Certain new applications of Faber polynomial expansion for some new subclasses of $\upsilon$-fold symmetric bi-univalent functions associated with $q$-calculus

1.   Introduction

As we all know, the study of variable exponent function space inspired by nonlinear elasticity theory and nonstandard growth differential equations is one of the key contents of harmonic analysis in the past three decades, attracting extensive attention from many scholars. In ^[19], the theory of function spaces with variable exponent was progressed since some elementary properties were established by Kováčik and Rákosník, and they studied many basic properties of variable exponent Lebesgue spaces and Sobolev spaces on $\mathbb{R}^{n}$. Later, the Lebesgue spaces with variable exponent $L^{p(\cdot)}(\mathbb{R}^{n})$ were extensively investigated; see ^[7,8,22]. In ^[14], Izuki first introduced the Herz spaces with variable exponent ${\rm{\dot K}} ^{\alpha, q}_{p(\cdot)}(\mathbb{R}^{n})$, which are generalizations of the Herz spaces ${\rm{\dot K}} ^{\alpha, q}_{p}(\mathbb{R}^{n})$, and considered the boundedness of commutators of fractional integrals on Herz spaces with variable exponent. In ^[13], Izuki introduced the Herz-Morrey spaces with variable exponent $\mathrm{M\dot{K}}_{q, p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})$, which are generalizations of the Herz-Morrey spaces $\mathrm{M\dot{K}}_{q, p}^{\alpha, \lambda}(\mathbb{R}^{n})$, and studied the boundedness of vector valued sublinear operators on Herz-Morrey spaces with variable exponent $\mathrm{M\dot{K}}_{q, p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})$. On the other hand, in the study of boundary value problems for the Laplace equation on Lipschitz domains, the classical theory of Muckenhoupt weights is a powerful tool in harmonic analysis; see ^[21]. Generalized Muckenhoupt weights with variable exponent have been intensively studied; see ^[4,5].

In ^[11], Hardy defined the classical Hardy operators as:

In ^[6], Christ and Grafakos defined the $n-$dimensional Hardy operators as:

and established the boundedness of $\mathcal{P}(f)(x)$ in $L^{p}(\mathbb{R}^{n})$, getting the best constants.

In ^[9], under the condition of $0\leq \beta < n$ and $|x| = \sqrt{\sum_{i = 1}^{n}x_{i}^{2}}$, Fu et al. defined the $n-$dimensional fractional Hardy operators and its adjoint operators as:

and established the boundedness of their commutators in Lebesgue spaces and homogeneous Herz spaces.

Let $L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ be the collection of all locally integrable functions on $\mathbb{R}^{n}$. Given a function $b\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ and $m\in \mathbb{N}$, Wang et al. ^[23] defined the $m$th order commutators of $n-$dimensional fractional Hardy operators and adjoint operators as:

and

Obviously, when $m = 0$, $\mathcal{H}^{0}_{\beta, b} = \mathcal{H}_{_{\beta}}$, $\mathcal{H}^{\ast 0}_{\beta, b} = \mathcal{H}^{\ast}_{\beta}$, and when $m = 1$, $\mathcal{H}^{1}_{\beta, b} = \mathcal{H}_{\beta, b}$, $\mathcal{H}^{\ast 1}_{\beta, b} = \mathcal{H}^{\ast}_{\beta, b}$. More important results with regard to these commutators, see ^[20,26,27].

Due to the need of future calculation in this paper, let $0 < \beta < n$, and the fractional integral operator $I_{\beta}$ is defined as:

Let $0\leq \beta < n$ and $f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})$, and the fractional maximal operator $M_{\beta}$ is defined as:

where the supremum is taken over all balls $B\subset\mathbb{R}^{n}$ containing $x$. When $\beta = 0$, we simply write $M$ instead of $M_{0}$, which is exactly the Hardy-Littlewood maximal function.

Let $f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ and $\mathrm{BMO}(\mathbb{R}^{n})$ consist of all $f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ with $\mathrm{BMO}(\mathbb{R}^{n}) < \infty$. $b$ is a bounded mean oscillation function if $\|b\|_{\mathrm{BMO}} < \infty$, and the $\|b\|_{\mathrm{BMO}}$ is defined as follow:

where the supremum is taken all over the balls $B\in \mathbb{R}^{n}$ and $b_{B}: = |B|^{-1}\int_{B}b(y)\mathrm{d}y$. For a comprehensive review of the bounded mean oscillation (BMO) space, please see the book ^[10].

