Lemmas 7 and 11, one establishes the lower and upper estimates

$$I_{\mathsf{\alpha }}\left({\mathcal{X}}_{{\mathcal{B}}_k}\right)\left(\xi \right)\ge C{2}^{k\mathsf{\alpha }}{\mathcal{X}}_{{\mathcal{B}}_k}\left(\xi \right)$$

$${\mathcal{X}}_{{\mathcal{B}}_k}\left(\xi \right)\le C{2}^{-k\mathsf{\alpha }}{I}_{\mathsf{\alpha }}\left({\mathcal{X}}_{{\mathcal{B}}_k}\right)\left(\xi \right)$$

whence it follows that

$$\begin{array}{c}\hfill \parallel {\mathcal{X}}_{{\mathcal{B}}_k}{\parallel }_{L^{l(·)}\left(\omega \right)}\le C{2}^{-k\mathsf{\alpha }}{\parallel I_{\mathsf{\alpha }}{\mathcal{X}}_{{\mathcal{B}}_k}\parallel }_{L^{l(·)}\left(\omega \right)}\\ \hfill \le C{2}^{-k\mathsf{\alpha }}{\parallel {\mathcal{X}}_{{\mathcal{B}}_k}\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\\ \hfill \le C{2}^{k(\mathit{n}-\mathsf{\alpha })}{\displaystyle {\parallel {\mathcal{X}}_{{\mathcal{B}}_k}\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}^{-1}}.\end{array}$$

(11)

Substituting inequality (11) into (10), one ultimately obtains

$$\begin{array}{cc}& \parallel H_{\mathsf{\alpha },\Omega }F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\le C{\displaystyle \sum _{m=-\infty }^k}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\parallel \Omega (\xi -\eta ){\mathcal{X}}_m{\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-1}}\right)}\omega {\left({\mathcal{B}}_m\right)}^{-{\displaystyle \frac{1}{s}}}\parallel {\mathcal{X}}_m{\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}{\displaystyle {\parallel {\mathcal{X}}_k\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}^{-1}}.\hfill \end{array}$$

(12)

Invoking the assertions of Lemma 5, one readily discerns the ensuing bound

$$\begin{array}{cc}& \parallel H_{\mathsf{\alpha },\Omega }F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\le C{\displaystyle \sum _{m=-\infty }^k}{2}^{(m-k)\mathit{n}{\partial }_2}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta ){\mathcal{X}}_m\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-1}}\right)}\omega {\left({\mathcal{B}}_m\right)}^{-{\displaystyle \frac{1}{s}}}.\hfill \end{array}$$

(13)

For indices $m\le k$, with $\eta \in C_m$ and $\xi \in C_k$, it is evident that $0\le |\xi -\eta |\le |\xi |+2^m\le 2·2^k$. Consequently, one has the integral estimate

$${\int }_{C_m}{|\Omega (\xi -\eta )|}^{s}d\eta \le {\displaystyle {\int }_0^{2^{k+1}}}{\int }_{s^{\mathit{n}-1}}{|\Omega \left({\xi }^{\prime }\right)|}^{s}d\sigma \left({\xi }^{\prime }\right)r^{\mathit{n}-1}dr\le C{2}^{k\mathit{n}}.$$

Hence, the preceding inequality may be further refined as

$$\begin{array}{cc}\hfill \parallel H_{\mathsf{\alpha },\Omega }F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}& \le C{\displaystyle \sum _{m=-\infty }^k}{2}^{(m-k)(\mathit{n}{\partial }_2-{\displaystyle \frac{\mathit{n}}{s}})}{\parallel F_m\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\hfill \\ & \le C{\displaystyle \sum _{m=-\infty }^k}{2}^{(m-k)(\mathit{n}{\partial }_2-{\displaystyle \frac{\mathit{n}}{s}})}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\omega {\left({\mathcal{B}}_m\right)}^{{\mathsf{\lambda }}_1}||{\mathcal{X}}_m{||}_{L^{l_1(·)}\left({\omega }_1\right)}.\hfill \end{array}$$

(14)

Recalling the asymptotic equivalence of the characteristic function norms, one observes

$$\parallel {\mathcal{X}}_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\le C\omega {\left(\mathcal{B}\right)}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}}{\parallel {\mathcal{X}}_m\parallel }_{L^{l(·)}\left(\omega \right)}.$$

Accordingly, one may recast the prior estimate in the form

$$\begin{array}{cc}& \parallel H_{\mathsf{\alpha },\Omega }F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\le C{\displaystyle \sum _{m=-\infty }^k}{2}^{(m-k)(\mathit{n}{\partial }_2-{\displaystyle \frac{\mathit{n}}{s}})}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\omega {\left({\mathcal{B}}_m\right)}^{{\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}}||{\mathcal{X}}_m{||}_{L^{l(·)}\left(\omega \right)}\hfill \\ & \le C{\displaystyle \sum _{m=-\infty }^k}{2}^{(m-k)(\mathit{n}{\partial }_2-{\displaystyle \frac{\mathit{n}}{s}})}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \frac{\omega {\left({\mathcal{B}}_m\right)}^{\mathsf{\lambda }}}{\omega {\left({\mathcal{B}}_k\right)}^{\mathsf{\lambda }}}}\omega {\left({\mathcal{B}}_k\right)}^{\mathsf{\lambda }}{\displaystyle \frac{||{\mathcal{X}}_m{||}_{L^{l(·)}}\left(\omega \right)}{||{\mathcal{X}}_k{||}_{L^{l(·)}}\left(\omega \right)}}||{\mathcal{X}}_k{||}_{L^{l(·)}\left(\omega \right)}.\hfill \end{array}$$

(15)

Exploiting the structural insights provided by part (ii) of Lemma 4, one immediately obtains the following boundedness property:

$$\begin{array}{cc}& \parallel H_{\mathsf{\alpha },\Omega }F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\le C{\displaystyle \sum _{m=-\infty }^k}{2}^{\mathit{n}(m-k)({\partial }_2-{\displaystyle \frac{1}{s}}+\mathsf{\lambda }+\partial )}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\omega {\left({\mathcal{B}}_k\right)}^{\mathsf{\lambda }}||{\mathcal{X}}_k{||}_{L^{l(·)}\left(\omega \right)}.\hfill \end{array}$$

