Boundedness and Two-Weight Estimates for Fractional Hardy Operators and Their Commutators on Morrey–Herz Spaces

Abstract

The principal objective of the present investigation is to rigorously establish the boundedness of fractional Hardy operators within the analytically intricate framework of two-weighted variable exponent Morrey–Herz spaces. As a technically demanding extension of these foundational results, analogous boundedness estimates are further derived for the associated commutator operators under the assumption that the symbol functions are members of the variable exponent bounded mean oscillation $\left(BMO\right)$ space, thereby significantly broadening the applicability and analytical depth of the underlying operator-theoretic framework.

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 ${\mathbb{R}}^{\mathfrak{n}}$ as

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

(1)

and established a sharp inequality associated with this operator. Subsequently, William G. Faris (see [4]) extended this construction by formulating the $\mathfrak{n}$-dimensional analog 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 $\mathfrak{n}$-dimensional fractional Hardy operator was introduced in the following form:

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

(2)

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

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

(3)

where the function $\mathfrak{b}$ 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 $\mathfrak{n}$-dimensional fractional Hardy operator were established by Shanzhen Fu, Zhaohui Liu, and Dachun Wang in [9].

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 [10]. Subsequently, the broader framework of Musielak–Orlicz spaces was introduced and rigorously developed in [11]. 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 ([12,13]). 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, [14,15]). Central bounded mean oscillation spaces, λ-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 ([16,17]). The theoretical origins of Morrey-type spaces can be traced back to the pioneering work of Charles B. Morrey [18], who introduced such spaces in the context of studying the regularity properties of solutions to partial differential equations. Subsequently, in [19], the authors introduced the notions of $\mathrm{\lambda }$-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 ([20,21]).

Moreover, owing to their extensive applicability in diverse scientific and applied disciplines—such as image processing [22], electrorheological fluid models [23], and the theory of partial differential equations [24]—function spaces with variable exponents have attracted considerable scholarly attention. Following the seminal contribution of Ondrej Kovacik [25], 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 [26]. More recently, Dachun Wang and collaborators introduced the notion of variable exponent central BMO spaces and established boundedness results for several operators in [27]. These findings were subsequently generalized by Zunwei Fu [28], who extended the analysis to the setting of variable exponent $\mathrm{\lambda }$-central Morrey spaces and the corresponding central BMO spaces. In ([29,30]), the 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, ([31,32,33]).

In particular, the analytical framework of variable exponent function spaces has undergone extensive and rigorous investigation, with considerable emphasis placed on establishing the boundedness properties of the Hardy–Littlewood maximal operator within these highly nonuniform and structurally intricate spaces (see, e.g., [12,15,34]). Consequently, the celebrated Muckenhoupt weight theory, originally formulated in [35], has naturally evolved into a considerably more sophisticated analytical framework through its extension to variable exponent settings, a development that has substantially enriched the modern theory of weighted inequalities and nonuniform function spaces (see [36,37,38]). Naturally, the classical theory has undergone substantial extensions through both weighted formulations and the considerably more intricate framework of variable exponent analysis ([39,40,41]). These far-reaching generalizations have given rise to a comprehensive body of profound analytical results encompassing, among other topics, duality theory, wavelet characterizations, the construction of unconditional and greedy bases, the boundedness of the Hardy–Littlewood maximal operator, a broad spectrum of sublinear operators, fractional integral operators, as well as commutator estimates for singular and fractional integral operators generated by $BMO$ symbols. More recently, this line of investigation has been further advanced through the systematic development of Herz spaces simultaneously incorporating variable exponent structures and Muckenhoupt weights, thereby furnishing a significantly more flexible and analytically sophisticated functional framework for the study of weighted operator inequalities (see [39,40,42]). For the first time, the two-weight framework was systematically incorporated into the study of Herz-type function spaces in ([41,43]), where the authors established the boundedness properties of fractional Hardy operators together with their associated commutators on two-weighted Herz spaces and two-weighted grand Morrey–Herz spaces.

Inspired by the remarkable advances documented in [28,38,39,41,43], this article undertakes a comprehensive and rigorous analysis of the boundedness properties of fractional Hardy operators and their corresponding commutators in the highly sophisticated setting of variable exponent two-weight Morrey–Herz spaces. The present investigation significantly broadens the existing theory by simultaneously incorporating variable exponent structures and two-weight techniques, thereby establishing a unified analytical framework that extends numerous classical and contemporary boundedness results in weighted harmonic analysis.

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 Hardy operators on Two-weighted Morrey–Herz 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 $\mathrm{\lambda }$-central BMO functions within the setting of Two-weighted Morrey–Herz spaces endowed with variable exponents.

Moreover, in this article, we use $\left|\mathcal{B}\right|$ to denote the Lebesgue measure of a measurable set $\mathcal{B}\subset {\mathbb{R}}^{\mathfrak{n}}$, 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 $2^{\mathfrak{n}}$ will be denoted by

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

$$A_n={\mathcal{B}}_n\setminus {\mathcal{B}}_{n-1}$$

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

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}}^i$$(i=1,2,3,\dots ,\mathfrak{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 collection $\mathcal{P}\left(\mathcal{B}\right)$ comprises all measurable functions $l(·)$ along with their conjugates $l^{\prime }(·$) satisfying the boundedness constraints

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

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

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

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

Endowed with the Luxemburg norm

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

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

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

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

$$\mathbb{M}F=\underset{r>0}{\sup }{\displaystyle \frac{1}{|{\mathcal{B}}_r|}}{\int }_{{\mathcal{B}}_r}\left|F\right|d\zeta$$

where ${\mathcal{B}}_r=\{\zeta \in {\mathbb{R}}^n:|\xi -\eta |<r\}.$

We assume that $0<\alpha <n$ and $l_1(·)\in \mathcal{P}\left({\mathbb{R}}^n\right)$ satisfies

$$1<{\left(l_1\right)}_{-}\le {\left(l_1\right)}_{+}<{\displaystyle \frac{n}{\alpha }}.$$

Define the variable exponent $l_2(·)$ by

$${\displaystyle \frac{1}{l_2\left(x\right)}}={\displaystyle \frac{1}{l_1\left(x\right)}}-{\displaystyle \frac{\alpha }{n}},x\in {\mathbb{R}}^n.$$

The above condition implies that $1<{\left(l_2\right)}_{-}\le {\left(l_2\right)}_{+}<\infty$. Moreover, we suppose that the function $l_1(·)$ is log-Hölder continuous with the global condition. Thus, the function $l_2(·)$ has the regularity properties needed to allow the estimates of the variable exponent fractional integral. Under these assumptions, the fractional integral operator $I_{\alpha }$ is bounded from $L^{l_1(·)}\left({\mathbb{R}}^{\mathfrak{n}}\right)$ to $L^{l_2(·)}\left({\mathbb{R}}^{\mathfrak{n}}\right)$.

We denote by $\mathfrak{B}$ the subclass of variable exponents $l(·)\in {\mathbb{R}}^i$ 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 $\mathfrak{B}\left(\mathcal{B}\right).$ The following inequalities have been established by Neugebauer, Cruz-Uribe and Fiorenza [34], as well as by Nakvinda [44], and are summarized in the ensuing proposition.

Proposition 1 

([44]). Let $\mathcal{B}$ be an open subset of $R^i$, 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

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

(4)

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

(5)

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

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 $\mathbb{M}\omega \left(\xi \right)\lesssim \omega \left(\xi \right),$ $\xi \in {\mathbb{R}}^i.$

Definition 2. 

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

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

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

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

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

Based on the above definition, Hitoshi Izuki and Takuma Noi [39] proved a significant monotonicity property for these classes of variable exponent weights, which is restated in the following proposition.

Definition 3. 

Let $\mathrm{\alpha }\in (0,i)$ and suppose that $l_2(·),l_1(·)\in \mathcal{P}\left({\mathbb{R}}^i\right)$ are variable exponent functions satisfying the structural relation ${\displaystyle \frac{1}{l_1(·)}}={\displaystyle \frac{1}{l_2(·)}}+{\displaystyle \frac{\mathrm{\alpha }}{i}}.$ A weight ω is said to belong to the class $A(l_2(·),l_1(·))$ if it fulfils the following integral inequality for every ball $\mathcal{B}\subset {\mathbb{R}}^i$

$$\parallel {\omega }^{-1}{\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l_1^{\prime }(·)}}^{{\displaystyle \frac{1}{1-\frac{\mathrm{\alpha }}{i}}}}\parallel \omega {\mathcal{X}}_{\mathcal{B}}{\parallel }_{L^{l_2(·)}}^{{\displaystyle \frac{1}{1-\frac{\mathrm{\alpha }}{i}}}}\le \left|\mathcal{B}\right|.$$

Definition 4. 