Recently, Muhammad Asim et al. established the estimates of fractional Hardy operators on weighted variable exponent Morrey-Herz spaces in ^[1]. Amjad Hussain et al. established the boundedness of the commutators of the Fractional Hardy operators on weighted variable Herz-Morrey spaces in ^[12]. Motivated by the mentioned work, in this paper, we will give the boundedness of the $m$th order commutators of $n-$dimensional fractional Hardy operators $\mathcal{H}^{m}_{_{\beta, b}}$ and its adjoint operators $\mathcal{H}_{\beta, b}^{\ast m}$ on weighted variable exponent Morrey-Herz space $\mathrm{M\dot{K}}_{q, p(\cdot)}^{\alpha, \lambda}(\omega)$.

The paper is organized as follows. In Section 2, we provide some preliminary knowledge. The main results and their proofs are given in Section 3. In Section 4, we provide the conclusion of this paper. Throughout this paper, we use the following symbols and notations:

(1) For a constant $R > 0$ and a point $x\in \mathbb{R}^{n}$, we write $B(x, R): = \{y\in \mathbb{R}^{n}: |x-y| < R\}$.

(2) For any measurable set $E\subset\mathbb{R}^{n}$, $|E|$ denotes the Lebesgue measure, and $\chi_{E}$ means the characteristic function.

(3) Given $k\in \mathbb{Z}$, we write $B_{k}: = \overline{B(0, 2^{k})} = \{x\in \mathbb{R}^{n}: |x|\leq 2^{k}\}$.

(4) We define a family $\{A_{k}\}_{k = -\infty}^{\infty}$ by $A_{k}: = B_{k}\setminus B_{k-1} = \{x\in \mathbb{R}^{n}: 2^{k-1} < |x|\leq 2^{k}\}$. Moreover $\chi_{k}$ denotes the characteristic function of $A_{k}$, namely, $\chi_{k}: = \chi_{A_{k}}$.

(5) For any index $1 < p(x) < \infty$, $p^{\prime}(x)$ denotes its conjugate index, namely, $\frac{1}{p(x)}+\frac{1}{p^{\prime}(x)} = 1$.

(6) If there exists a positive constant $C$ independent of the main parameters such that $A\leq CB$, then we write $A\lesssim B$. Additionally, $A\approx B$ means that both $A\lesssim B$ and $B\lesssim A$ hold.

2.   Preliminary knowledge

2.1.   Some definitions that need to be used in this paper

$\mathbf{Definition\; 2.1.}$ (^[7]) Let $p(\cdot): \mathbb{R}^{n}\rightarrow [1, \infty)$ be a measurable function.

(ⅰ) The Lebesgue space with variable exponent $L^{p(\cdot)}(\mathbb{R}^{n})$ is defined by

(ⅱ) The spaces with variable exponent $L_{\mathrm{loc}}^{p(\cdot)}(\mathbb{R}^{n})$ are defined by

The Lebesgue space $L^{p(\cdot)}(\mathbb{R}^{n})$ is a Banach space with the norm defined by

$\mathbf{Definition\; 2.2.}$ (^[7]) (ⅰ) The set $\mathcal{P}(\mathbb{R}^{n})$ consists of all measurable functions $p(\cdot): \mathbb{R}^{n}\rightarrow [1, \infty)$ satisfying

where

(ⅱ) The set $\mathcal{B}(\mathbb{R}^{n})$ consists of all measurable function $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ satisfying that the Hardy-Littlewood maximal operator $M$ is bounded on $L^{p(\cdot)}(\mathbb{R}^{n})$.

$\mathbf{Definition\; 2.3.}$ (^[7]) Suppose that $p(\cdot)$ is a real-valued function on $\mathbb{R}^{n}$. We say that

(ⅰ) $\mathcal{C}_{\mathrm{loc}}^{\mathrm{log}}(\mathbb{R}^{n})$ is the set of all local log-Hölder continuous functions $p(\cdot)$ satisfying

(ⅱ) $\mathcal{C}_{0}^{\mathrm{log}}(\mathbb{R}^{n})$ is the set of all local log-Hölder continuous functions $p(\cdot)$ satisfying at origin

(ⅲ) $\mathcal{C}_{\mathrm{\infty}}^{\mathrm{log}}(\mathbb{R}^{n})$ is the set of all local log-Hölder continuous functions satisfying at infinity

(ⅳ) $\mathcal{C}^{\mathrm{log}}(\mathbb{R}^{n}) = \mathcal{C}_{\mathrm{\infty}}^{\mathrm{log}}(\mathbb{R}^{n})\cap \mathcal{C}_{\mathrm{loc}}^{\mathrm{log}}(\mathbb{R}^{n})$ denotes the set of all global log-Hölder continuous functions $p(\cdot)$.