(16)

If,

$$\gamma =\mathit{n}({\partial }_2-{\displaystyle \frac{1}{s}}+\mathsf{\lambda }+\partial )$$

then

$$\begin{array}{cc}& \parallel H_{\mathsf{\alpha },\Omega }F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\le {C\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\omega {\left({\mathcal{B}}_k\right)}^{\mathsf{\lambda }}||{\mathcal{X}}_k{||}_{L^{l(·)}\left(\omega \right)}{\displaystyle \sum _{m=-\infty }^k}{2}^{-(k-m)\gamma }.\hfill \end{array}$$

(17)

Noting that the composite exponent $\gamma =\mathit{n}({\partial }_2-{\displaystyle \frac{1}{s}}+\mathsf{\lambda }+\partial )$ is strictly positive, one is immediately led to the telescopic summation argument, whereby the dyadic decay factor $2^{-(k-m)\gamma }$ ensures convergence of the series and furnishes a controlled passage from local to global scales. Consequently, one obtains the canonical embedding

$$\begin{array}{cc}& \parallel H_{\mathsf{\alpha },\Omega }{F\parallel }_{{\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)}\le C{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}.\hfill \end{array}$$

(18)

□

Theorem 2.

Let $\Omega \in L^s\left(S^{\mathit{n}-1}\right)$ with ${\displaystyle \frac{\mathit{n}}{\mathit{n}-1}}<s.$ Suppose that the variable exponents $l_1(·),l(·),l_2(·)\in \mathcal{P}\left({\mathbb{R}}^{\mathit{n}}\right)$ satisfy the inequalities (4) and (5) as stipulated in Proposition 1, and define the variable exponent $l(·)$ through the relation

$${\displaystyle \frac{1}{l_1(·)}}={\displaystyle \frac{1}{l(·)}}+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}.$$

Let ${\mathsf{\lambda }}_1$ be such that, for

$${\displaystyle \frac{1}{l_2(·)}}={\displaystyle \frac{1}{{\displaystyle l_1^{\prime }}(·)}}-{\displaystyle \frac{1}{s}},$$

there exists a constant ${\mathsf{\lambda }}_1>-{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}$, and define $\mathsf{\lambda }={\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}.$ Assume further that

$$\mathsf{\lambda }<{\displaystyle \frac{1}{s}}-{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1,$$

and that the weight functions ${\omega }^{l(·)},{\displaystyle {\omega }_1^{l_1(·)}}\in A_1.$ Under these assumptions, the adjoint fractional rough Hardy operator ${\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}$ is bounded as a mapping

$${\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}:{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)\to {\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)$$

and the following norm inequality is satisfied:

$$||{\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}{F||}_{{\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)}\le {C||F||}_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)},$$

where $C>0$ is a constant independent of F.

Proof of Theorem 2. 

Consider the following estimate

$$\begin{array}{cc}& |{\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}f\left(\xi \right)·{\mathcal{X}}_k|\le {\int }_{{\mathbb{R}}^{\mathit{n}}\setminus {\mathcal{B}}_k}{|F\left(\eta \right)\Omega (\xi -\eta )||\eta |}^{\mathsf{\alpha }-\mathit{n}}d\eta ·{\mathcal{X}}_k\left(\xi \right)\hfill \\ & \le C{\displaystyle \sum _{m=k+1}^{\infty }}{2}^{m(\mathsf{\alpha }-\mathit{n})}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta ){\mathcal{X}}_m\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}·{\mathcal{X}}_k\left(\xi \right).\hfill \end{array}$$

Invoking the relation

$${\displaystyle \frac{1}{l_2(·)}}+{\displaystyle \frac{1}{s}}={\displaystyle \frac{1}{{\displaystyle l_1^{\prime }}(·)}},$$

together with the generalized Holder inequality in the framework of variable exponent Lebesgue spaces, we deduce that

$$\begin{array}{cc}& \parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}f·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \le C{\displaystyle \sum _{m=k+1}^{\infty }}{2}^{m(\mathsf{\alpha }-\mathit{n})}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta )\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-1}}\right)}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}{\parallel {\mathcal{X}}_k\parallel }_{L^{l(·)}\left(\omega \right)}.\hfill \end{array}$$

Furthermore, by employing inequality (11), the preceding estimate can be reformulated as

$$\begin{array}{cc}& \parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}f·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \le C{\displaystyle \sum _{m=k+1}^{\infty }}{2}^{(m-k)(\mathsf{\alpha }-\mathit{n})}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta )\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-1}}\right)}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}{\displaystyle {\parallel {\mathcal{X}}_k\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}^{-1}}.\hfill \end{array}$$

Observe that the characteristic functions satisfy the well-known equivalence

$$\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le C\omega {\left({\mathcal{B}}_m\right)}^{-{\displaystyle \frac{1}{s}}}{\parallel {\mathcal{X}}_m\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}.$$

Consequently, substituting the above equivalence into the previous inequality yields

$$\begin{array}{cc}& \parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \le C{\displaystyle \sum _{m=k+1}^{\infty }}{2}^{(m-k)(\mathsf{\alpha }-\mathit{n})}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta )\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-1}}\right)}\omega {\left({\mathcal{B}}_m\right)}^{-{\displaystyle \frac{1}{s}}}\parallel {\mathcal{X}}_m{\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^1}\right)}{\displaystyle {\parallel {\mathcal{X}}_k\parallel }_{L^{{\displaystyle l_1^{\prime }}(·)}\left({\displaystyle {\omega }_1^{-1}}\right)}^{-1}}.\hfill \end{array}$$

Now, by invoking Lemma 6, we deduce that

$$\begin{array}{cc}& \parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \le C{\displaystyle \sum _{m=k+1}^{\infty }}{2}^{(m-k)(\mathsf{\alpha }-\mathit{n}+\mathit{n})}\parallel F_m{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta )\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-1}}\right)}\omega {\left({\mathcal{B}}_m\right)}^{-{\displaystyle \frac{1}{s}}}.\hfill \end{array}$$

