Analytical Investigations into Multilinear Fractional Rough Hardy Operators Within Morrey–Herz Spaces Characterized by Variable Exponents

Abstract

In this scholarly discourse, a rigorous examination is conducted on the boundedness properties of multilinear fractional rough Hardy operators within the structural framework of variable exponent Morrey–Herz spaces. Furthermore, analogous quantitative estimates are meticulously derived for their corresponding commutators, contingent upon the assumption that the governing symbol functions belong to the space of bounded mean oscillation $(\mathcal{BMO})$ with variable exponents.

1. Introduction

Amidst the continual advancement of theoretical paradigms in partial differential equations, such as nonlinear analysis and other interrelated mathematical disciplines, modern harmonic analysis, distinguished by its distinctive methodological framework and profound conceptual underpinnings, has emerged as an indispensable instrument for the intricate taxonomy and hierarchical decomposition of function spaces. Consequently, such analytical refinement has precipitated the methodical evolution of a wide-ranging corpus of specialized function space frameworks, comprising—albeit not exhaustively—Hardy spaces, Herz spaces, and Morrey spaces, each scrupulously formulated to embody delicate structural intricacies and subtle regularity attributes inherent to functions. The genesis of the theoretical edifice governing variable exponent $L^p$ spaces can be traced back to the pioneering exposition of Orlicz in 1931, as documented in [1], wherein the foundational constructs of this analytical framework were initially conceived and subjected to preliminary scrutiny. However, the substantive maturation and widespread academic recognition of the variable exponent function space paradigm did not materialize until the advent of 1991, catalyzed by the seminal contributions of Kováčik and Rákosník (see [2]), whose groundbreaking insights precipitated an era of intensified scholarly engagement and rigorous mathematical inquiry into the structural and functional intricacies of such spaces. In [2], the authors undertook an ambitious broadening of the classical Sobolev and Lebesgue spaces, thereby advancing their formulation into the domain of variable exponent Sobolev and Lebesgue spaces. Within this conceptual architecture, they provided a meticulous exposition and rigorous validation of the essential structural characteristics inherent to such variable exponent function spaces. Moreover, the theoretical advancements in this domain have engendered far-reaching applications, particularly in the analytical treatment of fluid mechanics and differential equations governed by non-standard growth conditions (cf. [3]), wherein conventional analytical methodologies prove inadequate. This intellectual impetus has engendered an extensive corpus of scholarly inquiry, wherein a multitude of researchers have expended significant effort in the meticulous examination of variable exponent spaces, thereby precipitating an extensive proliferation of theoretical advancements. Concomitantly, this investigative trajectory has catalyzed the formulation and rigorous scrutiny of an array of further generalizations within the ambit of variable exponent function spaces, encompassing, though not exhaustively, variable Besov spaces, variable Triebel–Lizorkin spaces, variable Hardy spaces, and variable Herz spaces (cf. [4,5,6]).

In contemporary mathematical discourse, substantial advancements have been realized in the rigorous establishment of boundedness criteria for a plethora of fundamental operators within the expansive framework of harmonic analysis on variable exponent function spaces. Notably, the scholarly endeavors delineated in [4,5,6,7,8] have meticulously scrutinized the boundedness of an array of integral operators on these variable function spaces, leveraging the nuanced structural properties intrinsic to variable $L^p$ spaces as a principal analytical apparatus. Conversely, since the groundbreaking contributions of Torchinsky (see [9]), the theoretical evolution of Morrey spaces has progressively asserted itself as a predominant paradigm within the realm of contemporary harmonic analysis. A multitude of scholars have engaged in meticulous investigations into the intrinsic structural and functional attributes of central Morrey spaces and central $\mathcal{BMO}$ spaces (refer to [6,10]), thereby effectuating a substantial refinement and augmentation of the theoretical framework governing operator boundedness within central Morrey spaces. Inspired by the burgeoning advancements in variable exponent function spaces, Fu et al., in their seminal work [11], undertook a profound generalization of classical central Morrey spaces by transcending the confines of the constant exponent paradigm and extending their applicability to the domain of variable exponents in the year 2019. They pioneered the conceptual edifice of central Morrey spaces and central $\mathcal{BMO}$ spaces endowed with variable exponents, wherein they meticulously derived comprehensive boundedness criteria for singular integral operators alongside their commutators. This scholarly endeavor engendered a profound advancement in the theoretical maturation and analytical profundity of central Morrey spaces. Given the foundational principles and inherent structural intricacies of variable function spaces, an inevitable and intellectually compelling trajectory emerges—namely, the rigorous exploration of operator boundedness within this generalized framework. During the interval spanning 2019 to 2022, a multitude of scholars undertook a meticulous and rigorous analytical investigation into the boundedness properties of diverse classes of operators, along with their associated commutators, within the theoretical construct of variable central Morrey spaces. For instance, Wang et al., in [12,13], embarked upon an exhaustive and profound investigation into the boundedness characteristics of multilinear singular integral operators and multilinear fractional integral operators, thereby extending and augmenting the foundational analysis delineated in [11]. Furthermore, in the year 2022, Hussain et al., in [14], undertook a rigorous examination of the boundedness properties of the Hardy operator within the structural framework of variable central Morrey spaces, thereby contributing substantively to the ongoing theoretical evolution of this domain.

In 1920, Hardy first introduced and elaborated upon the notion of the one-dimensional Hardy operator, as comprehensively presented in [15]. In the ensuing decades, an increasing contingent of researchers embarked upon the rigorous investigation and progressive refinement of its definition, concomitantly exploring an array of generalized formulations of Hardy-type operators. Notably, in the year 1995, Christ and Grafakos, in their seminal treatise [16], effectuated a profound extension of the Hardy operator paradigm from the one-dimensional framework to its higher-dimensional counterpart, rigorously ascertaining its boundedness within the theoretical construct of $L^p$ spaces. Subsequently, Fu et al., in [17], further propelled the generalization of Hardy operators by pioneering the introduction of n-dimensional fractional Hardy operators. Additionally, they undertook an intricate analytical exposition, wherein they established the boundedness properties of the commutators concomitant with these operators within the structural contexts of Lebesgue spaces and homogeneous Herz spaces. Subsequently, the boundedness properties of Hardy operators emerged as a focal point of profound scholarly inquiry, catalyzing a wave of rigorous investigations into their structural and functional characteristics. This intellectual pursuit has precipitated a wealth of intricate analyses, culminating in a more comprehensive understanding of these operators within diverse mathematical frameworks. In particular, Fu et al., in [18], embarked upon an extensive and meticulous exploration of the boundedness properties of n-dimensional rough Hardy operators and their corresponding commutators, thereby making a significant and substantive contribution to the theoretical edifice of this domain.

Moreover, in a complementary trajectory of research, Hussain et al., in [14], derived a series of precise and rigorous estimates concerning fractional Hardy operators and their associated commutators within the analytical framework of variable ƛ-central Morrey spaces. This endeavor not only expanded the theoretical purview of Hardy-type operators but also provided critical insights into their boundedness properties within the nuanced context of variable exponent function spaces, thereby reinforcing and deepening the prevailing mathematical discourse in this arena. These two seminal scholarly contributions constitute a profound wellspring of intellectual stimulation, serving as a catalyst for further mathematical inquiry. Given the burgeoning influence of variable exponent function spaces in the advancement of diverse scientific disciplines—including, but not limited to, physics, mathematics, and information science—the rigorous investigation into the boundedness of multilinear operators within this analytical framework has, in recent years, ascended to a position of paramount significance in contemporary research. This burgeoning field has not only enriched the structural understanding of function spaces but has also engendered novel methodologies for addressing complex mathematical phenomena. In consonance with this intellectual trajectory, the present treatise aspires to achieve a substantive breakthrough by meticulously scrutinizing the boundedness properties of n-dimensional multilinear fractional rough Hardy operators and their associated commutators within the theoretical construct of Morrey-Herz spaces endowed with variable exponents. Building upon the foundational analysis of the boundedness properties of multilinear fractional rough Hardy operators, this study extends its scope to a rigorous investigation of the boundedness of adjoint multilinear fractional rough Hardy operators and their corresponding commutators. By systematically exploring these operators within the framework of variable exponent function spaces, this research seeks to deepen the understanding of their structural and functional behavior.

Furthermore, a comprehensive comparative assessment is conducted, critically evaluating the methodologies and results associated with the boundedness of multilinear fractional rough Hardy operators. Through this analytical juxtaposition, the study derives significant conclusions regarding the broader class of multilinear operators, offering valuable insights into their theoretical properties and potential applications. In this treatise, the multilinear rough Hardy operator and its associated commutator are formally introduced and rigorously defined for the first time as fundamental constructs instrumental in the derivation of our principal results. Their precise formulation is articulated as follows.