It was proved in ^[7] that if $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\cap \mathcal{C}^{\mathrm{log}}(\mathbb{R}^{n})$, then the Hardy-Littlewood maximal operator $M$ is bounded on $L^{p(\cdot)}(\mathbb{R}^{n})$.

$\mathbf{Definition\; 2.4.}$ (^[21]) Given a non-negative, measure function $\omega$, for $1 < p < \infty$, $\omega\in A_{p}$ if

where the supremum is taken over all balls $B\subset \mathbb{R}^{n}$. Especially, we say $\omega\in A_{1}$ if

These weights characterize the weighted norm inequalities for the Hardy-Littlewood maximal operator, that is, $\omega\in A_{p}$, $1 < p < \infty$, if and only if $M:L^{p}(\omega)\rightarrow L^{p}(\omega)$.

$\mathbf{Definition\; 2.5.}$ (^[15]) Suppose that $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$. A weight $\omega$ is in the class $A_{p(\cdot)}$ if

Obviously, if $p(\cdot) = p, 1 < p < \infty$, then the above definition reduces to the classical Muckenhoupt $A_{p}$ class.

From ^[15], if $p(\cdot), q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$, and $p(\cdot)\leq q(\cdot)$, then $A_{1}\subset A_{p(\cdot)}\subset A_{q(\cdot)}$.

$\mathbf{Definition\; 2.6.}$ (^[15]) Let $0 < \beta < n$ and $p_{1}(\cdot), p_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{p_{2}(x)} = \frac{1}{p_{1}(x)}-\frac{\beta}{n}$. A weight $\omega$ is said to be an $A(p_{1}(\cdot), p_{2}(\cdot))$ weight if

$\mathbf{Definition\; 2.7.}$ (^[25]) Let $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ and $\omega\in A_{p(\cdot)}$. The weighted variable exponent Lebesgue space $L^{p(\cdot)}(\omega)$ denotes the set of all complex-valued measurable functions $f$ satisfying

This is a Banach space equipped with the norm

$\mathbf{Definition\; 2.8.}$ (^[1]) Let $\omega$ be a weight on $\mathbb{R}^{n}$, $0\leq \lambda < \infty$, $0 < q < \infty$, $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$, and $\alpha(\cdot): \mathbb{R}^{n}\rightarrow \mathbb{R}$ with $\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})$. The weighted variable exponent Morrey-Herz space $\mathrm{M\dot{K}}_{q, p(\cdot)}^{\alpha(\cdot), \lambda}(\omega)$ is the set of all measurable functions $f$ given by

where

It is noted that $\mathrm{M\dot{K}}_{q, p(\cdot)}^{\alpha(\cdot), 0}(\omega) = \mathrm{\dot{K}}_{q, p(\cdot)}^{\alpha(\cdot)}(\omega)$ is the variable exponent weighted Herz space defined in ^[2].

$\mathbf{Definition\; 2.9.}$ (^[15]) Let $\mathcal{M}$ be the set of all complex-valued measurable functions defined on $\mathbb{R}^{n}$ and $X$ be a linear subspace of $\mathcal{M}$.

(1) The space $X$ is said to be a Banach function space if there exists a function $\|\cdot\|_{X}:\mathcal{M}\rightarrow [0, \infty]$ satisfying the following properties: Let $f, g, f_{j}\in\mathcal{M}\; (j = 1, 2, \ldots)$. Then

(a) $f\in X$ holds if and only if $\|f\|_{X} < \infty$.

(b) Norm property:

ⅰ. Positivity: $\|f\|_{X}\geq 0$.

ⅱ. Strict positivity: $\|f\|_{X} = 0$ holds if and only if $f(x) = 0$ for almost every $x\in \mathbb{R}^{n}$.

ⅲ. Homogeneity: $\|\lambda f\|_{X} = |\lambda|\cdot\|f\|_{X}$ holds for all $\lambda\in\mathbb{C}$.

ⅳ. Triangle inequality: $\|f+g\|_{X}\leq \|f\|_{X}+\|g\|_{X}$.

(c) Symmetry: $\|f\|_{X} = \||f|\|_{X}$.

(d) Lattice property: If $0\leq g(x)\leq f(x)$ for almost every $x\in \mathbb{R}^{n}$, then $\|g\|_{X}\leq\|f\|_{X}$.

(e) Fatou property: If $0\leq f_{j}(x)\leq f_{j+1}(x)$ for all $j$, and $f_{j}(x)\rightarrow f(x)$ as $j\rightarrow \infty$ for almost every $x\in \mathbb{R}^{n}$, then $\lim\limits_{j\rightarrow \infty}\|f_{j}\|_{X} = \|f\|_{X}$.