Let $\nu ,\omega$ be weights on ${\mathbb{R}}^n$, $l(·)\in \mathcal{P}\left({\mathbb{R}}^n\right)$, $p\in (1,\infty )$, $\lambda \in [0,\infty )$, $\beta (·)\in L^{\infty }\left({\mathbb{R}}^n\right)$. The homogeneous two-weight Herz–Morrey space $\dot{M}{K}_{l(·),p}^{\beta (·),\mathrm{\lambda }}(\nu ,\omega )$ is defined by

$$\dot{M}{K}_{l(·),p}^{\beta (·),\mathrm{\lambda }}(\nu ,\omega ):=\left\{F\in L_{\mathrm{loc}}^{l(·)}({\mathbb{R}}^n\setminus \left\{0\right\},\omega ):{\parallel F\parallel }_{\dot{M}{K}_{l(·),p}^{\beta (·),\mathrm{\lambda }}(\nu ,\omega )}<\infty \right\},$$

where

$${\parallel F\parallel }_{\dot{M}{K}_{l(·),p}^{\beta (·),\mathrm{\lambda }}(\nu ,\omega )}:=\underset{k_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{k=-\infty }^{k_0}}{\parallel \nu {\left(B_k\right)}^{\beta (·)/n}F{\mathcal{X}}_k\parallel }_{L^{l(·)}\left(\omega \right)}^p\right)}^{{\displaystyle \frac{1}{p}}}.$$

The factor $(\nu {\left(B_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}})$ is, therefore, not to be regarded as part of the operator, but as a weight/normalization factor from the Morrey–Herz space.

Lemma 1 

([39]). Let $l_1(·),l_2(·)\in \mathcal{P}\left({\mathbb{R}}^i\right)\bigcap LH\left({\mathbb{R}}^i\right)$ 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 

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

Lemma 3 

([25]). Let ζ be a Banach function space. Then the following assertions hold:

1. 

The associate space ${\zeta }^{\prime }$ is itself a Banach function space. In particular, the norms ${|·|}_{{{(\zeta }^{\prime })}^{\prime }}$ and ${|·|}_{\zeta }$ are mutually equivalent, and consequently the double associate space coincides with the original space, that is, ${{(\zeta }^{\prime })}^{\prime }=\zeta$.

2. 

If $F\in \zeta$ and $G\in {\zeta }^{\prime }$, then the following integral estimate is valid:

$${\int }_{{\mathbb{R}}^n}\left|FG\right|dx\le {\parallel F\parallel }_{\zeta }{\parallel |G\parallel }_{{\zeta }^{\prime }}.$$

This essential inequality is widely known as Hölder’s inequality in the setting of Banach function spaces.

Lemma 4 

([41]). Let ω be a weight function defined on ${\mathbb{R}}^i$. 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:

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

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

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 

([41]). Let us consider $l(·)\in \mathcal{P}\cap LH\left({\mathbb{R}}^i\right)$. If ${\omega }^{l_2(·)}\in A_{l_2(·)}$ and ${\omega }^{l_1(·)}\in A_{l_1(·)}$, then this implies that ${\omega }^{-l_2^{\prime }(·)}\in A_{l_2^{\prime }(·)}$ and ${\omega }^{-l_1^{\prime }(·)}\in A_{l_1^{\prime }(·)}$, respectively. Consequently, the Hardy–Littlewood maximal operator $\mathbb{M}$ is bounded on $L^{l_2^{\prime }(·)}\left({\omega }^{-l_2^{\prime }(·)}\right)$. 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

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

(6)

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

(7)

Lemma 6 

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

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

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 

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

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

(8)

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

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

(9)

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

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 ${\omega }^{-1}\left(\xi \right)\in {\zeta }_{\mathrm{loc}}^{\prime }$. Here, ${\zeta }_{\mathrm{loc}}\left({\mathbb{R}}^i\right)$ 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 $\left|\mathcal{B}\right|<\infty$.

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

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

In this setting, the following lemma holds.

Lemma 8 

([45]). 1.  The weighted space $\zeta ({\mathbb{R}}^i,\omega )$ constitutes a Banach function space when equipped with the norm

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

2. 

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

Remark 1. 

Let $l(·)\in \mathcal{P}\left({\mathbb{R}}^i\right)$. By invoking the definition of the weighted Banach function space $\zeta ({\mathbb{R}}^i,\omega )$, 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 $\zeta =L^{l(·)}\left({\mathbb{R}}^i\right)$ and $W=\omega ,$ it follows immediately that $L^{l(·)}({\mathbb{R}}^i,\omega )=L^{l(·)}\left({\omega }^{l(·)}\right).$

2.

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

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

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

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

Lemma 9 

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

Lemma 10 

([40]). 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:

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

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

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

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

Lemma 11 

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

$${\displaystyle \frac{1}{l_2(·)}}={\displaystyle \frac{1}{l_1(·)}}-{\displaystyle \frac{\mathrm{\alpha }}{i}}.$$

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

Lemma 12 

([41]). Let $m,n\in \mathbb{Z}$, and suppose that $\omega \in A_l$ for some $l\in [1,\infty )$. Let $\epsilon \in (0,1)$ denote the constant introduced in Lemma 5, and define

$${\omega }^{-}=\left\{\begin{array}{cc}\epsilon ,\hfill & \mathit{if}{\alpha }^{-}\ge 0,\hfill \\ l,\hfill & \mathit{if}{\alpha }^{-}<0,\hfill \end{array}\right.{\omega }^{+}=\left\{\begin{array}{cc}l,\hfill & \mathit{if}{\alpha }^{+}\ge 0,\hfill \\ \epsilon ,\hfill & \mathit{if}{\alpha }^{+}<0.\hfill \end{array}\right.$$

Assume, furthermore, that the variable exponent function $\alpha (·)\in L^{\infty }\left({\mathbb{R}}^n\right)$ satisfies the log-Hölder continuity condition both at the origin and at infinity. Then there exists a positive constant C, independent of the integers $m,n$ and the points $\xi ,\eta$, such that, for every $\xi \in D_m$ and $\eta \in D_n$,

$$\omega {\left(B_m\right)}^{\alpha \left(\xi \right)}\le C\omega {\left(B_n\right)}^{\alpha \left(\eta \right)}\left\{\begin{array}{cc}{2}^{(m-n)i{\omega }^{-}{\alpha }^{-}},\hfill & \mathit{whenever}n\le m-1,\hfill \\ 1,\hfill & \mathit{whenever}m-1<n\le m+1,\hfill \\ 2^{(m-n)i{\omega }^{+}{\alpha }^{+}},\hfill & \mathit{whenever}n>m+1.\hfill \end{array}\right.$$

In particular, the above estimate provides a quantitative comparison between the weighted measures of dyadic balls corresponding to distinct scales, with the growth and decay rates being explicitly governed by the extremal values of the variable exponent $\alpha (·)$ and the structural characteristics of the weight ω.

ω denotes the weight function, whereas the quantities ${\omega }^{-}$ and ${\omega }^{+}$ are auxiliary constants and are not the essential infimum or essential supremum of the weight.

Lemma 13 

([41]). Let $\alpha (·)\in L^{\infty }\left({\mathbb{R}}^i\right)$ be a variable exponent satisfying the log-Hölder continuity condition both at the origin and at infinity. Then, for every integer $m\in \mathbb{Z}$ and every point $\xi \in D_m$, there exists a positive constant C, independent of m and ξ, such that

$$\omega {\left(D_m\right)}^{\alpha \left(\xi \right)}\approx \left\{\begin{array}{cc}\omega {\left(D_m\right)}^{{\alpha }_{\infty }},\hfill & \mathit{whenever}m\ge 0,\hfill \\ \omega {\left(D_m\right)}^{\alpha \left(0\right)},\hfill & \mathit{whenever}m\le -1,\hfill \end{array}\right.$$

where the equivalence constants depend solely on the log-Hölder continuity characteristics of $\alpha (·)$ and are entirely independent of the dyadic index m and the point ξ.