Proceeding further and following the line of reasoning employed in Theorem 1, the above inequality can be refined to yield

$$\begin{array}{cc}& \parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}F·{\mathcal{X}}_k{\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \le C{\displaystyle \sum _{m=k+1}^{\infty }}{2}^{(m-k)(\mathsf{\alpha }-{\displaystyle \frac{\mathit{n}}{s}})}{\parallel F_m\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}.\hfill \end{array}$$

Consequently, passing to the quasi-norm of the weighted central Morrey-type space, we arrive at

$$\begin{array}{c}\hfill \parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}F·{\mathcal{X}}_k{\parallel }_{{\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)}\le C{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=k+1}^{\infty }}{2}^{(m-k)(\mathsf{\alpha }-{\displaystyle \frac{\mathit{n}}{s}}+\mathit{n}\mathsf{\lambda }+\mathit{n})}.\end{array}$$

Finally, taking into account the condition

$$\gamma =\mathsf{\alpha }-{\displaystyle \frac{\mathit{n}}{s}}+\mathit{n}\mathsf{\lambda }+\mathit{n}<0,$$

$$\begin{array}{c}\hfill \parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}F·{\mathcal{X}}_k{\parallel }_{{\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)}\le C{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=k+1}^{\infty }}{2}^{(m-k)\gamma }.\end{array}$$

Since $(m-k>0)$ and $(\gamma <0)$, the above series is geometric and converges. Therefore, we immediately obtain the desired boundedness estimate

$$\begin{array}{c}\hfill \parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}{F\parallel }_{{\dot{\mathcal{B}}}^{l(·),\mathsf{\lambda }}\left(\omega \right)}\le C{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}.\end{array}$$

□

This completes the proof of the required estimate.

4. Commutators of Fractional Hardy Operators

Theorem 3.

Let $0<\mathsf{\alpha }<\mathit{n}$ and suppose that $\Omega \in L^s\left(S^{\mathit{n}-1}\right)$ with ${\displaystyle \frac{\mathit{n}}{\mathit{n}-1}}<s$. Assume that the variable exponent functions $l_1(·),\rho (·),l(·)\in \mathit{P}\left({\mathbb{R}}^{\mathit{n}}\right)$ satisfy the conditions (4) and (5) stated in Proposition 1. Furthermore, let the variable exponent $l_2(·)$ be defined through the relation

$${\displaystyle \frac{1}{l_1(·)}}={\displaystyle \frac{1}{l_2(·)}}-{\displaystyle \frac{1}{l(·)}}+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}.$$

Assume additionally that the parameter ${\mathsf{\lambda }}_1$ fulfills the following requirement. In the particular case when

$${\displaystyle \frac{1}{l(·)}}={\displaystyle \frac{1}{{\displaystyle l_1^{\prime }}(·)}}-{\displaystyle \frac{1}{s}},$$

there exists a constant ${\mathsf{\lambda }}_1>-\mathsf{\lambda }-{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}$ such that, upon setting $\mu ={\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}$ and ${\mathsf{\lambda }}_2={\mathsf{\lambda }}_1+\mathsf{\lambda }+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}},$ the following statement holds. If b belongs to the weighted central bounded mean oscillation space $CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)$, $\mathit{n}(\lambda +{\lambda }_1)+\partial >0,$ and the weights ${\omega }^{l(·)},{\displaystyle {\omega }_1^{l_1(·)}},{\displaystyle {\omega }_2^{l_2(·)}}$ are elements of the Muckenhoupt class $A_1,$ then the commutator generated by b and the fractional rough Hardy operator $H_{\mathsf{\alpha },\Omega }$ satisfies the estimate

$$||[b,H_{\mathsf{\alpha },\Omega }]{F||}_{{\dot{B}}^{l_2(·),{\mathsf{\lambda }}_2}\left({\omega }_2\right)}\le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{||F||}_{{\dot{B}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}.$$

Here, C denotes a positive constant independent of F.

Proof of Theorem 3. 

Consider the commutator generated by the function b and the fractional rough Hardy operator. For $\xi \in {\mathbb{R}}^{\mathit{n}}$, we have

$$\begin{array}{cc}\hfill |[b,H_{\mathsf{\alpha }}]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)|& \le {\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\int }_{\mathcal{B}(0,|\xi |)}|F\left(\eta \right)(b\left(\xi \right)-b\left(\eta \right))\Omega (\xi -\eta )|d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & \le {\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\int }_{\mathcal{B}(0,|\xi |)}|F\left(\eta \right)(b\left(\xi \right)-b_{\mathcal{B}})\Omega (\xi -\eta )|d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & +{\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\int }_{\mathcal{B}(0,|\xi |)}|F\left(\eta \right)(b\left(\eta \right)-b_{\mathcal{B}})\Omega (\xi -\eta )|d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & =A_1+A_2.\hfill \end{array}$$

We first proceed to estimate the term $A_1$. Let the auxiliary exponent function $\rho \left(\xi \right)$ be defined through the relation

$${\displaystyle \frac{1}{\rho \left(\xi \right)}}={\displaystyle \frac{1}{l_1\left(\xi \right)}}-{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}},$$

from which it follows that

$${\displaystyle \frac{1}{l_2\left(\xi \right)}}={\displaystyle \frac{1}{\rho \left(\xi \right)}}+{\displaystyle \frac{1}{l\left(\xi \right)}}.$$

Under this notation, the first component may be expressed as

$$\begin{array}{cc}\hfill A_1& =|(b\left(\xi \right)-b_{\mathcal{B}}){\mathcal{X}}_{\mathcal{B}}\left(\xi \right)||H_{\mathsf{\alpha },\Omega }F\left(\xi \right)|.\hfill \end{array}$$

Consequently, its norm in the weighted variable exponent Lebesgue space can be written as

$$\begin{array}{c}\hfill \parallel A_1{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}={\parallel (b\left(\xi \right)-b_{\mathcal{B}}){\mathcal{X}}_{\mathcal{B}}\left(\xi \right)H_{\mathsf{\alpha },\Omega }F\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}.\end{array}$$

Let the exponent function $\rho (·)$ satisfy

$${\displaystyle \frac{1}{\rho (·)}}={\displaystyle \frac{1}{l_2(·)}}-{\displaystyle \frac{1}{l(·)}}.$$