(f) For every measurable set $F\subset \mathbb{R}^{n}$ such that $|F| < \infty$, $\|\chi_{F}\|_{X}$ is finite. Additionally, there exists a constant $C_{F} > 0$ depending only on $F$ so that $\int_{F}|h(x)|\mathrm{d}x\leq C_{F}\|h\|_{X}$ holds for all $h\in X$.

(2) Suppose that $X$ is a Banach function space equipped with a norm $\|\cdot\|_{X}$. The associated space $X^{\prime}$ is defined by

where

2.2.   Some lemmas that need to be used in this paper

$\mathbf{Lemma\; 2.1.}$ (^[3]) Let $X$ be a Banach function space, and then we have the following:

(ⅰ) The associated space $X^{\prime}$ is also a Banach function space.

(ⅱ) $\|\cdot\|_{(X^{\prime})^{\prime}}$ and $\|\cdot\|_{X}$ are equivalent.

(ⅲ) If $g\in X$ and $f\in X^{\prime}$, then

is the generalized Hölder inequality.

$\mathbf{Lemma\; 2.2.}$ (^[15]) If $X$ is a Banach function space, then we have, for all balls $B$,

$\mathbf{Lemma\; 2.3.}$ (^[16]) Let $X$ be a Banach function space. Suppose that the Hardy-Littlewood maximal operator $M$ is weakly bounded on $X$, that is,

is true for all $f\in X$ and all $\lambda > 0$. Then, we have

$\mathbf{Lemma\; 2.4.}$ (^[15]) Given a function $W$ such that $0 < W(x) < \infty$ for almost every $x\in \mathbb{R}^{n}$, $W\in X_{\mathrm{loc}}(\mathbb{R}^{n})$ and $W^{-1}\in (X^{\prime})_{\mathrm{loc}}(\mathbb{R}^{n})$,

(ⅰ) $X(\mathbb{R}^{n}, W)$ is Banach function space equipped with the norm

where

(ⅱ) The associated space $X^{\prime}(\mathbb{R}^{n}, W^{-1})$ of $X(\mathbb{R}^{n}, W)$ is also a Banach function space.

$\mathbf{Lemma\; 2.5.}$ (^[15]) Let $X$ be a Banach function space and $M$ be bounded on $X^{\prime}$. Then, there exists a constant $\delta\in(0, 1)$ for all $B\subset \mathbb{R}^{n}$ and $E\subset B$,

The paper ^[19] shows that $L^{p(\cdot)}(\mathbb{R}^{n})$ is a Banach function space and the associated space $L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})$ with equivalent norm.

$\mathbf{Remark\; 2.6.}$ (^[1]) Let $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})$, and by comparing the $L^{p(\cdot)}(\omega^{p(\cdot)})$ and $L^{p^{\prime}(\cdot)}(\omega^{-p^{\prime}(\cdot)})$ with the definition of $X(\mathbb{R}^{n}, W)$, we have the following:

(1) If we take $W = \omega$ and $X = L^{p(\cdot)}(\mathbb{R}^{n})$, then we get $L^{p(\cdot)}(\mathbb{R}^{n}, \omega) = L^{p(\cdot)}(\omega^{p(\cdot)})$.

(2) If we consider $W = \omega^{-1}$ and $X = L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})$, then we get $L^{p^{\prime}(\cdot)}(\mathbb{R}^{n}, \omega^{-1}) = L^{p^{\prime}(\cdot)}(\omega^{-p^{\prime}(\cdot)})$.

By virtue of Lemma 2.4, we get

$\mathbf{Lemma\; 2.7.}$ (^[17]) Let $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\cap \mathcal{C}^{\mathrm{log}}(\mathbb{R}^{n})$ be a log-Hölder continuous function both at infinity and at origin, if $\omega^{p_{_{2}}(\cdot)}\in A_{p_{_{2}}(\cdot)}$ implies $\omega^{-p^{\prime}_{_{2}}(\cdot)}\in A_{p^{\prime}_{_{2}}(\cdot)}$. Thus, the Hardy-Littlewood operator is bounded on $L^{p^{\prime}_{_{2}}(\cdot)}(\omega^{-p^{\prime}_{_{2}}(\cdot)})$, and there exist constants $\delta_{1}, \delta_{2}\in (0, 1)$ such that

and

for all balls $B\subset \mathbb{R}^{n}$ and all measurable sets $E\subset B$.