Lemma 14 

([43]). Let $l(·)\in \mathcal{P}\left({\mathbb{R}}^n\right)$. Then, for every function $b\in BMO\left({\mathbb{R}}^n\right)$, there exists a positive constant C, independent of b, such that the following norm equivalence holds:

$$C^{-1}{\parallel b\parallel }_{BMO}\le \underset{\mathcal{B}\subset {\mathbb{R}}^n}{\sup }{\displaystyle \frac{1}{\parallel {\chi }_{\mathcal{B}}{\parallel }_{L^{l(·)}}}}{∥(b-b_{\mathcal{B}}){\chi }_{\mathcal{B}}∥}_{L^{l(·)}}\le C{\parallel b\parallel }_{BMO},$$

(10)

where the supremum is taken over all balls $\mathcal{B}\subset {\mathbb{R}}^n$. Moreover, for any integers $m,n\in \mathbb{Z}$ satisfying $n>m$, the oscillation of b over nested balls obeys the estimate

$${∥(b-b_{{\mathcal{B}}_m}){\chi }_{{\mathcal{B}}_n}∥}_{L^{l(·)}}\le {C(n-m)\parallel b\parallel }_{BMO}{\parallel {\chi }_{{\mathcal{B}}_n}\parallel }_{L^{l(·)}},$$

(11)

where the constant $C>0$ is independent of b, m, and n.

3. Boundedness of Fractional Hardy Operators

Theorem 1. 

Let $i=1,2,\dots ,\mathfrak{n}$, $0<\alpha <{\displaystyle \frac{i}{l^{+}}}$, $\lambda \in [0,\infty )$, $\beta (·)\in L^{\infty }\left({\mathbb{R}}^i\right)$, $1<p<\infty$, and let $l_2(·)\in LH\left({\mathbb{R}}^i\right)\cap \mathcal{P}\left({\mathbb{R}}^i\right)$. Assume that the variable exponent $l_1(·)$ is determined by the relation

$${\displaystyle \frac{1}{l_2(·)}}={\displaystyle \frac{1}{l_1(·)}}-{\displaystyle \frac{\alpha }{i}}.$$

Suppose further that $\nu \in A_r$ for some $r\in [1,\infty )$, $\omega \in A_{l_2(·)}$, and that

$${\omega }^{+}{\alpha }^{+}+\alpha -i+{\partial }_1<0,$$

where ${\partial }_1\in (0,1)$ denotes the constant introduced in Lemma 5 corresponding to the exponent function $l(·)$. In addition, assume that the exponent function $\beta (·)$ satisfies the log-Hölder continuity condition both at the origin and at infinity. Then the fractional Hardy operator $H_{\alpha }$ defines a bounded mapping from the two-weight variable exponent Morrey–Herz space $M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )$ into $M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )$. More precisely,

$$\parallel H_{\alpha }{F\parallel }_{M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )}\le C{\parallel F\parallel }_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )},$$

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

Proof of Theorem 1. 

For any $F\in M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )$, if we represent $F_m=F{\mathcal{X}}_m=F.{\mathcal{X}}_{{\mathcal{B}}_m}$ for each $m\in \mathbb{Z}$, then we write

$$F={\displaystyle \sum _{m=-\infty }^{\infty }}F{\mathcal{X}}_m={\displaystyle \sum _{m=-\infty }^{\infty }}{F}_m.$$

Invoking Lemma 12 and exploiting the dyadic decomposition of $\left(F\right)$, we obtain the following pointwise estimate:

$$\begin{array}{c}|\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}F\left(x\right)·{\mathcal{X}}_n\left(\xi \right)|\le \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\int }_{{\mathcal{B}}_n}\left|F\left(t\right)\right|dt·{\mathcal{X}}_n\left(\xi \right)\hfill \\ \le \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\displaystyle \sum _{m=-\infty }^n}{\int }_{{\mathcal{B}}_m}\left|F\left(t\right)\right|dt·{\mathcal{X}}_n\left(\xi \right)\hfill \\ \le 2^{n(\mathrm{\alpha }-i)}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{\int }_{{\mathcal{B}}_m}|\nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}F\left(t\right)|dt·{\mathcal{X}}_n\left(\xi \right)\hfill \\ \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel {\mathcal{X}}_m\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}{\mathcal{X}}_n\left(\xi \right).\hfill \end{array}$$

Consequently, upon taking the $\left(L^{l_2(·)}\left({\omega }^{l_2(·)}\right)\right)—$norm with respect to the variable $\left(\xi \right)$, together with the preceding pointwise estimate, we arrive at

$$\begin{array}{cc}& \parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}F·{\mathcal{X}}_n{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

(12)

Since

$$I_{\mathrm{\alpha }}\left({\mathcal{X}}_{{\mathcal{B}}_n}\right)\left(\xi \right)\ge C{2}^{n\mathrm{\alpha }}{\mathcal{X}}_{{\mathcal{B}}_n}\left(\xi \right),$$

and, equivalently,

$${\mathcal{X}}_{{\mathcal{B}}_n}\left(\xi \right)\le C{2}^{-n\mathrm{\alpha }}{I}_{\mathrm{\alpha }}\left({\mathcal{X}}_{{\mathcal{B}}_n}\right)\left(\xi \right).$$

Consequently, by taking the $\left(L^{l_2(·)}\left({\omega }^{l_2(·)}\right)\right)—$norm on both sides and subsequently employing the boundedness of $\left(I_{\mathrm{\alpha }}\right)$, we deduce that

$$\begin{array}{c}\\ \hfill \parallel {\mathcal{X}}_{{\mathcal{B}}_n}{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\le C{2}^{-n\mathrm{\alpha }}{\parallel I_{\mathrm{\alpha }}{\mathcal{X}}_{{\mathcal{B}}_n}\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\end{array}$$

By virtue of Lemmas 11 and 7, together with the corresponding lower and upper bounds for the fractional integral of the characteristic function of the dyadic ball, one obtains

$$\begin{array}{c}\hfill \parallel {\mathcal{X}}_{{\mathcal{B}}_n}{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\le C{2}^{-n\mathrm{\alpha }}{\displaystyle \frac{\parallel {\mathcal{X}}_{{\mathcal{B}}_n}{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}}{\parallel {\mathcal{X}}_{{\mathcal{B}}_n}{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}}}{\parallel {\mathcal{X}}_{{\mathcal{B}}_n}\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}\\ \hfill \le C{2}^{n(i-\mathrm{\alpha })}{\parallel {\mathcal{X}}_{{\mathcal{B}}_n}\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}^{-1}.\end{array}$$

(13)

Substituting the estimate in (13) into (12), and collecting the resulting dyadic factors, we consequently arrive at

$$\begin{array}{cc}& \parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}F·{\mathcal{X}}_n{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{m(i-\mathrm{\alpha })}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^{-1}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

In view of the assertions established in Lemma 5 $\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^{-1}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}=2^{(n-m){\partial }_1}$, the preceding estimate can be further refined to yield

$$\begin{array}{cc}& \parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}F·{\mathcal{X}}_n{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{m(i-\mathrm{\alpha })}{2}^{(n-m){\partial }_1}\hfill \\ & \le C{\displaystyle \sum _{m=-\infty }^n}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}.\hfill \end{array}$$

We now proceed to estimate the Herz–Morrey norm of $\left(H_{\mathrm{\alpha }}F\right)$. By Lemma 13, the norm under consideration admits the decomposition

$$\begin{array}{cc}\hfill \parallel H_{\mathrm{\alpha }}{F\left(x\right)\parallel }_{M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )}& =\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{H}_{\mathrm{\alpha }}{F}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & +\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{H}_{\mathrm{\alpha }}{F}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =A_1+A_2.\hfill \end{array}$$

We first estimate $\left(A_1\right)$. The convergence of the series appearing in the above estimates follows from the restrictions imposed on the parameters. Indeed, for the terms corresponding to the inner dyadic annuli, we obtain the factor

$$2^{k({\omega }^{+}{\alpha }^{+}+\alpha -i+{\partial }_1)}.$$

Set

$${\gamma }_1={\omega }^{+}{\alpha }^{+}+\alpha -i+{\partial }_1.$$

By the assumption

$${\omega }^{+}{\alpha }^{+}+\alpha -i+{\partial }_1<0,$$

we have ${\gamma }_1<0$. Hence,

$${\displaystyle \sum _{k=-\infty }^0}{2}^{k{\gamma }_1}={\displaystyle \sum _{m=0}^{\infty }}{2}^{-m{\gamma }_1}<\infty .$$

Consequently,

$${\displaystyle \sum _{k=-\infty }^0}{2}^{k({\omega }^{+}{\alpha }^{+}+\alpha -i+{\partial }_1)}\le C.$$

Thus, geometric series are convergent precisely under the stated conditions.