Employing Lemma 3, corresponding to the decomposition

$${\displaystyle \frac{1}{l_2(·)}}={\displaystyle \frac{1}{\rho (·)}}+{\displaystyle \frac{1}{l(·)}},$$

we obtain

$$\begin{array}{cc}\hfill \parallel A_1{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}& \le C\parallel H_{\mathsf{\alpha },\Omega }F{\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{\rho (·)}\left({\omega }^{-1}{\omega }_2\right)}{\parallel (b\left(\xi \right)-b_{\mathcal{B}}){\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & =C\parallel H_{\mathsf{\alpha },\Omega }{F\parallel }_{{\dot{\mathcal{B}}}^{\mu ,\rho (·)}\left({\omega }^{-1}{\omega }_2\right)}{|\mathcal{B}|}^{\mu }\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{\rho (·)}\left({\omega }^{-1}{\omega }_2\right)}{\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{|\mathcal{B}|}^{\mathsf{\lambda }}{\parallel {\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}.\hfill \end{array}$$

Recalling that $\mu ={\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}$, and invoking the boundedness result established in Theorem 1, we arrive at the estimate

$$\begin{array}{c}\hfill \parallel A_1{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le {C\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2}\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}{\parallel {\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{\rho (·)}\left({\omega }^{-1}{\omega }_2\right)}.\end{array}$$

$$\begin{array}{cc}\hfill A_2& ={\displaystyle \sum _{m=-\infty }^0}{\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\displaystyle \sum _{k=-\infty }^m}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}|F\left(\eta \right)(b\left(\eta \right)-b_{\mathcal{B}})\Omega (\xi -\eta )|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & \le {\displaystyle \sum _{m=-\infty }^0}{\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\displaystyle \sum _{k=-\infty }^m}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}|F\left(\eta \right)(b\left(\eta \right)-b_{2^k\mathcal{B}})\Omega (\xi -\eta )|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & +{\displaystyle \sum _{m=-\infty }^0}{\displaystyle \frac{1}{{|\xi |}^{\mathit{n}-\mathsf{\alpha }}}}{\displaystyle \sum _{k=-\infty }^m}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}|F\left(\eta \right)(b_{\mathcal{B}}-b_{2^k\mathcal{B}})\Omega (\xi -\eta )|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & =A_{21}+A_{22}\hfill \end{array}$$

$$\begin{array}{cc}\hfill A_{21}& ={\displaystyle \sum _{m=-\infty }^0}|2^m{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\displaystyle \sum _{k=-\infty }^m}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}|F\left(\eta \right)(b\left(\eta \right)-b_{2^k\mathcal{B}})\Omega (\xi -\eta )|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill A_{21}& \le C{\displaystyle \sum _{m=-\infty }^0}|2^m{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=-\infty }^m}{\parallel (b\left(\eta \right)-b_{2^k\mathcal{B}}){\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \parallel F{\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta ){\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)}\hfill \end{array}$$

$$\begin{array}{cc}\hfill A_{21}& \le C{\displaystyle \sum _{m=-\infty }^0}|2^m{\mathcal{B}|}^{{\displaystyle \frac{\alpha }{\mathit{n}}}-1}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=-\infty }^m}{\parallel (b\left(\eta \right)-b_{2^k\mathcal{B}}){\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \times \parallel F{\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta ){\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\displaystyle {\omega }_2^{-1}}\right)}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\lambda }\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\lambda }_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}{|2^m\mathcal{B}|}^{{\displaystyle \frac{\alpha }{\mathit{n}}}-1}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & \times {\displaystyle \sum _{k=-\infty }^m}|2^k{\mathcal{B}|}^{\lambda +{\lambda }_1}\parallel {\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}\parallel {\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -·)\mathcal{X}{2}^k\mathcal{B}\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)}.\hfill \end{array}$$

${\displaystyle \frac{1}{l_1(·)}}+{\displaystyle \frac{1}{s}},$ together with the characteristic-function estimates for the weighted variable exponent spaces, we obtain

$$\begin{array}{cc}\hfill & \parallel {\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}\parallel \mathcal{X}{2}^k\mathcal{B}{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -·){\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)}\hfill \\ & \le C{2}^{(k-m)\partial }|2^m\mathcal{B}|\parallel {\mathcal{X}}_{2^m\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}{\parallel {\mathcal{X}}_{2^m\mathcal{B}}\parallel }_{L^{l_1(·)}\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)},\hfill \end{array}$$

for $(k\le m\le 0)$, where $(\partial >0)$ is determined by the characteristic-function estimates.

Therefore,

$$\begin{array}{cc}\hfill A_{21}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\lambda }\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\lambda }_1}\left({\omega }_1\right)}\hfill \\ & \times {\displaystyle \sum _{m=-\infty }^0}|2^m{\mathcal{B}|}^{{\displaystyle \frac{\alpha }{\mathit{n}}}-1}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)|2^m\mathcal{B}|\parallel {\mathcal{X}}_{2^m\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}{\parallel {\mathcal{X}}_{2^m\mathcal{B}}\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\hfill \\ & \times {\displaystyle \sum _{k=-\infty }^m}{|2^k\mathcal{B}|}^{\lambda +{\lambda }_1}{2}^{(k-m)\partial }.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=-\infty }^m}{|2^k\mathcal{B}|}^{\lambda +{\lambda }_1}{2}^{(k-m)\partial }& =|2^m{\mathcal{B}|}^{\lambda +{\lambda }_1}{\displaystyle \sum _{k=-\infty }^m}{2}^{(k-m)\mathit{n}(\lambda +{\lambda }_1)+\partial }.\hfill \end{array}$$

Consequently, whenever $[\mathit{n}(\lambda +{\lambda }_1)+\partial >0,]$ the above geometric series converges and

$${\displaystyle \sum _{k=-\infty }^m}|2^k{\mathcal{B}|}^{\lambda +{\lambda }_1}{2}^{(k-m)\partial }\le C{|2^m\mathcal{B}|}^{\lambda +{\lambda }_1}.$$

Thus,

$$\begin{array}{cc}\hfill A_{21}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\lambda }\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\lambda }_1}\left({\omega }_1\right)}\hfill \\ & \times {\displaystyle \sum _{m=-\infty }^0}{|2^m\mathcal{B}|}^{{\displaystyle \frac{\alpha }{\mathit{n}}}+\lambda +{\lambda }_1}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & \times \parallel {\mathcal{X}}_{2^m\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}{\parallel \mathcal{X}{2}^m\mathcal{B}\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}.\hfill \end{array}$$