Let ${\xi }_1$ and ${\xi }_2$ be locally integrable functions defined on ${\mathbb{R}}^n$, and let the parameter $\mathcal{B}$ be constrained by the condition $0\le \mathcal{B}<mn$. The n-dimensional multilinear fractional rough Hardy operator, in conjunction with its adjoint counterpart, is formally defined as follows:

$$H_{\mathcal{B},\Omega }({\xi }_1,\dots ,{\xi }_m)={\displaystyle \frac{1}{{|t|}^{nm-\mathcal{B}}}}{\int }_{|({\tau }_1,\dots ,{\tau }_m)|<|t|}\prod _{j=1}^m{\xi }_j({\tau }_j)\Omega (x-{\tau }_j)d{\tau }_j.$$

$$H_{\mathcal{B},\Omega }^{*}({\xi }_1,\dots ,{\xi }_m)={\int }_{|({\tau }_1,\dots ,{\tau }_m)|>|t|}{\displaystyle \frac{1}{{|\tau |}^{nm-\mathcal{B}}}}\prod _{j=1}^m{\xi }_j({\tau }_j)\Omega (x-{\tau }_j)d{\tau }_i.$$

Furthermore, the commutators corresponding to the n-dimensional multilinear fractional rough Hardy operator and its adjoint counterpart are rigorously formulated as

$$[b,H_{\mathcal{B},\Omega }]({\xi }_1,\dots ,{\xi }_m)(\tau )=\sum _{j=1}^m[b_j,H_{\mathcal{B},\Omega }^j]({\xi }_1,\dots ,{\xi }_m)(\tau )$$

$$[b,H_{\mathcal{B},\Omega }^{*}]({\xi }_1,\dots ,{\xi }_m)(\tau )=\sum _{j=1}^m[b_j,H_{\mathcal{B},\Omega }^{*j}]({\xi }_1,\dots ,{\xi }_m)(\tau )$$

$$\begin{array}{cc}& [b_j,H_{\mathcal{B},\Omega }^j]({\xi }_1,\dots ,{\xi }_m)(\tau )=b_j(\tau )H_{\mathcal{B},\Omega }({\xi }_1,\dots ,{\xi }_m)(\tau )-H_{\mathcal{B},\Omega }({\xi }_1,\dots ,{\xi }_{j-1},{\xi }_j{b}_j,{\xi }_{j+1},\dots ,{\xi }_m)(\tau )·\hfill \end{array}$$

$$\begin{array}{cc}& [b_j,H_{\mathcal{B},\Omega }^{*j}]({\xi }_1,\dots ,{\xi }_m)(\tau )=b_j(\tau )H_{\mathcal{B},\Omega }^{*}({\xi }_1,\dots ,{\xi }_m)(\tau )-H_{\mathcal{B},\Omega }^{*}({\xi }_1,\dots ,{\xi }_{j-1},{\xi }_j{b}_j,{\xi }_{j+1},\dots ,{\xi }_m)(\tau )·\hfill \end{array}$$

Subsequently, we shall undertake a meticulous exposition of the structural composition of this treatise. In Section 2, we shall commence by furnishing a succinct yet rigorous recapitulation of certain fundamental notational conventions and pivotal lemmas that underpin the theoretical edifice of variable Lebesgue spaces. Concomitantly, we shall introduce the formal definitions of central bounded mean oscillation $\mathcal{BMO}$ spaces and Morrey–Herz spaces, both distinguished by their intrinsic dependence on variable exponents.

Thereafter, in Section 3, we shall rigorously establish the boundedness properties of the n-dimensional bilinear fractional Hardy operator and its adjoint operator when acting upon Morrey-Herz spaces endowed with variable exponents. Furthermore, we shall conduct an exhaustive examination of the boundedness properties of the commutators associated with the n-dimensional bilinear fractional Hardy operator and its adjoint operator within the analytical framework of Morrey–Herz spaces characterized by variable exponents.

To rigorously govern the continuity constraints requisite for the m-linear fractional rough Hardy operator, we shall invoke the boundedness properties of the fractional integral, which is formally delineated by the expression

$$I_{\mathcal{B}}(h)(\xi )={\mathit{\int }}_{{\mathbb{R}}^n}{\displaystyle \frac{h(\xi )}{{|\eta -\xi |}^{n-\mathcal{B}}}}d\xi .$$

Theorem 1.

Let $0<q_i,q<\infty ,$ where the function $q(·)$ resides within the intersection $\mathcal{P}\cap {\mathcal{Q}}^{log}$, and let $p(·)$ be such that the fundamental relation

$${\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\mathcal{B}}{n}}={\displaystyle \frac{1}{p(·)}}$$

holds, with the further decomposition

$${\displaystyle \frac{1}{p(·)}}=\sum _{i=1}^m{\displaystyle \frac{1}{p_i(·)}}.$$

Additionally, assume that the parameter ƛ satisfies the additive condition

$$\mathsf{ƛ}=\sum _{i=1}^m{\mathsf{ƛ}}_i,$$

and that the function $\mathfrak{a}(·)$, belonging to the space ${\mathcal{Q}}^{log}\cap L^{\infty }$, exhibits logarithmic Hölder continuity at the origin. Specifically, it satisfies the summation constraints

$$\mathfrak{a}(0)=\sum _{i=1}^m{\mathfrak{a}}_i(0),\mathfrak{a}(\infty )=\sum _{i=1}^m{\mathfrak{a}}_i(\infty ),$$

where the bounds

$$\mathfrak{a}(0)\le \mathfrak{a}(\infty )<n{\sigma }_{ii}+\mathsf{ƛ}-{\displaystyle \frac{n}{S}},$$

hold, with ${\sigma }_{ii}\in (0,1)$ being a sequence of constants emerging from Equation (3). Under these conditions, the following inequality holds:

$$\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m){\parallel }_{M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}\le C\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.$$

Theorem 2.

Let the parameters $q,q_i(·),p(·),p_i(·),\mathfrak{a}(·)$, and $\mathcal{B}$ preserve the identical definitions as those delineated in Theorem 1. Furthermore, assume that the inequality

$$\mathfrak{a}(\infty )\ge \mathfrak{a}(0)>\mathsf{ƛ}-n\sigma -{\displaystyle \frac{n}{S}}$$

is satisfied, wherein the parameter $\sigma \in (0,1)$ denotes the constant introduced in Lemma 3. Under these stipulations, the ensuing bound holds:

$$\parallel H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m){\parallel }_{M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}\le C\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.$$

Theorem 3.

Let $0<q_i,q<\infty$ for $i=1,2,\dots ,m$, where the function exponent $q(·)$ belongs to the intersection $\mathcal{P}({\mathbb{R}}^n)\cap {\mathcal{Q}}^{log}({\mathbb{R}}^n)$, and let $p(·)$ satisfy the structural relation

$${\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\mathcal{B}}{n}}={\displaystyle \frac{1}{p(·)}},with{\displaystyle \frac{1}{p(·)}}=\sum _{i=1}^m{\displaystyle \frac{1}{p_i(·)}}.$$

Define the parameter ƛ as

$$\mathsf{ƛ}=\sum _{i=1}^m{\mathsf{ƛ}}_i,$$

and assume that the function $\mathfrak{a}(·)$, which resides in the class ${\mathcal{Q}}^{log}({\mathbb{R}}^n)\cap L^{\infty }({\mathbb{R}}^n)$, exhibits logarithmic Hölder continuity at the origin. Suppose further that $\mathfrak{a}(0)$ satisfies the equalities

$$\mathfrak{a}(0)=\sum _{i=1}^m{\mathfrak{a}}_i(0),\mathfrak{a}(\infty )=\sum _{i=1}^m{\mathfrak{a}}_i(\infty ),$$

alongside the constraint

$$\mathfrak{a}(0)\le \mathfrak{a}(\infty )<\sum _{i=1}^{m}n{\sigma }_{ii}+\mathsf{ƛ}+{\displaystyle \frac{n}{S}},$$

where ${\sigma }_{ii}\in (0,1)$ are structural constants arising in (3). Under the aforementioned conditions, the commutator operator $[b,H_{\mathcal{B},\Omega }]$ is well-defined and exhibits boundedness in the following sense:

$$[b,H_{\mathcal{B},\Omega }]:\prod _{i=1}^{m}M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)\to M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n).$$

Here, the function tuple $b=(b_1,b_2,\dots ,b_m)$ is composed of elements $b_1,b_2,\dots ,b_m$ belonging to the space of bounded mean oscillations, $\mathcal{BMO}$.

Theorem 4.

Assume that the parameters $q,q_i,p(·),p_i(·),\mathfrak{a}(·)$, and $\mathcal{B}$ adhere to the same definitions and structural constraints as those delineated in the preceding Theorem 1. Furthermore, suppose that the function $\mathfrak{a}(·)$ satisfies the inequality

$$\mathfrak{a}(\infty )\ge \mathfrak{a}(0)>\mathsf{ƛ}-n\sigma -{\displaystyle \frac{n}{S}},$$

where $\sigma \in (0,1)$ is the constant emerging in Lemma 3. Under these premises, the commutator operator associated with the adjoint fractional Hardy operator, denoted by $[b,H_{\mathcal{B},\Omega }^{*}]$, is rigorously bounded from the product space ${\prod }_{i=1}^{m}M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)$ into the Herz–Morrey space $M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n).$

2. Established Nomenclature and Rigorous Formal Definitions

An extensive repertoire of terminological constructs and symbolic representations has been assimilated and operationalized within the conceptual framework of the present manuscript.