Hence, applying the preceding estimate and the discrete Hardy-type inequality, we obtain

$$\begin{array}{cc}\hfill A_1& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\left({\displaystyle \sum _{m=-\infty }^n}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{-1}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le {\parallel F\parallel }_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )}.\hfill \end{array}$$

We next turn our attention to the estimation of $\left(A_2\right)$. By applying the previously established pointwise estimate and subsequently separating the summation with respect to the dyadic index (m) into the ranges $(m\le -1)$ and $(m\ge 0),$ we obtain

$$\begin{array}{cc}\hfill A_2& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{k_0}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{H}_{\mathrm{\alpha }}{F}_m\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=-\infty }^n}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=-\infty }^{-1}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & +\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=0}^n}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =A_{1,2}+A_{2,2}.\hfill \end{array}$$

We first estimate $\left(A_{1,2}\right)$. Since ${\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1<0,$ the associated geometric series is convergent, and hence, by an application of the discrete Hardy inequality, we obtain

$$\begin{array}{cc}\hfill A_{1,2}& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=-\infty }^{-1}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{k_0}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le {\parallel F\parallel }_{M{\dot{K}}_{l_1{(}^{.}),p}^{\beta {(}^{.}),\lambda }(\nu ,\omega )}.\hfill \end{array}$$

To estimate $\left(A_{2,2}\right)$, the same decay condition ${\omega }^{+}{\alpha }^{+}+\alpha -i+{\partial }_1<0$ ensures the convergence of the corresponding dyadic series. Indeed,

$$\begin{array}{cc}\hfill A_{22}& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=-\infty }^n}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{k_0}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le {\parallel F\parallel }_{M{\dot{K}}_{l_1{(}^{.}),p}^{\beta {(}^{.}),\lambda }(\nu ,\omega )}.\hfill \end{array}$$

The second inequality follows from the convergence of the geometric series associated with ${\omega }^{+}{\alpha }^{+}+\alpha -i+{\partial }_1<0.$ By combining the estimates obtained for $\left(A_1\right)$, $\left(A_{1,2}\right)$, and $\left(A_{2,2}\right)$, we ultimately obtain

$$\begin{array}{cc}\hfill \parallel H_{\mathrm{\alpha }}{F\left(x\right)\parallel }_{M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )}& \le {C\parallel F\parallel }_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )}\hfill \end{array}$$

□

C denotes a generic constant that is independent of F and all dyadic indices.

Theorem 2. 

Let $i=1,2,\dots ,\mathfrak{n}$, $0<\alpha <{\displaystyle \frac{i}{l^{+}}}$, $\lambda \in [0,\infty )$, $\beta (·)\in L^{\infty }\left({\mathbb{R}}^i\right)$, $1<p<\infty$, and let $l_2(·)\in LH\left({\mathbb{R}}^i\right)\cap \mathcal{P}\left({\mathbb{R}}^i\right)$. Assume that the variable exponent $l_1(·)$ is determined by the relation

$${\displaystyle \frac{1}{l_2(·)}}={\displaystyle \frac{1}{l_1(·)}}-{\displaystyle \frac{\alpha }{i}}.$$

Suppose, furthermore, that $\nu \in A_r$ for some $r\in [1,\infty )$, $\omega \in A_{l_2(·)}$, and that

$${\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2>0,$$

where ${\partial }_2\in (0,1)$ denotes the constant introduced in Lemma 5 corresponding to the exponent function $l(·)$. In addition, assume that the exponent function $\beta (·)$ satisfies the log-Hölder continuity condition both at the origin and at infinity. Then the adjoint fractional Hardy operator $H_{\alpha }^{\ast }$ extends to a bounded linear operator from the two-weight variable exponent Morrey–Herz space

$$M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )$$

into

$$M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega ).$$

More explicitly, there exists a positive constant C, independent of the function F, such that

$$\parallel H_{\alpha }^{\ast }{F\parallel }_{M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )}\le C{\parallel F\parallel }_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )}.$$

Proof of Theorem 2. 

By virtue of Lemma 12, we have

$$\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}\le C\nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{2}^{(n-m){\omega }^{-}{\beta }^{-}}.$$

Upon decomposing the domain of integration into the corresponding dyadic annuli, we obtain the following pointwise estimate:

$$\begin{array}{cc}& |\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }F\left(\xi \right)·{\mathcal{X}}_n|\le \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\int }_{{\mathbb{R}}^i\setminus {\mathcal{B}}_n}|F\left(\eta \right){\left|\right|\eta |}^{\mathrm{\alpha }-i}d\eta ·{\mathcal{X}}_n\left(\xi \right)\hfill \\ & \le {\int }_{{\mathbb{R}}^i\setminus {\mathcal{B}}_n}|\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}F\left(\eta \right){\left|\right|\eta |}^{\mathrm{\alpha }-i}d\eta ·{\mathcal{X}}_n\left(\xi \right)\hfill \\ & \le {\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{\int }_{{\mathbb{R}}^i\setminus {\mathcal{B}}_n}|\nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}F\left(\eta \right){\left|\right|\eta |}^{\mathrm{\alpha }-i}d\eta ·{\mathcal{X}}_n\left(\xi \right)\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel {\mathcal{X}}_m\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}·{\mathcal{X}}_n\left(\xi \right).\hfill \end{array}$$

Taking the $\left(L^{l_2(·)}\left({\omega }^{l_2(·)}\right)\right)—$norm with respect to $\left(\xi \right),$ and subsequently applying the generalized Hölder inequality in the variable exponent setting, gives

$$\begin{array}{cc}& \parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }F·{\mathcal{X}}_n{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

(14)

Employing inequality (13) to control the characteristic-function factors appearing in (14), we consequently deduce

$$\begin{array}{cc}& \parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }F·{\mathcal{X}}_n{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}{2}^{n(i-\mathrm{\alpha })}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}^{-1}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{(m-n)(\mathrm{\alpha }-i)}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{{\partial }_2(m-n)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m)({\omega }^{-}{\mathrm{\alpha }}^{-}-{\partial }_2-\mathrm{\alpha }+i)}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}.\hfill \end{array}$$

The $(M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega ))—$norm of the adjoint fractional Hardy operator $\left(H_{\mathrm{\alpha }}F\right)$ can be decomposed into two complementary contributions, namely

$$\begin{array}{cc}\hfill \parallel H_{\mathrm{\alpha }}^{\ast }{F\left(x\right)\parallel }_{M{\dot{K}}_{l_2{(}^{.}),p}^{\beta {(}^{.}),\lambda }(\nu ,\omega )}& =\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }{F}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & +\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }{F}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =B_1+B_2.\hfill \end{array}$$

We first estimate the contribution $\left(B_1\right)$. The convergence of the series appearing in the above estimates follows from the restrictions imposed on the parameters.

For the terms corresponding to the outer dyadic annuli, we obtain

$$2^{-k({\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2)}.$$

Putting

$${\gamma }_2={\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2,$$

the assumption

$${\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2>0$$

gives ${\gamma }_2>0$. Therefore,

$${\displaystyle \sum _{m=0}^{\infty }}{2}^{-m{\gamma }_2}={\displaystyle \frac{1}{1-2^{-{\gamma }_2}}}<\infty ,$$

and hence

$${\displaystyle \sum _{m=0}^{\infty }}{2}^{-k({\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2)}\le C.$$

Thus, the geometric series are convergent precisely under the stated conditions.

Therefore, by applying the preceding estimate for $\left(H_{\mathrm{\alpha }}^{\ast }F\right)$ and invoking the corresponding discrete Hardy-type inequality, we obtain

$$\begin{array}{cc}\hfill B_1& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\left({\displaystyle \sum _{m=n+1}^{\infty }}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{-}{\mathrm{\alpha }}^{-}-\mathrm{\alpha }+i-{\partial }_2)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{-1}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le {\parallel F\parallel }_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )}.\hfill \end{array}$$

Hence, the negative-index contribution $\left(B_1\right)$ is uniformly controlled by the $(M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega ))—$ norm of $\left(F\right)$, with the constant (C > 0) independent of $\left(F\right)$ and the dyadic indices.