Finally, applying the characteristic-function estimate in $\left(L^{l_2(·)}\left({\omega }_2\right)\right)$, together with the defining relation among the Morrey-type parameters, gives

$$\begin{array}{cc}\hfill \parallel A_{21}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\lambda }\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\lambda }_1}\left({\omega }_1\right)}\hfill \\ & \times {\displaystyle \sum _{m=-\infty }^0}|2^m{\mathcal{B}|}^{{\displaystyle \frac{\alpha }{\mathit{n}}}+\lambda +{\lambda }_1}{\parallel {\mathcal{X}}_{2^m\mathcal{B}}\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\lambda }\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\lambda }_1}\left({\omega }_1\right)}{|\mathcal{B}|}^{{\lambda }_2}{\parallel {\mathcal{X}}_{2^m\mathcal{B}}\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}{\displaystyle \sum _{m=-\infty }^0}{2}^{m\mathit{n}{\lambda }_2},\hfill \end{array}$$

$$\begin{array}{c}\hfill \parallel A_{21}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2}\end{array}$$

$$\begin{array}{cc}\hfill A_{22}=& {\displaystyle \sum _{m=-\infty }^0}|2^m{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\displaystyle \sum _{k=-\infty }^m}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}|(b_{\mathcal{B}}-b_{2^k\mathcal{B}})F\left(\eta \right)\Omega (\xi -\eta )|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill |(b_{\mathcal{B}}-b_{2^n\mathcal{B}})|& ={\displaystyle \sum _{\mathit{n}=k}^{-1}}|(b_{2^{\mathit{n}+1}\mathcal{B}}-b_{2^{\mathit{n}}\mathcal{B}})\hfill \\ & ={\displaystyle \sum _{\mathit{n}=k}^{-1}}{\displaystyle \frac{1}{|2^{\mathit{n}}\mathcal{B}|}}{\int }_{2^{\mathit{n}}\mathcal{B}}|b\left(\eta \right)-b_{2^{\mathit{n}+1}\mathcal{B}}|d\eta \hfill \\ & \le C{\displaystyle \sum _{\mathit{n}=k}^{-1}}{\displaystyle \frac{1}{|2^{\mathit{n}}\mathcal{B}|}}\parallel (b-b_{2^{\mathit{n}+1}b}){\mathcal{X}}_{2^{\mathit{n}+1}\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}{\parallel {\mathcal{X}}_{2^{\mathit{n}+1}\mathcal{B}}\parallel }_{L^{l^{\prime }(·)}\left({\omega }^{-1}\right)}.\hfill \end{array}$$

In light of Lemma 7, the following inequality follows directly

$$\begin{array}{cc}\hfill |(b_{\mathcal{B}}-b_{2^k\mathcal{B}})|& \le C{\displaystyle \sum _{m=k}^{-1}}{\displaystyle \frac{1}{|2^m\mathcal{B}|}}{\parallel (b-b_{2^{m+1}b}){\mathcal{X}}_{2^{m+1}\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}{\displaystyle \frac{|2^{m+1}\mathcal{B}|}{\parallel {\mathcal{X}}_{2^{m+1}\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}}}\hfill \\ & \le C{\displaystyle \sum _{m=k}^{-1}}{\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{|2^{m+1}\mathcal{B}|}^{\mathsf{\lambda }}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\displaystyle \sum _{m=k}^{-1}}{|2^{m+1}\mathcal{B}|}^{\mathsf{\lambda }}.\hfill \end{array}$$

(19)

Since

$$|2^{m+1}\mathcal{B}|=2^{(m+1)\mathit{n}}|\mathcal{B}|,$$

we have

$$|2^{m+1}{\mathcal{B}|}^{\lambda }=2^{(m+1)\mathit{n}\lambda }{|\mathcal{B}|}^{\lambda }.$$

Hence,

$$\begin{array}{cc}\hfill {\displaystyle \sum _{m=k}^{-1}}{|2^{m+1}\mathcal{B}|}^{\lambda }& ={|\mathcal{B}|}^{\lambda }{\displaystyle \sum _{m=k}^{-1}}{2}^{(m+1)\mathit{n}\lambda }.\hfill \end{array}$$

Since $\lambda <0$, the sequence

$${\displaystyle {\left\{|2^{m+1}{\mathcal{B}|}^{\lambda }\right\}}_{m=k}^{-1}}$$

is decreasing as m increases. Consequently,

$$|2^{m+1}{\mathcal{B}|}^{\lambda }\le {|2^{k+1}\mathcal{B}|}^{\lambda },k\le m\le -1.$$

Moreover, for $k\le -1$, the sum contains exactly

$$(-1)-k+1=-k=|k|$$

terms. Therefore,

$${\displaystyle \sum _{m=k}^{-1}}|2^{m+1}{\mathcal{B}|}^{\lambda }\le |k||2^{k+1}{\mathcal{B}|}^{\lambda }.$$

$$\begin{array}{cc}\hfill |(b_{\mathcal{B}}-b_{2^k\mathcal{B}})|& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}|2^{k+1}{\mathcal{B}|}^{\mathsf{\lambda }}|k|\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill A_{22}\le & C{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)|2^m{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\displaystyle \sum _{k=-\infty }^m}{\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}|2^{k+1}{\mathcal{B}|}^{\mathsf{\lambda }}|k|\parallel F{\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\hfill \\ & \parallel \Omega (\xi -\eta ){\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)}{\parallel {\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)|2^m{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\displaystyle \sum _{k=-\infty }^m}|2^k{\mathcal{B}|}^{\mathsf{\lambda }+{\mathsf{\lambda }}_1}{|k|\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\hfill \\ & \parallel {\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\parallel \Omega (\xi -\eta ){\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)}{\parallel {\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)|2^m{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\displaystyle \sum _{k=-\infty }^m}|2^k{\mathcal{B}|}^{\mathsf{\lambda }+{\mathsf{\lambda }}_1}{|k|\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{|2^k\mathcal{B}|}^{{\displaystyle \frac{1}{l_1(·)}}+{\displaystyle \frac{1}{s}}+{\displaystyle \frac{1}{l(·)}}}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)|2^m{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\displaystyle \sum _{k=-\infty }^m}|2^k{\mathcal{B}|}^{\mathsf{\lambda }+{\mathsf{\lambda }}_1+1}|k|\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)|2^m{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}|2^m{\mathcal{B}|}^{\mathsf{\lambda }+{\mathsf{\lambda }}_1+1}|m|\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}|m||2^m{\mathcal{B}|}^{\mathsf{\lambda }+{\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right).\hfill \end{array}$$