(i)

The notation C is ubiquitously utilized throughout the present treatise to signify a constant, the specific magnitude of which is subject to variation across distinct occurrences.

(ii)

The function spaces $L_{loc}^1({\mathbb{R}}^n)$, $L^{\infty }({\mathbb{R}}^n)$, and $L_{loc}^{p(·)}({\mathbb{R}}^n)$ are rigorously delineated in the work of Grafakos [19].

(iii)

Let S be an arbitrary nonempty measurable subset of ${\mathbb{R}}^n$.

(iv)

The notation ${\mathcal{X}}_S$ is employed to designate the characteristic function associated with the set S, where $|S|$ represents its corresponding Lebesgue measure.

(v)

For any integer $\ell \in \mathbb{Z}$, we define ${\mathcal{X}}_{\ell }={\mathcal{X}}_{A_{\ell }},$ where the sets $O_{\ell }$ and $A_{\ell }$ are given by

$$O_{\ell }=\{y\in {\mathbb{R}}^n:|y|\le 2^{\ell }\},A_{\ell }=O_{\ell }\setminus O_{\ell -1}=\{y\in {\mathbb{R}}^n:2^{\ell -1}<|y|<2^{\ell }\}.$$

Definition 1.

Let $q(·):{\mathbb{R}}^n\to [1,\infty ]$ be a measurable function. The variable exponent Lebesgue space, formally denoted as $L^{q(·)}({\mathbb{R}}^n)$, is defined as the collection of all measurable functions ς for which the integral functional ${\mathcal{F}}_q(\varsigma )$, explicitly formulated as

$${\mathcal{F}}_q(\varsigma )={\int }_{{\mathbb{R}}^n}{\left(|\varsigma (\xi )|\right)}^{q(\xi )}d\mu ,$$

is constrained to a finite magnitude. The function space $L^{q(·)}({\mathbb{R}}^n)$ is intrinsically furnished with the norm delineated below, which endows it with a complete normed structure, thereby ensuring its Banach space property:

$${\parallel \varsigma \parallel }_{L^{q(·)}}=inf\left\{\delta >0:{\mathcal{F}}_q({\displaystyle \frac{\varsigma }{\delta }})={\int }_{{\mathbb{R}}^n}{\left({\displaystyle \frac{|\varsigma (\xi )|}{\delta }}\right)}^{q(\xi )}d\xi \le 1\right\}.$$

We introduce the notation $\mathcal{P}({\mathbb{R}}^n)$ to signify the totality of all measurable functions $q(·):{\mathbb{R}}^n\to (1,\infty )$ that satisfy the stringent boundedness condition

$$1<q^{-}\le q(\xi )\le q^{+}<\infty ,$$

where the critical lower and upper bounds, denoted respectively by $q^{-}$ and $q^{+}$, are defined through the essential infimum and supremum operators as follows:

$$q^{-}:=\underset{\xi \in {\mathbb{R}}^n}{essinf}q(\xi ),q^{+}:=\underset{\xi \in {\mathbb{R}}^n}{esssup}q(\xi ).$$

In the subsequent analysis, we shall invoke certain well-established structural constraints, prominently recognized as the logarithmic condition and the decay conditions.

The notation ${\mathcal{Q}}_{\mathrm{loc}}^{log}({\mathbb{R}}^n)$ is employed to designate the class of all functions $q(·)$ that exhibit local logarithmic Hölder continuity [20], thereby adhering to the following asymptotic constraint:

$$|q(\xi )-q(\mu )|\lesssim {\displaystyle \frac{-C}{log(|\xi -\mu |)}},|\mu -\xi |<{\displaystyle \frac{1}{2}},\xi ,\mu \in {\mathbb{R}}^n.$$

If $q(·)\in {\mathcal{Q}}_0^{log}({\mathbb{R}}^n),$ then it adheres to the asymptotic decay constraint at the origin, which is rigorously expressed as

$$|q(\xi )-q(0)|\lesssim {\displaystyle \frac{C}{log(|\frac{1}{|\xi |}+e|)}},\xi \in {\mathbb{R}}^n.$$

If $q(·)\in {\mathcal{Q}}_{\infty }^{log}({\mathbb{R}}^n)$, then it satisfies the prescribed decay condition at infinity, specifically for $2<|\xi |$, wherein the following asymptotic bound holds:

$$|q(\xi )-q_{\infty }|\le {\displaystyle \frac{C_{\infty }}{log(|\xi |+e)}}.$$

The notation ${\mathcal{Q}}^{log}={\mathcal{Q}}_{\mathrm{loc}}^{log}\bigcap {\mathcal{Q}}_{\infty }^{log}$ is employed to designate the collection of all functions $q(·)$ that exhibit global logarithmic Hölder continuity.

We introduce the notation $\mathcal{D}({\mathbb{R}}^n)$ to denote the class of functions $q(·)$ that belong to the intersection $\mathcal{P}({\mathbb{R}}^n)\cap {\mathcal{Q}}^{log}({\mathbb{R}}^n)$ and additionally satisfy the fundamental criterion that the Hardy–Littlewood maximal operator exhibits boundedness on the variable exponent Lebesgue space $L^{\mathrm{q}(·)}({\mathbb{R}}^n)$. It has been rigorously established in [21] that whenever $q(·)\in \mathcal{P}({\mathbb{R}}^n)\cap {\mathcal{Q}}^{log}({\mathbb{R}}^n)$, the maximal operator $\mathcal{M}$ is indeed bounded on $L^{q(·)}({\mathbb{R}}^n).$

Definition 2

([19]). Let $b\in L_{loc}^1.$ The functional norm associated with the space of functions of bounded mean oscillation is rigorously defined as

$${\parallel b\parallel }_{\mathcal{BMO}({\mathbb{R}}^n)}=\underset{O}{sup}{\displaystyle \frac{1}{|O|}}{\int }_O|b(\mu )-b_O|d\mu ,$$

where the supremum is taken over all balls $O\subset {\mathbb{R}}^n$, and the mean value of b over O is given by

$$b_O={\displaystyle \frac{1}{|O|}}{\int }_{O}f(y)dy.$$

A function b is said to possess bounded mean oscillation if and only if ${\parallel b\parallel }_{\mathcal{BMO}({\mathbb{R}}^n)}<\infty .$ The space $\mathcal{BMO}({\mathbb{R}}^n)$ is thus defined as the set of all functions $b\in L_{loc}^1({\mathbb{R}}^n)$ for which the aforementioned norm remains finite.

Definition 3

([22]). For given parameters $0<q<\infty ,$ $0\le \mathsf{ƛ}<\infty ,$ and a function $\mathfrak{a}(·):{\mathbb{R}}^n\to \mathbb{R}$ satisfying $\mathfrak{a}(·)\in L^{\infty }({\mathbb{R}}^n),$ along with an exponent function $p(·)\in \mathcal{P}({\mathbb{R}}^n)$, we rigorously define the space $M{\dot{K}}_{q,p(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)$ as follows:

$$M{\dot{K}}_{q,p(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)=\left\{f\in L_{loc}^{p(·)}({\mathbb{R}}^n\setminus \left\{0\right\}):{\parallel f\parallel }_{M{\dot{K}}_{q,p(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}<\infty \right\},$$

where the associated quasi-norm is expressed as

$${\parallel f\parallel }_{M{\dot{K}}_{q,p(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}=\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(·)q}{\parallel f{\mathcal{X}}_k\parallel }_{L^{{}^p(·)}({\mathbb{R}}^n)}^q\right)}^{1/q}.$$

3. Fundamental Theorems and Principal Findings

Within this section, we shall embark upon the rigorous substantiation of the fundamental results articulated in the introductory segment. However, prior to delving into these proofs, it is essential to establish a collection of auxiliary lemmas, which will serve as indispensable analytical tools in this pursuit.

Lemma 1

([23]). Let $\mathcal{F}$ be a Banach function space. Then the following fundamental properties hold:

(1) 

The corresponding dual space ${\mathcal{F}}^{\prime }$ necessarily constitutes a Banach function space.

(2) 

The norms ${\parallel ·\parallel }_{\mathcal{F}}$ and ${\parallel ·\parallel }_{{({\mathcal{F}}^{\prime })}^{\prime }}$ exhibit equivalence, ensuring structural consistency between $\mathcal{F}$ and its bidual.

(3) 

(generalized Hölder inequality) If $g_1\in {\mathcal{F}}^{\prime }$ and $g_2\in \mathcal{F}$, then the integral bound

$${\int }_{{\mathbb{R}}^n}|g_1{g}_2|\le \parallel g_1{\parallel }_{{\mathcal{F}}^{\prime }}{\parallel g_2\parallel }_{\mathcal{F}}$$

holds universally, encapsulating a crucial duality relationship in the functional framework.