We next proceed to estimate the second contribution $\left(B_2\right)$. By applying the estimate previously established for $\left(H_{\mathrm{\alpha }}^{\ast }F\right)$ and subsequently decomposing the resulting dyadic summation according to the sign of the index $\left(m\right)$, we obtain

$$\begin{array}{cc}\hfill B_2& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{k_0}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{H}_{\mathrm{\alpha }}{F}_m\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\left({\displaystyle \sum _{m=n+1}^{\infty }}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{-}{\mathrm{\alpha }}^{-}-\mathrm{\alpha }+i-{\partial }_2)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\left({\displaystyle \sum _{m=n+1}^{-1}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{-}{\mathrm{\alpha }}^{-}-\mathrm{\alpha }+i-{\partial }_2)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & +\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\left({\displaystyle \sum _{m=0}^{\infty }}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{-}{\mathrm{\alpha }}^{-}-\mathrm{\alpha }+i-{\partial }_2)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =B_{1,2}+B_{2,2}.\hfill \end{array}$$

Here, the decomposition $(B_2=B_{1,2}+B_{2,2})$ separates the contributions corresponding to the negative and nonnegative dyadic components of $\left(F\right)$. This separation permits the two terms to be treated independently by exploiting the geometric decay implied by ${\omega }^{-}{\mathrm{\alpha }}^{-}-\mathrm{\alpha }+i-{\partial }_2>0.$

We first estimate $\left(B_{1,2}\right)$. In view of the structural condition ${\omega }^{-}{\mathrm{\alpha }}^{-}-\mathrm{\alpha }+i-{\partial }_2>0,$ the dyadic factor appearing in the inner summation exhibits sufficient geometric decay. Consequently, an application of the discrete Hardy inequality ${\left({\sum }_{n=-\infty }^{K_0}{\left({\sum }_{m=n+1}^{\infty }{2}^{-\gamma (m-n)}{a}_m\right)}^p\right)}^{1/p}\le C{\left({\sum }_{m=-\infty }^{K_0}{a}_m^p\right)}^{1/p},$ yields

$$\begin{array}{cc}\hfill B_{1,2}& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=n+1}^{-1}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{-}{\mathrm{\alpha }}^{-}-\mathrm{\alpha }+i-{\partial }_2)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{k_0}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le {\parallel F\parallel }_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )}.\hfill \end{array}$$

The remaining term $\left(B_{2,2}\right)$ can be estimated in an entirely analogous manner to $\left(B_1\right)$, since the same positivity condition ${\omega }^{-}{\mathrm{\alpha }}^{-}-\mathrm{\alpha }+i-{\partial }_2>0$ ensures the convergence of the corresponding dyadic series. Therefore, combining the estimates obtained for $\left(B_1\right),$$\left(B_{1,2}\right),$ and $\left(B_{2,2}\right)$, we arrive at

$$\begin{array}{cc}\hfill \parallel H_{\mathrm{\alpha }}^{\ast }{F\left(x\right)\parallel }_{M{\dot{K}}_{l_2{(}^{.}),p}^{\beta {(}^{.}),\lambda }(\nu ,\omega )}& \le {C\parallel F\parallel }_{M{\dot{K}}_{l_1{(}^{.}),p}^{\beta {(}^{.}),\lambda }(\nu ,\omega )}.\hfill \end{array}$$

Thus, the adjoint fractional Hardy operator $\left(H_{\mathrm{\alpha }}^{\ast }\right)$ is bounded from $(M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega ))$ into $(M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )),$ which establishes the desired estimate and completes the proof. □

4. Commutators of Fractional Hardy Operators

Theorem 3. 

Let $i=1,2,\dots ,\mathfrak{n}$, $0<\alpha <{\displaystyle \frac{i}{l^{+}}}$, $\lambda \in [0,\infty )$, $\beta (·)\in L^{\infty }\left({\mathbb{R}}^i\right)$, and $1<p<\infty$. Suppose that

$$l_2(·)\in LH\left({\mathbb{R}}^i\right)\cap \mathcal{P}\left({\mathbb{R}}^i\right),$$

and that the variable exponents $l_1(·)$ and $l_2(·)$ satisfy

$${\displaystyle \frac{1}{l_2(·)}}={\displaystyle \frac{1}{l_1(·)}}-{\displaystyle \frac{\alpha }{i}}.$$

Let $\nu \in A_r$ for some $r\in [1,\infty )$, $\omega \in A_{l_2(·)}$, and $b\in BMO\left({\mathbb{R}}^i\right)$. Assume that

$${\omega }^{+}{\alpha }^{+}+\alpha -i+{\partial }_1<0,$$

where ${\partial }_1\in (0,1)$ is the constant associated with the exponent function $l(·)$ as given in Lemma 5. Furthermore, suppose that $\beta (·)$ satisfies the log-Hölder continuity conditions both at the origin and at infinity. Then the commutator operator $[b,H_{\alpha }]$ is bounded from the two-weight variable exponent Morrey–Herz space

$$M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )$$

into

$$M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega ).$$

More precisely, for every

$$F\in M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega ),$$

there exists a constant $C>0$, independent of F, such that

$${∥[b,H_{\alpha }]F∥}_{M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )}\le C{\parallel b\parallel }_{BMO}{∥F∥}_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )}.$$

Consequently,

$$[b,H_{\alpha }]:M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )⟶M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )$$

is a bounded linear operator.

Proof of Theorem 3. 

Consider the commutator induced by the symbol b associated with the fractional Hardy operator. For each $\xi \in {\mathbb{R}}^i$, by inserting and subtracting the mean value $b_{\mathcal{B}m}$, we obtain

$$\begin{array}{cc}\hfill |[b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)|& \le \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\int }_{\mathcal{B}(0,|\xi \left|\right)}\left|F\left(\eta \right)(b\left(\xi \right)-b\left(\eta \right))\right|d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & \le \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\int }_{\mathcal{B}(0,|\xi \left|\right)}\left|F\left(\eta \right)(b\left(\xi \right)-b_{{\mathcal{B}}_m})\right|d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & +\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\int }_{\mathcal{B}(0,|\xi \left|\right)}\left|F\left(\eta \right)(b\left(\eta \right)-b_{2^n\mathcal{B}}+b_{2^n\mathcal{B}}-b_{{\mathcal{B}}_m})\right|d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & =X_1+X_2.\hfill \end{array}$$

We first turn our attention to the estimation of the term $X_1$. To this end, we shall exploit the structural properties of the symbol b, together with the appropriate integral estimates and the relevant norm inequalities, in order to derive a suitable upper bound for $X_1$.

$$\begin{array}{cc}\hfill X_1& =\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\int }_{\mathcal{B}(0,|\xi \left|\right)}\left|F\left(t\right)(b\left(\xi \right)-b_{{\mathcal{B}}_m})\right|dt·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & \le \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\displaystyle \sum _{m=-\infty }^n}{\int }_{{\mathcal{B}}_m}\left|F\left(t\right)\right|dt·(b\left(\xi \right)-b_{{\mathcal{B}}_m}){\mathcal{X}}_n\left(\xi \right)\hfill \\ & \le 2^{n(\mathrm{\alpha }-i)}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{\int }_{{\mathcal{B}}_m}|\nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}F\left(t\right)|dt·(b\left(\xi \right)-b_{{\mathcal{B}}_m}){\mathcal{X}}_n\left(\xi \right)\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel {\mathcal{X}}_m\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}(b\left(\xi \right)-b_{{\mathcal{B}}_m})·{\mathcal{X}}_n\left(\xi \right).\hfill \end{array}$$

Consequently, by applying the generalized Hölder inequality in the corresponding weighted variable exponent Lebesgue spaces, one obtains

$$\begin{array}{cc}& \parallel X_1{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}{\parallel (b\left(\xi \right)-b_{{\mathcal{B}}_m})·{\mathcal{X}}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

(15)

Invoking Lemma 14 (${∥(b-b_{{\mathcal{B}}_m}){\chi }_{{\mathcal{B}}_n}∥}_{L^{l(·)}}\le {C(n-m)\parallel b\parallel }_{BMO}{\parallel {\chi }_{{\mathcal{B}}_n}\parallel }_{L^{l(·)}}$), which provides the requisite control of the oscillation of the symbol b in terms of its $BMO$ norm, the preceding estimate yields

$$\begin{array}{cc}& \parallel X_1{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}(n-m)\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel {\mathcal{X}}_m\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}\hfill \\ & {\parallel b\parallel }_{BMO}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

(16)

Furthermore, employing the relation

$$\begin{array}{c}\\ \hfill \parallel {\mathcal{X}}_{{\mathcal{B}}_n}{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\le C{2}^{n(i-\mathrm{\alpha })}{\parallel {\mathcal{X}}_{{\mathcal{B}}_n}\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}^{-1}.\end{array}$$