Using

$$\parallel {\mathcal{X}}_{2^m\mathcal{B}}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le C{|2^m\mathcal{B}|}^{{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}},$$

where

$${\displaystyle l_2^{-}}:=\underset{x\in {\mathbb{R}}^{\mathit{n}}}{ess\; inf}{l}_2\left(\xi \right).$$

$$\begin{array}{cc}\hfill \parallel A_{22}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}|m||2^m{\mathcal{B}|}^{\mathsf{\lambda }+{\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}}{\parallel {\mathcal{X}}_{2^m\mathcal{B}}\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}|m||2^m{\mathcal{B}|}^{\mathsf{\lambda }+{\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}}{|2^m\mathcal{B}|}^{{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}|m||2^m{|}^{{\mathsf{\lambda }}_2+{\displaystyle \frac{1}{l_2(·)}}}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2+{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2}{\parallel {\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}.\hfill \end{array}$$

Collecting the bounds derived for $A_1$, $A_2$, $A_{21}$, and $A_{22},$ we deduce the desired estimate

$$\begin{array}{c}\hfill \parallel [b,H_{\mathsf{\alpha },\Omega }]f{\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2}{\parallel {\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\end{array}$$

$$\begin{array}{c}\hfill \parallel [b,H_{\mathsf{\alpha },\Omega }]{F\parallel }_{{\dot{\mathcal{B}}}^{l_2(·),{\mathsf{\lambda }}_2}\left({\omega }_2\right)}\le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\end{array}$$

□

Theorem 4.

Let $0<\mathsf{\alpha }<\mathit{n}$, and let $\Omega \in L^s\left(S^{\mathit{n}-1}\right)$ for some ${\displaystyle \frac{\mathit{n}}{\mathit{n}-1}}<s$. Suppose that the variable exponent functions $l_1(·),\rho (·),l(·)\in \mathit{P}\left({\mathbb{R}}^{\mathit{n}}\right)$ satisfy the conditions (4) and (5) stipulated in Proposition 1. Moreover, define the variable exponent $l_2(·)$ by the pointwise relation

$${\displaystyle \frac{1}{l_1(·)}}={\displaystyle \frac{1}{l_2(·)}}-{\displaystyle \frac{1}{l(·)}}+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}.$$

Furthermore, assume that the parameter ${\mathsf{\lambda }}_1$ satisfies the prescribed admissibility condition. In particular, in the case where

$${\displaystyle \frac{1}{l(·)}}={\displaystyle \frac{1}{{\displaystyle l_1^{\prime }}(·)}}-{\displaystyle \frac{1}{s}},$$

suppose that there exists a constant

$${\mathsf{\lambda }}_1<-\mathsf{\lambda }-{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}.$$

Upon introducing the parameters

$$\mu ={\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}},{\mathsf{\lambda }}_2={\mathsf{\lambda }}_1+\mathsf{\lambda }+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}},$$

the following boundedness assertion is valid.

More precisely, if ${\mathsf{\lambda }}_2<0,$${\mathsf{\lambda }}_2+{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}>0,$ $b\in CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right),$ and the weights ${\omega }^{l(·)},{\displaystyle {\omega }_1^{l_1(·)}},{\displaystyle {\omega }_2^{l_2(·)}}$ belong to the Muckenhoupt class $A_1$, then the commutator associated with b and the adjooint fractional rough Hardy operator ${\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}$ extends boundedly from ${\dot{B}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)$ into ${\dot{B}}^{l_2(·),{\mathsf{\lambda }}_2}\left({\omega }_2\right)$. In particular, there exists a positive constant C, independent of F, such that

$$\begin{array}{c}\hfill \parallel [b,{\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}]{F\parallel }_{{\dot{B}}^{l_2(·),{\mathsf{\lambda }}_2}\left({\omega }_2\right)}\le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{B}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}.\end{array}$$

Proof of Theorem 4.

$$\begin{array}{cc}\hfill |[b,{\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)|& \le {\int }_{\mathcal{B}{(0,|\xi |)}^c}{\displaystyle \frac{|F(\eta \left)\right(b\left(\xi \right)-b\left(\eta \right))\Omega (\xi -\eta )|}{{|\eta |}^{\mathit{n}-\mathsf{\alpha }}}}d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & \le {\int }_{\mathcal{B}{(0,|\xi |)}^c}{\displaystyle \frac{|F\left(\eta \right)(b\left(\xi \right)-b_{\mathcal{B}})\Omega (\xi -\eta )|}{{|\eta |}^{\mathit{n}-\mathsf{\alpha }}}}d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & +{\int }_{\mathcal{B}{(0,|\xi |)}^c}{\displaystyle \frac{|F\left(\eta \right)(b\left(\eta \right)-b_{\mathcal{B}})\Omega (\xi -\eta )|}{{|\eta |}^{\mathit{n}-\mathsf{\alpha }}}}d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & =D_1+D_2.\hfill \end{array}$$

For the first component, one readily observes that

$$\begin{array}{c}\hfill D_1=|(b\left(\xi \right)-b_{\mathcal{B}}){\mathcal{X}}_{\mathcal{B}}\left(\xi \right)||{\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}F\left(\xi \right)|.\end{array}$$

Consequently, its norm in the weighted variable exponent Lebesgue space can be expressed as

$$\begin{array}{c}\hfill \parallel D_1{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}={\parallel (b\left(\xi \right)-b_{\mathcal{B}}){\mathcal{X}}_{\mathcal{B}}\left(\xi \right){\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}F\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}.\end{array}$$