Lemma 2

([11]). If the function $q(·)$ is an element of the class $\mathcal{P}({\mathbb{R}}^n)$, then for any arbitrary ball $O\subset {\mathbb{R}}^n$, there exists a positive constant $C>0$ such that the following two-sided inequality holds:

$$C^{-1}<{|O|}^{-1}\parallel {\mathcal{X}}_O{\parallel }_{L^{q^{\prime }(·)}}{\parallel {\mathcal{X}}_O\parallel }_{L^{q(·)}}<C.$$

This establishes a fundamental norm equivalence within the variable exponent framework, ensuring the controlled interplay between dual norms over characteristic functions of measurable subsets.

Lemma 3

([24]). If $\mathcal{F}$ constitutes a Banach function space and the Hardy–Littlewood maximal operator $\mathcal{M}$ exhibits boundedness on the dual space ${\mathcal{F}}^{\prime },$ then for any measurable set $O\subset {\mathbb{R}}^n$ and any subset $S\subset O,$ there exists a constant $\sigma \in (0,1),$ such that the following decay estimate holds:

$${\displaystyle \frac{\parallel {\mathcal{X}}_S{\parallel }_{\mathcal{F}}}{\parallel {\mathcal{X}}_O{\parallel }_{\mathcal{F}}}}\lesssim {\left({\displaystyle \frac{|S|}{|O|}}\right)}^{\sigma }.$$

This inequality signifies a crucial embedding property within the functional framework, demonstrating a power-type control over the relative norm ratios of characteristic functions corresponding to nested measurable sets.

Proposition 1

([25]). Consider an open set E, and assume that $p(·)\in \mathcal{P}(E)$ fulfills the following conditions:

$$|p(\gamma )-p(\xi )|\le {\displaystyle \frac{-c}{log(|\gamma -\xi |)}},{\displaystyle \frac{1}{2}}\ge |\gamma -\xi |$$

(1)

$$|p(\gamma )-p(\xi )|\le {\displaystyle \frac{-c}{log(|\gamma |+e)}},|\gamma |\le |\xi |$$

(2)

then it follows that $p(·)\in \mathcal{D}({\mathbb{R}}^n),$ where C denotes a positive constant that does not depend on γ and ξ.

Lemma 4

([26]). If $q(·)\in \mathcal{D}({\mathbb{R}}^n)$, then there exists a constant $0<\sigma <1$ and a positive constant C such that for any ball B in ${\mathbb{R}}^n$ and any measurable subset $A\subset B$, the subsequent inequalities are satisfied:

$${\displaystyle \frac{\parallel {\mathcal{X}}_A{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}}{\parallel {\mathcal{X}}_B{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}}}\le C{\left({\displaystyle \frac{|A|}{|B|}}\right)}^{\sigma }.$$

Remark 1.

Assume $q(·)\in \mathcal{P}({\mathbb{R}}^n)$ and fulfills the conditions (1) and (2) as articulated in Proposition 1. Consequently, $q^{\prime }(·)$ adheres to these conditions as well, indicating that both $q(·)$ and $q^{\prime }(·)$ are members of $\mathcal{D}({\mathbb{R}}^n)$. By invoking Lemma 4 and utilizing [27], we derive the existence of constants ${\sigma }_{ii}\in (0,{\displaystyle \frac{1}{(q_i)+}})$ such that the inequalities

$${\displaystyle \frac{\parallel {\mathcal{X}}_A{\parallel }_{L^{q_i^{\prime }(·)}({\mathbb{R}}^n)}}{\parallel {\mathcal{X}}_B{\parallel }_{L^{{q^{\prime }}_i(·)}({\mathbb{R}}^n)}}}\le C{\left({\displaystyle \frac{|A|}{|B|}}\right)}^{{\sigma }_{ii}}$$

(3)

are satisfied for all balls $B\subset {\mathbb{R}}^n$ and for all subsets $A\subset B$.

Lemma 5

([28]). Let $q(·)$ be an element of the class $\mathcal{P}({\mathbb{R}}^n)$, where the parameter $\mathcal{B}$ satisfies the constraint

$$0<\mathcal{B}<{\displaystyle \frac{n}{{(q_1)}^{+}}}.$$

Define the conjugate exponent function $q^{\prime }(·)$ via the relation

$${\displaystyle \frac{1}{q^{\prime }(·)}}+{\displaystyle \frac{\mathcal{B}}{n}}={\displaystyle \frac{1}{q(·)}}.$$

Then, for any ball $O_k=\{x\in {\mathbb{R}}^n:|x|\le 2^k\}$ with $k\in \mathbb{Z}$, the following norm inequality holds:

$$\parallel {\mathcal{X}}_{O_k}{\parallel }_{L^{q^{\prime }}(·)}\le C{2}^{-k\mathcal{B}}{\parallel {\mathcal{X}}_{O_k}\parallel }_{L^{q(·)}}.$$

This inequality encapsulates a crucial scaling property in the context of variable exponent Lebesgue spaces, revealing the interplay between weighted norm estimates and the underlying geometric structure of the domain.

Lemma 6

([28]). Suppose that $q(·)$ belongs to the class $\mathcal{P}({\mathbb{R}}^n)$. Then, for any function $b\in \mathcal{BMO}$ and for all integers $l,i\in \mathbb{Z}$ with $l>i,$ the following fundamental norm inequalities hold:

$$C^{-1}{\parallel b\parallel }_{\mathcal{BMO}}\le \underset{0:Ball}{sup}{\displaystyle \frac{1}{\parallel {\mathcal{X}}_0{\parallel }_{L^{q(·)}}}}\parallel (b-b_0){\mathcal{X}}_B{\parallel }_{L^{q(·)}}\le C{\parallel b\parallel }_{\mathcal{BMO}}$$

$$\parallel (b-b_{O_i}){\mathcal{X}}_{O_l}{\parallel }_{L^{q(·)}}\le C(l-i){\parallel b\parallel }_{\mathcal{BMO}}{\parallel {\mathcal{X}}_{0_l}\parallel }_{L^{q(·)}}.$$

Here, C denotes a positive absolute constant, independent of the indices l and i.

Proposition 2

([29]). If the function $\mathfrak{a}(·)$ is an element of the intersection $L^{\infty }$ and ${\mathcal{Q}}^{log},$ and suppose that $p(·)$ belongs to the class $\mathcal{P},$ while the parameters satisfy $0<q<\infty$ and $0\le \mathsf{ƛ}<\infty ,$ then the norm of the operator within the space $M{\dot{K}}_{{}^q,{}^p{(}^{·})}^{\mathfrak{a}(·),\mathsf{ƛ}}$ satisfies the following equivalence relation:

$$\begin{array}{cc}& \parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k{\parallel }_{M{\dot{K}}_{{}^q,{}^p(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}}^q\hfill \\ & =\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\mathsf{ƛ}q}\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(·)q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{{}^p(·)}}^q\hfill \\ & \approx max\{{\displaystyle \underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}}{2}^{-k_0\mathsf{ƛ}q}\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(0)q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{{}^p(·)}}^q,\hfill \\ & \underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}q}(\sum _{k=-\infty }^{-1}{2}^{k\mathfrak{a}(0)q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{{}^p(·)}}^q\hfill \\ & +\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{{}^p(·)}}^q\left)\right\}.\hfill \end{array}$$

Proof of Theorem 1.

For every sequence of functions $g_1,g_2,\dots ,g_m$ residing within the function space $M{\dot{K}}_{q_i,p_i(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)$, we introduce the notation

$$g_{il}=g_i·{\mathcal{X}}_l=g_i·{\mathcal{X}}_{A_l}$$

for any index $l\in \mathbb{Z}$ and $i\in {\mathbb{Z}}^{+}.$ Consequently, the function $g_i(x)$ can be represented as the decomposition

$$g_i(x)=\sum _{l=-\infty }^{\infty }{g}_i(x)·{\mathcal{X}}_l(x)=\sum _{l=-\infty }^{\infty }{g}_{il}(x).$$

Leveraging the generalized Hölder inequality, we derive the subsequent estimation for the fractional multilinear Hardy operator acting on the product of these functions:

$$\begin{array}{cc}\hfill |H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)(x)·{\mathcal{X}}_k(x)|& \le {\displaystyle \frac{1}{{|x|}^{mn-\mathcal{B}}}}{\int }_{O_k}{\int }_{O_k}\dots {\int }_{O_k}(|g_1(t_1)||g_2(t_2)|\dots |g_m(t_m)|\hfill \\ & \times {\Omega }_1(x-t_1){\Omega }_2(x-t_2){\Omega }_3(x-t_3)\dots {\Omega }_m(x-t_m))d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(x)\hfill \\ & ={\displaystyle \frac{1}{{|x|}^{mn-\mathcal{B}}}}{\int }_{O_k}|g_1(t_1){\Omega }_1(x-t_1)|d{t}_1\hfill \\ & \times {\int }_{O_k}|g_2(t_2){\Omega }_2(x-t_2)|d{t}_2\dots {\int }_{O_k}|g_m(t_m){\Omega }_m(x-t_m)|d{t}_m·{\mathcal{X}}_k(x)\hfill \\ & \le C{2}^{-kmn}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}{\parallel {\Omega }_i(x-t_i){\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{2}^{k\mathcal{B}}{\mathcal{X}}_k(x),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{k\mathcal{B}}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{2}^{-kmn}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