(17)

Thus, the contribution of $X_1$ is reduced to a dyadic summation involving the oscillation factor $(n-m)$, the scaling factor $2^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}$, and the corresponding weighted variable exponent norms. This formulation is particularly convenient for subsequently invoking the assumed relations among the exponents and establishing the convergence of the resulting series.

$$\begin{array}{cc}& \parallel X_1{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{(n-m)\parallel b\parallel }_{BMO}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\hfill \\ & 2^{m(i-\mathrm{\alpha })}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^{-1}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

Invoking the assertions of Lemma 5, we readily obtain the following estimate:

$$\begin{array}{cc}& \parallel X_1{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{(n-m)\parallel b\parallel }_{BMO}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{m(i-\mathrm{\alpha })}{2}^{(n-m){\partial }_1}\hfill \\ & \le C{\displaystyle \sum _{m=-\infty }^n}{(n-m)\parallel b\parallel }_{BMO}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}.\hfill \end{array}$$

We next turn our attention to the estimation of $X_2$. By decomposing the ball $\mathcal{B}(0,|\xi \left|\right)$ into the dyadic annuli $\mathcal{B}m$, we have

$$\begin{array}{cc}& X_2=\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\int }_{\mathcal{B}(0,|\xi \left|\right)}\left|F\left(\eta \right)(b\left(\eta \right)-b_{{\mathcal{B}}_m})\right|d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & \le \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \sum _{m=-\infty }^0}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\displaystyle \sum _{n=-\infty }^m}{\int }_{2^n\mathcal{B}\setminus 2^{n-1}\mathcal{B}}\left|F\left(\eta \right)(b\left(\eta \right)-b_{2^n\mathcal{B}}+b_{2^n\mathcal{B}}-b_{{\mathcal{B}}_m})\right|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & =\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \sum _{m=-\infty }^0}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\displaystyle \sum _{n=-\infty }^m}{\int }_{2^n\mathcal{B}\setminus 2^{n-1}\mathcal{B}}\left|F\left(\eta \right)(b\left(\eta \right)-b_{2^n\mathcal{B}})\right|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & +\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \sum _{m=-\infty }^0}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\displaystyle \sum _{n=-\infty }^m}{\int }_{2^n\mathcal{B}\setminus 2^{n-1}\mathcal{B}}\left|F\left(\eta \right)(b_{2^n\mathcal{B}}-b_{{\mathcal{B}}_m})\right|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & =Y_{11}+Y_{22}.\hfill \end{array}$$

$$\begin{array}{cc}& Y_{11}=\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \sum _{m=-\infty }^0}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\displaystyle \sum _{n=-\infty }^m}{\int }_{2^n\mathcal{B}\setminus 2^{n-1}\mathcal{B}}\left|F\left(\eta \right)(b\left(\eta \right)-b_{2^n\mathcal{B}})\right|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^0}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & {\displaystyle \sum _{n=-\infty }^m}\parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_n{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel (b\left(\eta \right)-b_{{\mathcal{B}}_n}){\mathcal{X}}_n\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^0}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & {\displaystyle \sum _{n=-\infty }^m}\parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_n{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel b\parallel }_{BMO}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}.\hfill \end{array}$$

Consequently, upon taking the $L^{l_2(·)}\left({\omega }^{l_2(·)}\right)$-norm on both sides and invoking the generalized Hölder inequality in the corresponding weighted variable exponent spaces, we obtain

$$\begin{array}{cc}\hfill \parallel Y_{11}{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\le & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^0}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{\parallel {\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & {\displaystyle \sum _{n=-\infty }^m}\parallel \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_n{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel b\parallel }_{BMO}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}.\hfill \end{array}$$

Furthermore, by employing the relation

$$\begin{array}{c}\\ \hfill \parallel {\mathcal{X}}_{{\mathcal{B}}_n}{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\le C{2}^{n(i-\mathrm{\alpha })}{\parallel {\mathcal{X}}_{{\mathcal{B}}_n}\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}^{-1}\end{array}$$

(18)

and substituting this estimate into the preceding inequality, it follows that

$$\begin{array}{cc}& \parallel Y_{11}{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{\parallel b\parallel }_{BMO}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\hfill \\ & 2^{m(i-\mathrm{\alpha })}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^{-1}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

Hence, the contribution arising from $X_2$ is reduced to a dyadic summation whose convergence can be subsequently established by exploiting the prescribed relationships among the exponents and the decay properties encoded in the factor $2^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}$. This estimate constitutes the requisite intermediate bound for completing the analysis of the commutator.

Invoking the assertions of Lemma 5, together with the corresponding estimates established above, we readily deduce the following bound:

$$\begin{array}{cc}& \parallel Y_{11}{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{2}^{-n(i-\mathrm{\alpha })}{\displaystyle \sum _{m=-\infty }^n}{2}^{(n-m){\omega }^{+}{\mathrm{\alpha }}^{+}}{\parallel b\parallel }_{BMO}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{m(i-\mathrm{\alpha })}{2}^{(n-m){\partial }_1}\hfill \\ & \le C{\displaystyle \sum _{m=-\infty }^n}{\parallel b\parallel }_{BMO}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}.\hfill \end{array}$$

Now, to estime $Y_{22}$, we use the standard telescoping estimate for BMO averages,

$$|b_{B_n}-b_{B_m}{|\le C|n-m|\parallel b\parallel }_{BMO},$$

$$\begin{array}{cc}& Y_{22}=\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\displaystyle \sum _{m=-\infty }^0}{\displaystyle \frac{1}{{\left|\xi \right|}^{i-\mathrm{\alpha }}}}{\displaystyle \sum _{n=-\infty }^m}{\int }_{2^n\mathcal{B}\setminus 2^{n-1}\mathcal{B}}\left|F\left(\eta \right)(b_{2^n\mathcal{B}}-b_{{\mathcal{B}}_m})\right|d\eta ·{\mathcal{X}}_{2^m\mathcal{B}\setminus 2^{m-1}\mathcal{B}}\left(\xi \right)\hfill \\ & \le C(n-m){\displaystyle \sum _{m=-\infty }^n}{\parallel b\parallel }_{BMO}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}.\hfill \end{array}$$

Combining the estimates obtained for $X_1$ and $X_2$, and recalling the decomposition of the commutator into these two constituent terms, we consequently arrive at

$$\begin{array}{cc}& \parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right){\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{\displaystyle \sum _{m=-\infty }^n}{(n-m)\parallel b\parallel }_{BMO}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}.\hfill \end{array}$$

Therefore, the preceding estimates collectively establish the desired dyadic bound for the commutator. In particular, the factor $(n-m)$ arising from the oscillatory behavior of the $BMO$ symbol is accompanied by the geometric factor $2^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)},$ whose decay, under the prescribed assumptions on the parameters, guarantees the convergence of the resulting series and thereby completes the essential estimate required for the boundedness argument.

$$\begin{array}{cc}& \parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right){\parallel }_{M{\dot{K}}_{l_2{(}^{.}),p}^{\beta {(}^{.}),\lambda }(\nu ,\omega )}\hfill \\ & =\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & +\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =X_{11}+X_{22}.\hfill \end{array}$$

We first estimate the term $X_{11}$. Since, by assumption, ${\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1<0,$ the resulting geometric series is convergent. Consequently, using the estimate established above, we obtain

$$\begin{array}{cc}\hfill X_{11}& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\left({\displaystyle \sum _{m=-\infty }^n}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}(n-m){\parallel b\parallel }_{BMO}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{-1}}{\parallel b\parallel }_{BMO}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le {\parallel b\parallel }_{BMO}{\parallel F\parallel }_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )}.\hfill \end{array}$$

We now proceed to estimate the remaining contribution $X_{22}$. By applying the estimate established above and subsequently decomposing the inner summation according to the ranges $m\le -1$ and $0\le m\le n$, we obtain

$$\begin{array}{cc}\hfill X_{22}& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{k_0}}{\parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=-\infty }^n}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}(n-m){\parallel b\parallel }_{BMO}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=-\infty }^{-1}}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}(n-m){\parallel b\parallel }_{BMO}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & +\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=0}^n}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}(n-m){\parallel b\parallel }_{BMO}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =X_{11}^{\prime }+X_{22}^{\prime }.\hfill \end{array}$$