Invoking the generalized Hölder inequality in the variable exponent setting, namely ${\displaystyle \frac{1}{l_2(·)}}={\displaystyle \frac{1}{p(·)}}+{\displaystyle \frac{1}{l(·)}},$ we deduce that

$$\begin{array}{cc}\hfill \parallel D_1{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}& \le C\parallel (b\left(\xi \right)-b_{\mathcal{B}}){\mathcal{X}}_{\mathcal{B}}{\left(\xi \right)\parallel }_{L^{l(·)}\left(\omega \right)}{\parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}F\left(\xi \right){\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{p(·)}\left({\omega }^{-1}{\omega }_2\right)}\hfill \\ & ={C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{|\mathcal{B}|}^{\mathsf{\lambda }}\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}{|\mathcal{B}|}^{\mu }\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{p(·)}\left({\omega }^{-1}{\omega }_2\right)}{\parallel {\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}F\parallel }_{{\dot{\mathcal{B}}}^{\mu ,p(·)}\left({\omega }^{-1}{\omega }_2\right)}.\hfill \end{array}$$

Given the relation $\mu ={\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}$, an application of the boundedness result established in Theorem 2 immediately yields the estimate

$$\begin{array}{c}\hfill \parallel D_1{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2}\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}\parallel {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{p(·)}\left({\omega }^{-1}{\omega }_2\right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}.\end{array}$$

We now turn our attention to the second component, which is given by

$$\begin{array}{c}\hfill D_2={\int }_{\mathcal{B}{(0,|\xi |)}^c}{\displaystyle \frac{|F\left(\eta \right)(b\left(\eta \right)-b_{\mathcal{B}})\Omega (\xi -\eta )|}{{|\eta |}^{\mathit{n}-\mathsf{\alpha }}}}d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right).\end{array}$$

To facilitate the subsequent estimation, we decompose the domain of integration into a sequence of dyadic annuli. Accordingly, we obtain

$$\begin{array}{cc}\hfill D_2& ={\displaystyle \sum _{k=-\infty }^0}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}{\displaystyle \frac{|F\left(\eta \right)(b\left(\eta \right)-b_{\mathcal{B}})\Omega (\xi -\eta )|}{{|\eta |}^{\mathit{n}-\mathsf{\alpha }}}}d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & \le {\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}|2^k{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}|F\left(\eta \right)(b\left(\eta \right)-b_{2^k\mathcal{B}})\Omega (\xi -\eta )|d\eta \hfill \\ & +{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}|2^k{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}|F\left(\eta \right)(b_{\mathcal{B}}-b_{2^k\mathcal{B}})\Omega (\xi -\eta )|d\eta \hfill \\ & =D_{21}+D_{22}.\hfill \end{array}$$

We first estimate the term $D_{12}$, which can be written as

$$\begin{array}{cc}\hfill D_{21}& ={\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}|2^k{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}|F\left(\eta \right)(b\left(\eta \right)-b_{2^k\mathcal{B}})\Omega (\xi -\eta )|d\eta .\hfill \end{array}$$

Employing the generalized Holder inequality corresponding to the exponents satisfying ${\displaystyle \frac{1}{l(·)}}+{\displaystyle \frac{1}{l_1(·)}}+{\displaystyle \frac{1}{s}}=1,$ we obtain the estimate

$$\begin{array}{cc}\hfill D_{21}& \le C{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}|2^k{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\parallel (b\left(\eta \right)-b_{2^k\mathcal{B}}){\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \parallel F{\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta ){\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)}.\hfill \end{array}$$

By exploiting the defining properties of the weighted central BMO space together with the norm characterization of the central Morrey space, we deduce

$$\begin{array}{cc}\hfill D_{21}& \le C{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}|2^k{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}|2^k{\mathcal{B}|}^{\mathsf{\lambda }}{\parallel {\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ \hfill & \times {\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}|2^k{\mathcal{B}|}^{{\mathsf{\lambda }}_1}\parallel {\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}{\parallel \Omega (\xi -\eta ){\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)}\hfill \\ \hfill & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\hfill \\ \hfill & \times {\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}{|2^k\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1+\mathsf{\lambda }+{\mathsf{\lambda }}_1+{\displaystyle \frac{1}{s}}+{\displaystyle \frac{1}{l_1(·)}}+{\displaystyle \frac{1}{l(·)}}}.\hfill \end{array}$$

Using

$${\displaystyle \frac{1}{l(·)}}+{\displaystyle \frac{1}{l_1(·)}}+{\displaystyle \frac{1}{s}}=1,$$

we have

$${\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1+\mathsf{\lambda }+{\mathsf{\lambda }}_1+{\displaystyle \frac{1}{s}}+{\displaystyle \frac{1}{l_1(·)}}+{\displaystyle \frac{1}{l(·)}}={\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}+\mathsf{\lambda }+{\mathsf{\lambda }}_1.$$

Hence, setting

$${\mathsf{\lambda }}_2={\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}+\mathsf{\lambda }+{\mathsf{\lambda }}_1,$$

we obtain

$$\begin{array}{cc}\hfill D_{21}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\hfill \\ & \times {\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}{|2^k\mathcal{B}|}^{{\mathsf{\lambda }}_2}.\hfill \end{array}$$

If ${\mathsf{\lambda }}_2<0$, then

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=m+1}^{\infty }}{|2^k\mathcal{B}|}^{{\mathsf{\lambda }}_2}& ={|\mathcal{B}|}^{{\mathsf{\lambda }}_2}{\displaystyle \sum _{k=m+1}^{\infty }}{2}^{k\mathit{n}{\mathsf{\lambda }}_2}\hfill \\ \hfill & \le {C|\mathcal{B}|}^{{\mathsf{\lambda }}_2}{2}^{(m+1)\mathit{n}{\mathsf{\lambda }}_2}\hfill \\ \hfill & =C|2^{m+1}{\mathcal{B}|}^{{\mathsf{\lambda }}_2}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill D_{21}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\hfill \\ \hfill & \times {\displaystyle \sum _{m=-\infty }^0}{|2^{m+1}\mathcal{B}|}^{{\mathsf{\lambda }}_2}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right).\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill \parallel D_{21}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\hfill \\ \hfill & \times {\displaystyle \sum _{m=-\infty }^0}|2^{m+1}{\mathcal{B}|}^{{\mathsf{\lambda }}_2}{\parallel {\mathcal{X}}_{2^m\mathcal{B}}\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}.\hfill \end{array}$$