(4)

Utilizing the fundamental identity

$${\displaystyle \frac{1}{r_i}}+{\displaystyle \frac{1}{S}}={\displaystyle \frac{1}{p_i^{\prime }}},$$

we proceed to establish the following estimate:

$$\begin{array}{cc}\hfill & \parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{k\mathcal{B}}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}\parallel {\mathcal{X}}_l{\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}{2}^{-kmn}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

(5)

Subsequently, we observe that the characteristic function norms satisfy the relation

$$\parallel {\mathcal{X}}_l{\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}\approx |B_l{|}^{{\displaystyle \frac{1}{r_i(·)}}}\approx |B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}$$

Employing this asymptotic equivalence in our previous inequality, we refine the estimate as follows:

$$\begin{array}{cc}\hfill & \parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{k\mathcal{B}}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}\parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{2}^{-kmn}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

(6)

We commence by considering the case where $g={\mathcal{X}}_{O_k}$ and invoking the definition of the fractional integral operator $I_{\mathcal{B}}$, yielding the inequality

$$I_{\mathcal{B}}({\mathcal{X}}_{O_k})(x)\ge C{2}^{k\mathcal{B}}{\mathcal{X}}_{O_k}(x),$$

which immediately furnishes the upper bound

$${\mathcal{X}}_{O_k}(x)\le C{2}^{-k\mathcal{B}}{I}_{\mathcal{B}}({\mathcal{X}}_{O_k})(x).$$

Upon taking norms and invoking the conclusions from Lemmas 2 and 5, we obtain

$$\begin{array}{cc}\hfill \parallel {\mathcal{X}}_{O_k}{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C{2}^{-k\mathcal{B}}||I_{\mathcal{B}}{\mathcal{X}}_{O_k}{||}_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{-k\mathcal{B}}||{\mathcal{X}}_{O_k}{||}_{L^{p(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{-k\mathcal{B}}\prod _{i=1}^m||{\mathcal{X}}_{O_k}{||}_{L^{p_i(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{k(mn-\mathcal{B})}\prod _{i=1}^m{\parallel {\mathcal{X}}_{O_k}\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}^{-1}.\hfill \end{array}$$

(7)

This result serves as a pivotal component for our subsequent analysis. Substituting this inequality into the previously established bound (6), we deduce

$$\begin{array}{cc}\hfill \parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}{|B_l|}^{{\displaystyle \frac{-1}{S}}}\hfill \\ & \times \parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}{({\mathbb{R}}^n)}^{\prime }}{\parallel {\mathcal{X}}_k\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}^{-1}\hfill \\ & \le C\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}{|B_l|}^{{\displaystyle \frac{-1}{S}}}{\displaystyle \frac{\parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}}{\parallel {\mathcal{X}}_k{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}}}.\hfill \end{array}$$

Further refining this inequality, we examine the behavior of $\Omega (x-t)$ for $x\in C_k$ and $t\in C_l$ with $l\le k$, yielding

$$0\le |t-x|\le 2^l+|x|\le 2^k.2.$$

This condition facilitates the integral bound

$${\int }_{C_l}{|\Omega (x-t)|}^{s}dt\le {\int }_0^{2^k.2}{\int }_{s^{n-1}}{|\Omega (x^{\prime })|}^{s}d\sigma (x^{\prime })r^{n-1}dr\le C{2}^{kn}$$

Leveraging these estimates, we conclude

$$\begin{array}{c}\hfill \parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\le C\sum _{l=-\infty }^k\prod _{i=1}^m{2}^{(n{\sigma }_{ii}-{\displaystyle \frac{n}{S}})(l-k)}\end{array}{\parallel g_{il}\parallel }_{L^{p_i(·)}({\mathbb{R}}^n).}$$

In the subsequent stage of the proof, our primary objective is to rigorously estimate the quantity $\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}$. To this end, we delineate our analysis into two distinct scenarios.

Case 1: For $l<0,$ we proceed as follows:

$$\begin{array}{cc}\hfill \parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}& =\prod _{i=1}^m{2}^{-l{\mathfrak{a}}_i(0)}{\left(2^{l{\mathfrak{a}}_i(0)q_i}{\parallel g_{il}\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}^{q_i}\right)}^{{\displaystyle \frac{1}{q_i}}}\hfill \\ & \le \prod _{i=1}^m{2}^{-l{\mathfrak{a}}_i(0)}{\left(\sum _{\ell =-\infty }^l{2}^{\ell {\mathfrak{a}}_i(0)q_i}{\parallel g_{i\ell }\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}^{q_i}\right)}^{{\displaystyle \frac{1}{q_i}}}\hfill \\ & \le \prod _{i=1}^m{2}^{l({\mathsf{ƛ}}_i-{\mathfrak{a}}_i(0))}{2}^{-l{\mathsf{ƛ}}_i}{\left(\sum _{\ell =-\infty }^l{2}^{\ell {\mathfrak{a}}_i(·)q_i}{\parallel g_{i\ell }\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}^{q_i}\right)}^{{\displaystyle \frac{1}{q_i}}}\hfill \\ & \le C\prod _{i=1}^m{2}^{l({\mathsf{ƛ}}_i-{\mathfrak{a}}_i(0))}{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\hfill \end{array}$$

At this juncture, we employ the structural identities

$$\mathfrak{a}(0)=\sum _{i=1}^m{\mathfrak{a}}_i(0)and\mathsf{ƛ}=\sum _{i=1}^m{\mathsf{ƛ}}_i,$$

which subsequently yield the refined estimate

$$\begin{array}{cc}\hfill \parallel g_{il}{\parallel }_{L^{p_1(·)}({\mathbb{R}}^n)}& \le C{2}^{l(\mathsf{ƛ}-\mathfrak{a}(0))}\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\hfill \end{array}$$

Case 2: For $l\ge 0,$ the analysis proceeds as follows:

$$\begin{array}{cc}\hfill \parallel g_{il}{\parallel }_{L^{p_1(·)}({\mathbb{R}}^n)}& =\prod _{i=1}^m{2}^{-l{\mathfrak{a}}_i(\infty )}{\left(2^{l{\mathfrak{a}}_i(\infty )q_i}{\parallel g_{il}\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}^{q_i}\right)}^{{\displaystyle \frac{1}{q_i}}}\hfill \\ & \le \prod _{i=1}^m{2}^{-l{\mathfrak{a}}_i(\infty )}{\left(\sum _{\ell =0}^l{2}^{\ell {\mathfrak{a}}_i(\infty )q_i}{\parallel g_{i\ell }\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}^{q_i}\right)}^{{\displaystyle \frac{1}{q_i}}}\hfill \\ & \le \prod _{i=1}^m{2}^{l({\mathsf{ƛ}}_i-{\mathfrak{a}}_i(\infty ))}{2}^{-l{\mathsf{ƛ}}_i}{\left(\sum _{\ell =-\infty }^l{2}^{\ell {\mathfrak{a}}_i(·)q_i}{\parallel g_{i\ell }\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}^{q_i}\right)}^{{\displaystyle \frac{1}{q_i}}}\hfill \\ & \le C\prod _{i=1}^m{2}^{l({\mathsf{ƛ}}_i-{\mathfrak{a}}_i(\infty ))}{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\hfill \end{array}$$

In this context, by invoking the summation identities

$$\mathfrak{a}(\infty )=\sum _{i=1}^m{\mathfrak{a}}_i(\infty )and\mathsf{ƛ}=\sum _{i=1}^m{\mathsf{ƛ}}_i$$

we derive the refined estimate

$$\begin{array}{c}\hfill \parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}=C{2}^{l(\mathsf{ƛ}-\mathfrak{a}(\infty ))}\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\end{array}$$

By invoking the intrinsic structural formulation of variable exponent Herz–Morrey spaces and leveraging the analytical machinery encapsulated in Proposition 2, we rigorously establish the ensuing fundamental inequality:

$$\begin{array}{cc}\hfill \parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m){\parallel }_{M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}& =\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(·)q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le max\{\underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(0)q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}},\hfill \\ & \underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}({\left(\sum _{k=-\infty }^{-1}{2}^{k\mathfrak{a}(0)q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}})\}\hfill \\ & =:max\{{\mathcal{Y}}_1,{\mathcal{Y}}_2+{\mathcal{Y}}_3\},\hfill \end{array}$$

(8)

where the terms ${\mathcal{Y}}_1$, ${\mathcal{Y}}_2$, ${\mathcal{Y}}_3$ are explicitly delineated as follows:

$${\mathcal{Y}}_1=\underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(0)q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}},$$

$${\mathcal{Y}}_2=\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{-1}{2}^{k\mathfrak{a}(0)q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}},$$

$${\mathcal{Y}}_3=\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\parallel H_{\mathcal{B},\Omega }(g_1,g_2,\dots ,g_m)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}}.$$