We first consider the term $X_{11}^{\prime }$. Since ${\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1<0,$ the associated dyadic factor exhibits sufficient decay, and hence the corresponding geometric series is convergent. Therefore, by an application of the discrete Hardy inequality ${\left({\sum }_{n=-\infty }^{K_0}{\left({\sum }_{m=n+1}^{\infty }{2}^{-\gamma (m-n)}{a}_m\right)}^p\right)}^{1/p}\le C{\left({\sum }_{m=-\infty }^{K_0}{a}_m^p\right)}^{1/p}$, we obtain

$$\begin{array}{cc}\hfill X_{11}^{\prime }& \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\left({\displaystyle \sum _{m=-\infty }^{-1}}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}(n-m){\parallel b\parallel }_{BMO}{2}^{(n-m)({\omega }^{+}{\mathrm{\alpha }}^{+}+\mathrm{\alpha }-i+{\partial }_1)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le \underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{m=-\infty }^{k_0}}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(t\right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}{\parallel b\parallel }_{BMO}\hfill \\ & \le {\parallel F\parallel }_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )}{\parallel b\parallel }_{BMO}.\hfill \end{array}$$

The estimation of $X_{22}^{\prime }$ proceeds analogously to that of $X_1$, since the same combination of weighted variable exponent estimates and dyadic summability arguments applies in this case. Accordingly, by combining the estimates established for $X_1$ and $X_2$, together with the preceding bounds for $X_{11}^{\prime }$ and $X_{22}^{\prime }$, we ultimately arrive at the following inequality:

$$\begin{array}{cc}\hfill \parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right){\parallel }_{M{\dot{K}}_{l_2{(}^{.}),p}^{\beta {(}^{.}),\lambda }(\nu ,\omega )}& \le {C\parallel b\parallel }_{BMO}{\parallel F\parallel }_{M{\dot{K}}_{l_1{(}^{.}),p}^{\beta {(}^{.}),\lambda }(\nu ,\omega )}.\hfill \end{array}$$

Thus, the commutator generated by the symbol b and the fractional Hardy operator admits the desired boundedness estimate from $M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )$ into $M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )$, with the operator norm controlled linearly by ${\parallel b\parallel }_{BMO}$. □

Theorem 4. 

Let $i=1,2,\dots ,\mathfrak{n}$, $0<\alpha <{\displaystyle \frac{i}{l^{+}}}$, $\lambda \in [0,\infty )$, $\beta (·)\in L^{\infty }\left({\mathbb{R}}^i\right)$, and $1<p<\infty$. Assume that

$$l_2(·)\in LH\left({\mathbb{R}}^i\right)\cap \mathcal{P}\left({\mathbb{R}}^i\right),$$

and that the variable exponents $l_1(·)$ and $l_2(·)$ satisfy the fractional scaling relation

$${\displaystyle \frac{1}{l_2(·)}}={\displaystyle \frac{1}{l_1(·)}}-{\displaystyle \frac{\alpha }{i}}.$$

Let $\nu \in A_r$ for some $r\in [1,\infty )$, $\omega \in A_{l_2(·)}$, and $b\in BMO\left({\mathbb{R}}^i\right)$. Suppose, moreover, that

$${\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2>0,$$

where ${\partial }_2\in (0,1)$ denotes the constant associated with the exponent function $l(·)$ as specified in Lemma 5. In addition, assume that the variable smoothness function $\beta (·)$ satisfies the prescribed log-Hölder continuity conditions both at the origin and at infinity.

Under the aforementioned assumptions, the commutator associated with the adjoint fractional Hardy operator,

$$[b,H_{\alpha }^{\ast }]F=b{H}_{\alpha }^{\ast }F-H_{\alpha }^{\ast }\left(bF\right),$$

admits a bounded extension from the two-weight variable exponent Morrey–Herz space

$$M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )$$

into

$$M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega ).$$

More precisely, for every

$$F\in M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega ),$$

there exists a constant $C>0$, independent of F, such that

$${∥[b,H_{\alpha }^{\ast }]F∥}_{M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )}\le C{\parallel b\parallel }_{BMO}{∥F∥}_{M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )}.$$

Consequently, the commutator operator

$$[b,H_{\alpha }^{\ast }]:M{\dot{K}}_{l_1(·),p}^{\beta (·),\lambda }(\nu ,\omega )⟶M{\dot{K}}_{l_2(·),p}^{\beta (·),\lambda }(\nu ,\omega )$$

is a bounded linear operator, with its operator norm controlled by a constant multiple of the $BMO$-norm of the symbol b.

C denotes a generic constant that is independent of F, b and all dyadic indices.

Proof of Theorem 4. 

For the commutator associated with the adjoint fractional Hardy operator, we proceed in an analogous manner. For every $\xi \in {\mathbb{R}}^i$, we have

$$\begin{array}{cc}\hfill |[b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)|& \le \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\int }_{\mathcal{B}{(0,|\xi \left|\right)}^c}{\displaystyle \frac{\left|F\right(\eta \left)\right(b\left(\xi \right)-b\left(\eta \right))}{{\left|\eta \right|}^{i-\mathrm{\alpha }}}}d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & \le \nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\int }_{\mathcal{B}{(0,|\xi \left|\right)}^c}{\displaystyle \frac{\left|F\left(\eta \right)\right(b\left(\xi \right)-b_{{\mathcal{B}}_m}\left)\right|}{{\left|\eta \right|}^{i-\mathrm{\alpha }}}}d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & +\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\int }_{\mathcal{B}{(0,|\xi \left|\right)}^c}{\displaystyle \frac{\left|F\left(\eta \right)\right(b\left(\eta \right)-b_{{\mathcal{B}}_m}\left)\right|}{{\left|\eta \right|}^{i-\mathrm{\alpha }}}}d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & =Y_1+Y_2.\hfill \end{array}$$

We first estimate the contribution $Y_1$. By decomposing the exterior region $\mathcal{B}{(0,|\xi \left|\right)}^c$ into the dyadic annular regions $\mathcal{B}m$ for $m\ge n+1$ and applying Lemma 12, together with the appropriate estimates for the variable exponent $\beta (·)$, we obtain

$$\begin{array}{cc}\hfill Y_1& =\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\int }_{\mathcal{B}{(0,|\xi \left|\right)}^c}{\displaystyle \frac{\left|F\left(\eta \right)\right(b\left(\xi \right)-b_{{\mathcal{B}}_m}\left)\right|}{{\left|\eta \right|}^{i-\mathrm{\alpha }}}}d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & \le {\int }_{{\mathbb{R}}^i\setminus {\mathcal{B}}_n}|\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}F\left(\eta \right)(b\left(\xi \right)-b_{{\mathcal{B}}_m}){\left|\right|\eta |}^{\mathrm{\alpha }-i}d\eta ·{\mathcal{X}}_n\left(\xi \right)\hfill \\ & \le {\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{\int }_{{\mathbb{R}}^i\setminus {\mathcal{B}}_n}|\nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}F\left(\eta \right)(b\left(\xi \right)-b_{{\mathcal{B}}_m}){\left|\right|\eta |}^{\mathrm{\alpha }-i}d\eta ·{\mathcal{X}}_n\left(\xi \right)\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel {\mathcal{X}}_m\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}(b\left(\xi \right)-b_{{\mathcal{B}}_m})·{\mathcal{X}}_n\left(\xi \right).\hfill \end{array}$$

Taking the $L^{l_2(·)}\left({\omega }^{l_2(·)}\right)$-norm and employing the generalized Hölder inequality in the corresponding weighted variable exponent spaces, we consequently arrive at

$$\begin{array}{cc}& \parallel Y_1{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}\parallel b\left(\xi \right)-b_{{\mathcal{B}}_m}{)·{\mathcal{X}}_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}(n-m){\parallel b\parallel }_{BMO}{\parallel X_n\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

(19)

In the final step, we have invoked the standard $BMO$ oscillation estimate to control the difference $b\left(\xi \right)-b_{{\mathcal{B}}_m}$. The factor $(n-m)$ naturally emerges from the comparison of the averages of the symbol b over the corresponding dyadic balls. This estimate provides the essential decay structure required for the subsequent summation over $m>n$.