Using

$$\parallel {\mathcal{X}}_{2^m\mathcal{B}}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le C{|2^m\mathcal{B}|}^{{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}},$$

where

$${\displaystyle l_2^{-}}:=\underset{x\in {\mathbb{R}}^{\mathit{n}}}{ess\; inf}{l}_2\left(x\right).$$

we obtain

$$\begin{array}{cc}\hfill \parallel D_{21}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\hfill \\ & \times {\displaystyle \sum _{m=-\infty }^0}|2^{m+1}{\mathcal{B}|}^{{\mathsf{\lambda }}_2}{|2^m\mathcal{B}|}^{{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2+{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}}\hfill \\ & \times {\displaystyle \sum _{m=-\infty }^0}{2}^{mi\left({\mathsf{\lambda }}_2+{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}\right)}.\hfill \end{array}$$

Since

$${\mathsf{\lambda }}_2+{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}>0,$$

the last geometric series is convergent. Thus,

$$\begin{array}{cc}\hfill \parallel D_{21}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2+{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill D_{22}& ={\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}|2^k{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\int }_{2^k\mathcal{B}\setminus 2^{k-1}\mathcal{B}}|(b_{\mathcal{B}}-b_{2^k\mathcal{B}})F\left(\eta \right)|d\eta .\hfill \end{array}$$

In the subsequent analysis, the argument proceeds through the deployment of inequality (20) as a pivotal instrument

$$\begin{array}{cc}\hfill D_{22}& \le C{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}|2^k{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}{\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}|2^{k+1}{\mathcal{B}|}^{\mathsf{\lambda }}|k|\parallel F{\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\hfill \\ & \parallel \Omega (\xi -\eta ){\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)}{\parallel {\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^{l(·)}\left(\omega \right)}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}|2^k{\mathcal{B}|}^{{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}-1}|2^k{\mathcal{B}|}^{\mathsf{\lambda }+{\mathsf{\lambda }}_1}{|k|\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}\hfill \\ & \parallel \Omega (\xi -\eta ){\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^s\left({\displaystyle {\omega }_1^{-1}}{\omega }^{-1}\right)}\parallel {\mathcal{X}}_{2^k\mathcal{B}}{\parallel }_{L^{l(·)}\left(\omega \right)}{\parallel {\mathcal{X}}_{2^k\mathcal{B}}\parallel }_{L^{l_1(·)}\left({\omega }_1\right)}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right){\displaystyle \sum _{k=m+1}^{\infty }}|2^k{\mathcal{B}|}^{\mathsf{\lambda }+{\mathsf{\lambda }}_1+{\displaystyle \frac{\mathsf{\alpha }}{\mathit{n}}}}|k|\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)|2^{m+1}{\mathcal{B}|}^{{\mathsf{\lambda }}_2}|m+1|\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}|m+1||2^{m+1}{\mathcal{B}|}^{{\mathsf{\lambda }}_2}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel D_{22}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}|m+1||2^{m+1}{\mathcal{B}|}^{{\mathsf{\lambda }}_2}{\parallel {\mathcal{X}}_{2^m\mathcal{B}}\parallel }_{L^{l_2(·)}\left(\omega \right)}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}|m+1||2^{m+1}{\mathcal{B}|}^{{\mathsf{\lambda }}_2}{|2^m\mathcal{B}|}^{{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}}\hfill \\ & \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{\displaystyle \sum _{m=-\infty }^0}|m+1||2^{m+1}{|}^{{\mathsf{\lambda }}_2+{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2+{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}}.\hfill \end{array}$$

Since

$${\mathsf{\lambda }}_2+{\displaystyle \frac{1}{{\displaystyle l_2^{-}}}}>0,$$

$$\begin{array}{cc}\hfill \parallel D_{22}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}& \le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2}{\parallel {\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}.\hfill \end{array}$$

Synthesizing the preceding estimates for $D_1$, $D_2$, $D_{21}$, and $D_{22},$ we ultimately deduce the principal boundedness result:

$$\begin{array}{c}\hfill \parallel [b,{\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}]F{\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}{|\mathcal{B}|}^{{\mathsf{\lambda }}_2}{\parallel {\mathcal{X}}_{\mathcal{B}}\parallel }_{L^{l_2(·)}\left({\omega }_2\right)}\end{array}$$

which, upon passage to the central Morrey-type norm, furnishes the global bound

$$\begin{array}{c}\hfill \parallel [b,{\displaystyle H_{\mathsf{\alpha },\Omega }^{*}}]{F\parallel }_{{\dot{\mathcal{B}}}^{l_2(·),{\mathsf{\lambda }}_2}\left({\omega }_2\right)}\le {C\parallel b\parallel }_{CBM{O}^{l(·),\mathsf{\lambda }}\left(\omega \right)}{\parallel F\parallel }_{{\dot{\mathcal{B}}}^{l_1(·),{\mathsf{\lambda }}_1}\left({\omega }_1\right)}.\end{array}$$

□

5. Conclusions

Within the ambit of the present treatise, we have undertaken a meticulous and analytically rigorous investigation into the boundedness characteristics of the rough Hardy operator, together with its attendant commutators, situated in the highly nuanced and structurally elaborate setting of variable exponent weighted central Morrey spaces. Our discourse is predicated upon the foundational assumption that the underlying symbol functions are confined to the class of functions of bounded mean oscillation (BMO), thereby ensuring an appropriate degree of regularity and oscillatory control.

By the imposition of a carefully calibrated ensemble of admissibility conditions, we have succeeded in deriving affirmative boundedness assertions, which, in turn, illuminate the subtle and nontrivial interdependencies between these operators and the ambient functional architecture. The results obtained herein not only consolidate existing theoretical frameworks but also provide a deeper conceptual understanding of operator behavior in variable exponent contexts.

It is anticipated that the analytical perspectives and methodological constructs developed in this study will furnish a fertile groundwork for subsequent scholarly endeavors, particularly those directed toward the establishment of analogous boundedness phenomena in alternative function spaces governed by variable exponent dynamics and related nonstandard growth conditions.