Commencing with an incisive examination of ${\mathcal{Y}}_1,$ we shall rigorously derive an upper bound for its magnitude. Exploiting the fundamental constraint

$$\mathfrak{a}(0)\le \mathfrak{a}(\infty )<n{\sigma }_{ii}-{\displaystyle \frac{n}{S}}+\mathsf{ƛ}$$

we proceed to the sequence of refined estimates:

$$\begin{array}{cc}\hfill {\mathcal{Y}}_1& \le C\underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(0)q}{\left(\sum _{l=-\infty }^k\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}-n{\sigma }_{ii})(k-l)}{\parallel g_i\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le C\underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(0)q}{\left(\sum _{l=-\infty }^k\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}-n{\sigma }_{ii})(k-l)}{2}^{l(\mathsf{ƛ}-\mathfrak{a}(0))}{\parallel g_i\parallel }_{M{\dot{k}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le C\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}\underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathsf{ƛ}q}{\left(\sum _{l=-\infty }^k{2}^{(n{\sigma }_{ii}+\mathsf{ƛ}-\mathfrak{a}(0)-{\displaystyle \frac{n}{S}})(l-k)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le C\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\hfill \end{array}$$

The estimation of ${\mathcal{Y}}_2$ is conducted analogously to that of ${\mathcal{Y}}_1$, adhering to an identical methodological framework. Consequently, we proceed to the approximation of ${\mathcal{Y}}_3.$

$$\begin{array}{cc}\hfill {\mathcal{Y}}_3& =C\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\left(\sum _{l=-\infty }^k\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}-n{\sigma }_{ii})(k-l)}{\parallel g_i\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & =C\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\left(\sum _{l=-\infty }^0\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}-n{\sigma }_{ii})(k-l)}{\parallel g_i\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +C\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\left(\sum _{l=1}^k\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}-n{\sigma }_{ii})(k-l)}{\parallel g_i\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & ={\mathcal{Y}}_3^{\prime }+{\mathcal{Y}}_3^{″}\hfill \\ & {\mathcal{Y}}_3^{\prime }=C\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\left(\sum _{l=-\infty }^0\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}-n{\sigma }_{ii})(k-l)}{\parallel g_i\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & {\mathcal{Y}}_3^{″}=\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\left(\sum _{l=1}^k\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}-n{\sigma }_{ii})(k-l)}{\parallel g_i\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{Y}}_3^{\prime }& \le C\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\left(\sum _{l=-\infty }^0\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}-n{\sigma }_{ii})(k-l)}{2}^{l(\mathsf{ƛ}-\mathfrak{a}(\infty ))}{\parallel g_i\parallel }_{M{\dot{k}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le C\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathsf{ƛ}q}{\left(\sum _{l=-\infty }^0{2}^{(n{\sigma }_{ii}+\mathsf{ƛ}-\mathfrak{a}(\infty )-{\displaystyle \frac{n}{S}})(l-k)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le C\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{Y}}_3^{″}& \le C\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\left(\sum _{l=1}^k\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}-n{\sigma }_{ii})(k-l)}{2}^{l(\mathsf{ƛ}-\mathfrak{a}(\infty ))}{\parallel g_i\parallel }_{M{\dot{k}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le C\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathsf{ƛ}q}{\left(\sum _{l=1}^k{2}^{(n{\sigma }_{ii}+\mathsf{ƛ}-\mathfrak{a}(\infty )-{\displaystyle \frac{n}{S}})(l-k)}\right)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le C\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\hfill \end{array}$$

Consequently, we deduce the following upper bound:

$$\begin{array}{cc}\hfill {\mathcal{Y}}_3& \le C\prod _{i=1}^m{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\hfill \end{array}$$

Incorporating the derived estimations for ${\mathcal{Y}}_1,{\mathcal{Y}}_2$, and ${\mathcal{Y}}_3$ into inequality (8) culminates in the anticipated assertion, thereby establishing the desired conclusion. □

Proof of Theorem 2.

By leveraging Hölder’s inequality, we establish the ensuing upper bound:

$$\begin{array}{cc}\hfill |H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m)(x)·{\mathcal{X}}_k(x)|& \le {\int }_{{\mathbb{R}}^n\setminus O_k}{\int }_{{\mathbb{R}}^n\setminus O_k}\dots {\int }_{{\mathbb{R}}^n\setminus O_k}{\displaystyle \frac{1}{{|t|}^{mn-\mathcal{B}}}}|g_1(t_1){\Omega }_1(x-t_1)|\hfill \\ & \times |g_2(t_2){\Omega }_2(x-t_2)|\dots |g_m(t_m){\Omega }_m(x-t_m)|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(x)\hfill \\ & \le C\sum _{l=k+1}^{\infty }\prod _{i=1}^m{2}^{-l(mn-\mathcal{B})}\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}{\parallel {\Omega }_i(x-t_i){\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n))}{\mathcal{X}}_k(x).\hfill \end{array}$$

Exploiting the identity

$${\displaystyle \frac{1}{r_i}}+{\displaystyle \frac{1}{S}}={\displaystyle \frac{1}{p_i^{\prime }}},$$

$$\begin{array}{cc}\hfill |H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m)(x)·{\mathcal{X}}_k(x)|& \le C\sum _{l=k+1}^{\infty }\prod _{i=1}^m{2}^{-l(mn-\mathcal{B})}\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}{\parallel {\mathcal{X}}_l\parallel }_{L^{r_i}({\mathbb{R}}^n))}{\mathcal{X}}_k(x).\hfill \end{array}$$

This, in turn, culminates in the ensuing norm estimate:

$$\begin{array}{cc}\hfill \parallel H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m)(x)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C\sum _{l=k+1}^{\infty }\prod _{i=1}^m{2}^{-l(mn-\mathcal{B})}\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}{\parallel {\Omega }_i(x-t_i)\parallel }_{L^S}\hfill \\ & \times \parallel {\mathcal{X}}_l{\parallel }_{L^{r_i}({\mathbb{R}}^n))}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Accordingly, we establish the asymptotic relation

$$\parallel {\mathcal{X}}_l{\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}\approx |B_l{|}^{{\displaystyle \frac{1}{r_i(·)}}}\approx |B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)},$$

which enables us to refine the prior estimate into

$$\begin{array}{cc}\hfill \parallel H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m)(x)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C\sum _{l=k+1}^{\infty }\prod _{i=1}^m{2}^{-l(mn-\mathcal{B})}\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}{|B_l|}^{{\displaystyle \frac{-1}{S}}}\hfill \\ & \times \parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Further invoking the inequality established in (7), we derive the refined upper bound

$$\begin{array}{cc}\hfill \parallel H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m)(x)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C\sum _{l=k+1}^{\infty }\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}{|B_l|}^{{\displaystyle \frac{-1}{S}}}\hfill \\ & \times \parallel {\mathcal{X}}_l{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^{-1}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C\sum _{l=k+1}^{\infty }\prod _{i=1}^m{2}^{({\displaystyle \frac{n}{S}}+n\sigma )(k-l)}{\parallel g_{il}\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)},\hfill \end{array}$$

where, in the concluding step, we have applied the outcome of Lemma 4.

Employing an analogous methodological framework as delineated in Theorem 1, we arrive at the subsequent formulation:

$$\parallel H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m){\parallel }_{M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}^q=max\{{\mathcal{Z}}_1,{\mathcal{Z}}_2+{\mathcal{Z}}_3\},$$

where the constituent terms are explicitly characterized as follows:

$${\mathcal{Z}}_1=\underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}q}\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(0)q}{\parallel H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m)(x)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q.$$

$${\mathcal{Z}}_2=\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}q}\sum _{k=-\infty }^{-1}{2}^{k\mathfrak{a}(0)q}{\parallel H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m)(x)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q.$$

$${\mathcal{Z}}_3=\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}q}\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\parallel H_{\mathcal{B},\Omega }^{*}(g_1,g_2,\dots ,g_m)(x)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q.$$

Notably, the boundedness characteristics of each individual component ${\mathcal{Z}}_{\ell }$ for ℓ = 1, 2, 3 are inherently analogous to the properties of their corresponding counterparts ${\mathcal{Y}}_{\ell }$, as meticulously expounded in Theorem 1. This brings us to the threshold of the conclusive result. □

Proof of Theorem 3.

$$\begin{array}{cc}& |[b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k(z)|\le {\displaystyle \frac{1}{{|z|}^{mn-\mathcal{B}}}}{\int }_{O_k}{\int }_{O_k}\dots {\int }_{O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m)\hfill \\ & \times {\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)\times (b_1(z)-b(t_1))|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k{\int }_{O_k}{\int }_{O_k}\dots {\int }_{O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m){\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)\hfill \\ & \times (b_1(z)-{(b_1)}_{B_l}+{(b_1)}_{B_l}-b_1(t_1))|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k{\int }_{O_k}{\int }_{O_k}\dots {\int }_{O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m)\hfill \\ & \times {\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)(b_1(z)-{(b_1)}_{B_l})|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & +C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k{\int }_{O_k}{\int }_{O_k}\dots {\int }_{O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m)(b_1(t_1)-{(b_1)}_{B_l})|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & =\mathcal{I}+\mathcal{II},\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{I}& =C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k{\int }_{O_k}{\int }_{O_k}\dots {\int }_{O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m)\hfill \\ & \times {\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)(b_1(z)-{(b_1)}_{B_l})|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z).\hfill \end{array}$$