Substituting inequality (18) into (19) and subsequently invoking the relevant estimates for the characteristic functions of the dyadic balls, we obtain

$$\begin{array}{cc}& \parallel Y_1{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel {\mathcal{X}}_m\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}\hfill \\ & 2^{n(i-\mathrm{\alpha })}{(n-m)\parallel b\parallel }_{BMO}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}^{-1}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{(m-n)(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}(n-m){\parallel b\parallel }_{BMO}{2}^{{\partial }_2(m-n)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m)({\omega }^{-}{\mathrm{\alpha }}^{-}-{\partial }_2-\mathrm{\alpha }+i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}(n-m){\parallel b\parallel }_{BMO}.\hfill \end{array}$$

We next turn to the estimation of the second component $Y_2$. By decomposing the exterior region $\mathcal{B}{(0,|\xi \left|\right)}^c$ into the dyadic annular regions and applying the corresponding estimates for the variable exponent, we obtain

$$\begin{array}{cc}& Y_2=\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{\int }_{\mathcal{B}{(0,|\xi \left|\right)}^c}{\displaystyle \frac{\left|F\left(\eta \right)\right(b\left(\eta \right)-b_{{\mathcal{B}}_m}\left)\right|}{{\left|\eta \right|}^{i-\mathrm{\alpha }}}}d\eta ·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\hfill \\ & \le {\int }_{{\mathbb{R}}^i\setminus {\mathcal{B}}_n}|\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}F\left(\eta \right)(b\left(\eta \right)-b_{{\mathcal{B}}_m}){\left|\right|\eta |}^{\mathrm{\alpha }-i}d\eta ·{\mathcal{X}}_n\left(\xi \right)\hfill \\ & \le {\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{\int }_{{\mathbb{R}}^i\setminus {\mathcal{B}}_n}|\nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}F\left(\eta \right)(b\left(\eta \right)-b_{{\mathcal{B}}_m}){\left|\right|\eta |}^{\mathrm{\alpha }-i}d\eta ·{\mathcal{X}}_n\left(\xi \right)\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel (b\left(\eta \right)-b_{{\mathcal{B}}_m})·{\mathcal{X}}_m\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}{\mathcal{X}}_n\left(\xi \right).\hfill \end{array}$$

Taking the $L^{l_2(·)}\left({\omega }^{l_2(·)}\right)$-norm and employing the generalized Hölder inequality, we consequently have

$$\begin{array}{cc}\hfill \parallel Y_2{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}& \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\hfill \\ & \parallel (b\left(\eta \right)-b_{{\mathcal{B}}_m})·{\mathcal{X}}_m{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}{\parallel {\mathcal{X}}_n\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

By Lemma 14, the local oscillation of the symbol b is controlled by its $BMO$ norm. Hence,

$$\begin{array}{cc}\hfill \parallel Y_2{\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}& \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\hfill \\ & {\parallel b\parallel }_{BMO}\parallel {\mathcal{X}}_m{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}{\parallel {\mathcal{X}}_n\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{m(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel b\parallel }_{BMO}{\parallel {\mathcal{X}}_m\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}\hfill \\ & 2^{n(i-\mathrm{\alpha })}{\parallel {\mathcal{X}}_n\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}^{-1}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}{2}^{(m-n)(\mathrm{\alpha }-i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel b\parallel }_{BMO}{2}^{{\partial }_2(m-n)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m)({\omega }^{-}{\mathrm{\alpha }}^{-}-{\partial }_2-\mathrm{\alpha }+i)}\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m{\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}{\parallel b\parallel }_{BMO}.\hfill \end{array}$$

We get the following terms for those corresponding to the outer dyadic annuli:

$$2^{-k({\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2)}.$$

Putting

$${\gamma }_2={\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2,$$

the assumption

$${\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2>0$$

gives ${\gamma }_2>0$. Therefore,

$${\displaystyle \sum _{m=0}^{\infty }}{2}^{-m{\gamma }_2}={\displaystyle \frac{1}{1-2^{-{\gamma }_2}}}<\infty ,$$

and hence

$${\displaystyle \sum _{m=0}^{\infty }}{2}^{-k({\omega }^{-}{\alpha }^{-}-\alpha +i-{\partial }_2)}\le C.$$

Hence, the geometric series are convergent only when mentioned.

Thus, both $Y_1$ and $Y_2$ are governed by the same dyadic decay factor, $2^{(n-m)({\omega }^{-}{\mathrm{\alpha }}^{-}-{\partial }_2-\mathrm{\alpha }+i)},$ which, under the prescribed assumptions on the parameters, provides the requisite summability over the range $m\ge n+1$. This establishes the fundamental estimates necessary for controlling the commutator associated with the adjoint fractional Hardy operator.

Combining the estimates obtained for $Y_1$ and $Y_2$, and recalling the decomposition of the commutator into these two constituent terms, we consequently obtain

$$\begin{array}{cc}& \parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right){\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}\hfill \\ & \le C{\displaystyle \sum _{m=n+1}^{\infty }}{2}^{(n-m){\omega }^{-}{\mathrm{\alpha }}^{-}}((n-m)2^{m(\mathrm{\alpha }-i)}{\parallel \nu {\left({\mathcal{B}}_m\right)}^{{\displaystyle \frac{\beta \left(\eta \right)}{i}}}{F}_m\parallel }_{L^{l_1(·)}\left({\omega }^{l_1(·)}\right)}\hfill \\ & \parallel (b\left(\eta \right)-b_{{\mathcal{B}}_m})·{\mathcal{X}}_m{\parallel }_{L^{l_1^{\prime }(·)}\left({\omega }^{-l_1^{\prime }(·)}\right)}{\parallel {\mathcal{X}}_n\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}.\hfill \end{array}$$

We now proceed to estimate the resulting expression in the weighted variable exponent Morrey–Herz framework. By Lemma 13, together with the definition of the corresponding quasi-norm, we have

$$\begin{array}{cc}& \parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right){\parallel }_{M{\dot{K}}_{l_2{(}^{.}),p}^{\beta {(}^{.}),\lambda }(\nu ,\omega )}\hfill \\ & =\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{-1}}{\parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & +\underset{K_0\in \mathbb{Z}}{\sup }{2}^{-k_0\lambda }{\left({\displaystyle \sum _{n=-\infty }^{k_0}}{\parallel [b,\nu {\left({\mathcal{B}}_n\right)}^{{\displaystyle \frac{\beta \left(\xi \right)}{i}}}{H}_{\mathrm{\alpha }}^{\ast }]F\left(\xi \right)·{\mathcal{X}}_{\mathcal{B}}\left(\xi \right)\parallel }_{L^{l_2(·)}\left({\omega }^{l_2(·)}\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =Y_{11}+Y_{22}.\hfill \end{array}$$

The estimates for $Y_{11}$ and $Y_{22}$ can be established by following, mutatis mutandis, the same sequence of arguments employed in the treatment of $X_{11}$ and $X_{22}$ in Theorem 3. More precisely, by invoking the corresponding dyadic decomposition, the weighted variable exponent Hölder inequality, the $BMO$ oscillation estimate, and the requisite summability conditions on the resulting geometric series, one obtains analogous bounds for both contributions.

Consequently, the arguments developed in the proof of Theorem 3 can be transferred directly to the present setting, with the appropriate modifications corresponding to the adjoint fractional Hardy operator $H_{\mathrm{\alpha }}^{\ast }$. This completes the required estimation of the commutator in the weighted variable exponent Morrey–Herz space. □

As a special case, by taking $l_1(·)=q_1,l_2(·)=q_2,\beta (·)=\beta ,$ and setting $\nu =\omega =1,$ we recover the following well-known boundedness result established in the literature: $\parallel H_{\alpha }{F\parallel }_{M{\dot{K}}_{q_2,p}^{\beta ,\lambda }}\le C{\parallel F\parallel }_{M{\dot{K}}_{q_1,p}^{\beta ,\lambda }}.$ Thus, our result extends this known constant-exponent unweighted boundedness result to the more general setting of variable exponents and weights.

5. Conclusions

In the present study, we have conducted a comprehensive and rigorous analysis of the boundedness properties of Hardy operators and their associated commutators within the intricate framework of two-weighted Morrey–Herz Spaces with variable exponents. The analysis is developed under the assumption that the corresponding symbols belong to the class of functions of bounded mean oscillation $\left(BMO\right)$, thereby providing the requisite control over their local oscillatory behavior.

Under a suitable collection of structural and admissibility assumptions, we establish a series of boundedness results that elucidate the intricate interaction between the operators under consideration and the underlying variable exponent functional framework. The conclusions obtained not only extend and reinforce several existing results but also contribute to a broader understanding of the behavior of Hardy-type operators in nonstandard functional settings.

The techniques and analytical principles developed herein are expected to provide a useful foundation for further investigations into related operator-theoretic problems. In particular, they may serve as a starting point for extending the present theory to other variable exponent function spaces and to functional frameworks characterized by more general forms of nonstandard growth and weighted structures.