By judiciously leveraging Hölder inequality, we establish the ensuing upper bound:

$$\begin{array}{cc}\hfill \mathcal{I}\le & C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m|(b_1(z)-{(b_1)}_{B_l})·{\mathcal{X}}_k(z)|\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}{\parallel {\Omega }_i(x-t_i){\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Utilizing the relation ${\displaystyle \frac{1}{r_i}}+{\displaystyle \frac{1}{S}}={\displaystyle \frac{1}{p_i^{\prime }}}$, we refine the aforementioned estimate to

$$\begin{array}{cc}\hfill \mathcal{I}\le & C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m|(b_1(z)-{(b_1)}_{B_l})·{\mathcal{X}}_k(z)|\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}{\parallel {\mathcal{X}}_l\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Furthermore, invoking the established asymptotic equivalences

$$\parallel {\mathcal{X}}_l{\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}\approx |B_l{|}^{{\displaystyle \frac{1}{r_i(·)}}}\approx |B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}$$

we ultimately deduce

$$\begin{array}{cc}\hfill \mathcal{I}\le & C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m|(b_1(z)-{(b_1)}_{B_l})·{\mathcal{X}}_k(z)|\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Through a meticulous sequential application of Lemmas 2, 4 and 6, we deduce the ensuing bound:

$$\begin{array}{cc}\hfill & {\parallel \mathcal{I}\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{k\mathcal{B}}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_1(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}\parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{2}^{-mkn}{\parallel (b_1(z)-{(b_1)}_{B_l}){\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ \hfill & \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_1(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}\parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}(k-l)\parallel b_1{\parallel }_{\mathcal{BMO}}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \mathcal{II}& =2^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k{\int }_{O_k}{\int }_{O_k}\dots {\int }_{O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m)\hfill \\ & \times {\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)(b_1(t_1)-{(b_1)}_{B_l})|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & =2^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k{\int }_{O_k}{g}_1(t_1){\Omega }_1(x-t_1)(b_1(t_1){(b_1)}_{B_l})d{t}_1\hfill \\ & \times {\int }_{O_k}{g}_2(t_2){\Omega }_2(x-t_2)d{t}_2\dots {\int }_{O_k}{\Omega }_m(x-t_m)g_m(t_m)d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}{\parallel (b_1(t_1)-{(b_1)}_{B_l}){\Omega }_i(x-t_i){\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}·{\mathcal{X}}_k(z)\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\parallel \mathcal{II}\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel (b_1(t_1)-{(b_1)}_{B_l}){\Omega }_i(x-t_i){\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{\parallel {\mathcal{X}}_k(x)\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel b_1{\parallel }_{\mathcal{BMO}}\parallel {\Omega }_i(x-t_i){\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{\parallel {\mathcal{X}}_k(x)\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Leveraging the fundamental identity

$${\displaystyle \frac{1}{r_i}}+{\displaystyle \frac{1}{S}}={\displaystyle \frac{1}{p_i^{\prime }}}$$

we derive the ensuing upper bound:

$$\begin{array}{cc}\hfill {\parallel \mathcal{II}\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel b_1{\parallel }_{\mathcal{BMO}}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}\parallel {\mathcal{X}}_l{\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}{\parallel {\mathcal{X}}_k(x)\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Observing the equivalence

$$\parallel {\mathcal{X}}_l{\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}\approx |B_l{|}^{{\displaystyle \frac{1}{r_i(·)}}}\approx |B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)},$$

we substitute this into the preceding bound, yielding

$$\begin{array}{cc}\hfill {\parallel \mathcal{II}\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel b_1{\parallel }_{\mathcal{BMO}}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}{|B_l|}^{{\displaystyle \frac{-1}{S}}}\hfill \\ & \times \parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{\parallel {\mathcal{X}}_k(x)\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

(10)

By synthesizing the preceding inequalities, namely (9) and (10), we deduce the subsequent refined estimate:

$$\begin{array}{cc}& \parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_1(·)}({\mathbb{R}}^n)}\parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}(k-l)\parallel b_1{\parallel }_{\mathcal{BMO}}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Moreover, by considering the geometric constraints imposed by $x\in C_k$ and $t\in C_l$ for $l\le k$, it follows that $0\le |t-x|\le 2^l+|x|\le 2^k.2,$ leading to the integral bound

$${\int }_{C_l}{|\Omega (x-t)|}^{s}dt\le {\int }_0^{2^k.2}{\int }_{s^{n-1}}{|\Omega (x^{\prime })|}^{s}d\sigma (x^{\prime })r^{n-1}dr\le C{2}^{kn}$$

Substituting this into our earlier bound, we obtain

$$\begin{array}{cc}& \parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C{2}^{-k(mn-\mathcal{B})}\sum _{l=-\infty }^k\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_1(·)}({\mathbb{R}}^n)}\parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}(k-l)\parallel b_1{\parallel }_{\mathcal{BMO}}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

To progress further in our analysis, we now adhere to the methodology delineated in Theorem 1, which culminates in the refined formulation:

$$\begin{array}{cc}& \parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C\sum _{l=-\infty }^k\prod _{i=1}^m{2}^{(n{\sigma }_{ii}+{\displaystyle \frac{n}{S}})(l-k)}(k-l)\parallel b_1{\parallel }_{\mathcal{BMO}}{\parallel g_{il}\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

By invoking the fundamental characterization of the variable exponent Herz–Morrey space in conjunction with Proposition 2, we rigorously derive the ensuing inequality:

$$\begin{array}{cc}\hfill & \parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z){\parallel }_{M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}\hfill \\ & =\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(·)q}{\parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le max\{\underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(0)q}{\parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}},\hfill \\ & \underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}({\left(\sum _{k=-\infty }^{-1}{2}^{k\mathfrak{a}(0)q}{\parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}})\}\hfill \\ & =max\{{\mathcal{A}}_1,{\mathcal{A}}_2+{\mathcal{A}}_3\},\hfill \end{array}$$

where we define the auxiliary terms

$${\mathcal{A}}_1=\underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(0)q}{\parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}},$$

$${\mathcal{A}}_2=\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{-1}{2}^{k\mathfrak{a}(0)q}{\parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}},$$

$${\mathcal{A}}_3=\underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}}.$$

Employing an analogous methodological framework to the intricate derivations expounded in Theorem 1, we deduce the following refined upper bound:

$$\begin{array}{cc}& \parallel [b_1,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m){\parallel }_{M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}\le C\prod _{i=1}^m\parallel b_1{\parallel }_{\mathcal{BMO}}{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\hfill \end{array}$$

Furthermore, by virtue of an entirely symmetric argument, one may promptly establish the ensuing inequality:

$$\begin{array}{ccc}\parallel [b_2,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m){\parallel }_{M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}& \le & C\prod _{i=1}^m\parallel b_2{\parallel }_{\mathcal{BMO}}{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\\ ,& ,& ,\\ ⋮& ⋮& ⋮\\ ,& ,& ,\\ \parallel [b_m,H_{\mathcal{B},\Omega }](g_1,g_2,\dots ,g_m){\parallel }_{M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}& \le & C\prod _{i=1}^m\parallel b_m{\parallel }_{\mathcal{BMO}}{\parallel g_i\parallel }_{M{\dot{K}}_{q_i,p_i(·)}^{{\mathfrak{a}}_i(·),{\mathsf{ƛ}}_i}({\mathbb{R}}^n)}.\end{array}$$

□

Proof of Theorem 4.

$$\begin{array}{cc}& |[b_1,H_{\mathcal{B},\Omega }^{*}](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k(z)|\le {\int }_{{\mathbb{R}}^n\setminus O_k}{\int }_{{\mathbb{R}}^n\setminus O_k}\dots {\int }_{{\mathbb{R}}^n\setminus O_k}{\displaystyle \frac{1}{{|t|}^{mn-\mathcal{B}}}}|g_1(t_1)\hfill \\ & \times g_2(t_2)\dots g_m(t_m){\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)(b_1(z)-b(t_1))|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & \le C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}{\int }_{{\mathbb{R}}^n\setminus O_k}{\int }_{{\mathbb{R}}^n\setminus O_k}\dots {\int }_{{\mathbb{R}}^n\setminus O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m){\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)\hfill \\ & \times (b_1(z)-{(b_1)}_{B_l}+{(b_1)}_{B_l}-b_1(t_1))|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & \le C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}{\int }_{{\mathbb{R}}^n\setminus O_k}{\int }_{{\mathbb{R}}^n\setminus O_k}\dots {\int }_{{\mathbb{R}}^n\setminus O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m){\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)\hfill \\ & \times (b_1(z)-{(b_1)}_{B_l})|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & +C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}{\int }_{{\mathbb{R}}^n\setminus O_k}{\int }_{{\mathbb{R}}^n\setminus O_k}\dots {\int }_{{\mathbb{R}}^n\setminus O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m){\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)\hfill \\ & \times (b_1(t_1)-{(b_1)}_{B_l})|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & =\mathcal{III}+\mathcal{IV},\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{III}& =C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}{\int }_{{\mathbb{R}}^n\setminus O_k}{\int }_{{\mathbb{R}}^n\setminus O_k}\dots {\int }_{{\mathbb{R}}^n\setminus O_k}|g_1(t_1)\hfill \\ & \times g_2(t_2)\dots g_m(t_m){\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)(b_1(z)-{(b_1)}_{B_l})|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \end{array}$$

By leveraging the celebrated Hölder inequality, we rigorously derive the subsequent upper bound:

$$\begin{array}{cc}\hfill \mathcal{III}\le & C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}\prod _{i=1}^m|(b_1(z)-{(b_1)}_{B_l})·{\mathcal{X}}_k(z)|\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}{\parallel {\Omega }_i(x-t_i){\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Exploiting the fundamental identity

$${\displaystyle \frac{1}{r_i}}+{\displaystyle \frac{1}{S}}={\displaystyle \frac{1}{p_i^{\prime }}},$$

we obtain the refined estimate:

$$\begin{array}{cc}\hfill \mathcal{III}\le & C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}\prod _{i=1}^m|(b_1(z)-{(b_1)}_{B_l})·{\mathcal{X}}_k(z)|\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}{\parallel {\mathcal{X}}_l\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Furthermore, utilizing the structural approximation

$$\parallel {\mathcal{X}}_l{\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}\approx |B_l{|}^{{\displaystyle \frac{1}{r_i(·)}}}\approx |B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)},$$

we refine our bound as follows:

$$\begin{array}{cc}\hfill \mathcal{III}\le & C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}\prod _{i=1}^m|(b_1(z)-{(b_1)}_{B_l})·{\mathcal{X}}_k(z)|\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

By successively invoking Lemmas 2, 4 and 6, we rigorously establish the following estimate:

$$\begin{array}{cc}\hfill & {\parallel \mathcal{III}\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C\sum _{l=k+1}^{\infty }\prod _{i=1}^m{2}^{-l(mn-\mathcal{B})}\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}\parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{\parallel (b_1(z)-{(b_1)}_{B_l}){\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ \hfill & \le C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}\parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}(k-l)\parallel b_1{\parallel }_{\mathcal{BMO}}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \end{array}$$

(11)

Furthermore, consider the integral expression associated with $\mathcal{IV}$, given by:

$$\begin{array}{cc}\hfill \mathcal{IV}& =\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}{\int }_{{\mathbb{R}}^n\setminus O_k}{\int }_{{\mathbb{R}}^n\setminus O_k}\dots {\int }_{{\mathbb{R}}^n\setminus O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m){\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)\hfill \\ & \times (b_1(t_1)-{(b_1)}_{B_l})|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & =\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}{\int }_{{\mathbb{R}}^n\setminus O_k}{\int }_{{\mathbb{R}}^n\setminus O_k}\dots {\int }_{{\mathbb{R}}^n\setminus O_k}|g_1(t_1)g_2(t_2)\dots g_m(t_m){\Omega }_1(x-t_1){\Omega }_2(x-t_2)\dots {\Omega }_m(x-t_m)\hfill \\ & \times (b_1(t_1)-{(b_1)}_{B_l})|d{t}_{1}d{t}_2\dots d{t}_m·{\mathcal{X}}_k(z)\hfill \\ & \le C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}{\parallel (b_1(t_1)-{(b_1)}_{B_l}){\Omega }_i(x-t_i){\mathcal{X}}_l\parallel }_{L^{p_1^{\prime }(·)}({\mathbb{R}}^n)}·{\mathcal{X}}_k(x)\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\parallel \mathcal{IV}\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}{\parallel (b_1(t_1)-{(b_1)}_{B_l}){\Omega }_i(x-t_i){\mathcal{X}}_l\parallel }_{L^{p_1^{\prime }(·)}({\mathbb{R}}^n)}\hfill \\ & \times \parallel {\mathcal{X}}_k{(x)\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel b_1{\parallel }_{\mathcal{BMO}}\parallel {\Omega }_i(x-t_i){\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{\parallel {\mathcal{X}}_k(x)\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

By leveraging the identity ${\displaystyle \frac{1}{r_i}}+{\displaystyle \frac{1}{S}}={\displaystyle \frac{1}{p_i^{\prime }}}$, we arrive at the following refined bound:

$$\begin{array}{c}\hfill {\parallel \mathcal{IV}\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\le C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel b_1{\parallel }_{\mathcal{BMO}}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}\parallel {\mathcal{X}}_l{\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}{\parallel {\mathcal{X}}_k(x)\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\end{array}$$

Exploiting the asymptotic equivalence

$$\parallel {\mathcal{X}}_l{\parallel }_{L^{r_i(·)}({\mathbb{R}}^n)}\approx |B_l{|}^{{\displaystyle \frac{1}{r_i(·)}}}\approx |B_l{|}^{{\displaystyle \frac{-1}{S}}}{\parallel {\mathcal{X}}_l\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)},$$

we refine the aforementioned inequality into its final form:

$$\begin{array}{cc}\hfill {\parallel \mathcal{IV}\parallel }_{L^{q(·)}({\mathbb{R}}^n)}& \le C\sum _{l=k+1}^{\infty }{2}^{-l(mn-\mathcal{B})}\prod _{i=1}^m\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel b_1{\parallel }_{\mathcal{BMO}}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}{|B_l|}^{{\displaystyle \frac{-1}{S}}}\hfill \\ & \times \parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{\parallel {\mathcal{X}}_k(x)\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

(12)

By synthesizing the estimates encapsulated in inequalities (11) and (12), we rigorously deduce the ensuing bound:

$$\begin{array}{cc}& \parallel [b_1,H_{\mathcal{B},\Omega }^{*}](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k{\parallel }_{L^{q(·)}({\mathbb{R}}^n)}\hfill \\ & \le C\sum _{l=k+1}^{\infty }\prod _{i=1}^m{2}^{-l(mn-\mathcal{B})}(k-l)\parallel b_1{\parallel }_{\mathcal{BMO}}\parallel g_{il}{\parallel }_{L^{p_i(·)}({\mathbb{R}}^n)}\parallel {\Omega }_i(x-t_i){\parallel }_{L^S}|B_l{|}^{{\displaystyle \frac{-1}{S}}}\parallel {\mathcal{X}}_l{\parallel }_{L^{p_i^{\prime }(·)}({\mathbb{R}}^n)}{\parallel {\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}.\hfill \end{array}$$

Upon invoking Proposition 2, we further refine our analysis and obtain the following norm inequality:

$$\begin{array}{cc}\hfill & \parallel [b_1,H_{\mathcal{B},\Omega }^{*}](g_1,g_2,\dots ,g_m){\parallel }_{M{\dot{K}}_{q,q(·)}^{\mathfrak{a}(·),\mathsf{ƛ}}({\mathbb{R}}^n)}\hfill \\ & =\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(·)q}{\parallel [b_1,H_{\mathcal{B},\Omega }^{*}](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le max\{\underset{k_0<0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}{\left(\sum _{k=-\infty }^{k_0}{2}^{k\mathfrak{a}(0)q}{\parallel [b_1,H_{\mathcal{B},\Omega }^{*}](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}},\hfill \\ & \underset{k_0\ge 0}{\underset{k_0\in \mathbb{Z}}{sup}}{2}^{-k_0\mathsf{ƛ}}({\left(\sum _{k=-\infty }^{-1}{2}^{k\mathfrak{a}(0)q}{\parallel [b_1,H_{\mathcal{B},\Omega }^{*}](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left(\sum _{k=0}^{k_0}{2}^{k\mathfrak{a}(\infty )q}{\parallel [b_1,H_{\mathcal{B},\Omega }^{*}](g_1,g_2,\dots ,g_m)(z)·{\mathcal{X}}_k\parallel }_{L^{q(·)}({\mathbb{R}}^n)}^q\right)}^{{\displaystyle \frac{1}{q}}}\left)\right\}\hfill \\ & =max\{Q_1,Q_2+Q_3\}.\hfill \end{array}$$

The quantitative bounds associated with $Q_i$ (for i = 1, 2, 3) can be systematically ascertained through the deployment of analytical techniques that closely parallel those meticulously delineated within the framework of Theorem 3. By invoking analogous methodological constructs, one may derive precise upper bounds for these terms, thereby reinforcing the established estimates within the variable exponent Herz–Morrey space setting. □

4. Results

In the present exposition, we have conducted a rigorous examination of the boundedness properties of the multilinear rough Hardy operator and its associated commutators within the intricate structural landscape of variable exponent Herz–Morrey spaces. Our analysis operates under the fundamental premise that the symbol functions reside within the realm of $\mathcal{BMO}$ spaces. By imposing a set of judiciously chosen conditions, we have successfully established affirmative boundedness results, thereby elucidating the intricate interplay between these operators and the underlying functional framework. The insights garnered from this investigation may serve as a catalyst for further mathematical inquiry, particularly in the pursuit of analogous boundedness criteria in other function spaces governed by variable exponent structures.