$\mathbf{Lemma\; 2.8.}$ (^[15]) Let $p_{1}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\cap \mathcal{C}^{\mathrm{log}}(\mathbb{R}^{n})$ and $0 < \beta < \frac{n}{p_{1}^{+}}$. Define $p_{2}(\cdot)$ by $\frac{1}{p_{1}(x)}-\frac{1}{p_{2}(\cdot)} = \frac{\beta}{n}$. If $\omega\in A(p_{1}(\cdot), p_{2}(\cdot))$, then $I_{\beta}$ is bounded from $L^{p_{1}(\cdot)}(\omega^{p_{1}(\cdot)})$ to $L^{p_{2}(\cdot)}(\omega^{p_{2}(\cdot)})$.

$\mathbf{Lemma\; 2.9.}$ ([24, Corollary 3.11]) Let $b\in \mathrm{BMO}(\mathbb{R}^{n}), m\in \mathbb{N}$, and $k, j\in \mathbb{Z}$ with $k > j$. Then, we have

3.   Main results and their proofs

$\mathbf{Proposition\; 3.1.}$ (^[12] Let $q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$, $0 < p < \infty$, and $0\leq \lambda < \infty$. If $\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\cap \mathcal{C}^{\mathrm{log}}(\mathbb{R}^{n})$, then

$\mathbf{Theorem\; 3.1.}$ Let $0 < q_{1}\leq q_{2} < \infty$, $p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\cap \mathcal{C}^{\mathrm{log}}(\mathbb{R}^{n})$ and $p_{1}(\cdot)$ be such that $\frac{1}{p_{2}(\cdot)} = \frac{1}{p_{1}(\cdot)}-\frac{\beta}{n}$. Also, let $\omega^{p_{2}(\cdot)}\in A_{1}$, $b\in \mathrm{BMO}(\mathbb{R}^{n})$, $\lambda > 0$ and $\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\cap \mathcal{C}^{\mathrm{log}}(\mathbb{R}^{n})$ be log-Hölder continuous at the origin, with $\alpha(0)\leq \alpha(\infty) < \lambda+n\delta_{2}-\beta$, where $\delta_{2}\in(0, 1)$ is the constant appearing in (2.13). Then,

Proof. For arbitrary $f\in \mathrm{M\dot{K}}_{q_{1}, p_{1}(\cdot)}^{\alpha(\cdot), \lambda}(\omega^{p_{1}(\cdot)})$, let $f_{j} = f\cdot\chi_{j} = f\cdot\chi_{A_{j}}$ for every $j\in \mathbb{Z}$, and then

By the inequality of $C_{p}$, it is not difficult to see that

For $E_{1}$, by the generalized Hölder inequality, we have

By taking the $(L^{p_{2}(\cdot)}(\omega^{p_{2}(\cdot)}))$-norm on both sides of (3.4) and using (2.15) of Lemma 2.9, we get

For $E_{2}$, by the generalized Hölder inequality, we have

By taking the $(L^{p_{2}(\cdot)}(\omega^{p_{2}(\cdot)}))$-norm on both sides of (3.6) and using (2.14) of Lemma 2.9, we get

Hence, from inequalities (3.3), (3.5) and (3.7), we get

By virtue of Lemma 2.5, we have

Note that $\|\chi_{j}\|_{L^{p_{2}(\cdot)}(\omega^{p_{2}(\cdot)})} \leq\|\chi_{B_{j}}\|_{L^{p_{2}(\cdot)}(\omega^{p_{2}(\cdot)})}$ and $\chi_{B_{j}}(x)\lesssim 2^{-j\beta}I_{\beta}(\chi_{B_{j}})$ (see [18, p. 350]). By applying (2.8), (3.9) and Lemma 2.8, we obtain

By virtue of (2.7) and (2.8), combining (2.13) and (3.10), we have

Hence by virtue of (3.8) and (3.11), we have

In order to estimate $\|f_{j}\|_{L^{p_{1}(\cdot)}(\omega^{p_{1}(\cdot)})}$, we consider two cases as below.

Case 1: For $j < 0$, we get

Case 2: For $j\geq0$, we get

Now, by virtue of the condition $q_{1}\leq q_{2}$ and the definition of weighted variable exponent Morrey-Herz space along with the use of Proposition 3.1, we get

where

First, we estimate $J_{1}$. Since $\alpha(0)\leq \alpha(\infty) < n\delta_{2}+\lambda-\beta$, combining (3.12) and (3.13), we get

The estimate of $J_{2}$ is similar to that of $J_{1}$.

Lastly, we estimate $J_{3}$. Since $\alpha(0)\leq \alpha(\infty) < n\delta_{2}+\lambda-\beta$, combining (3.12) and (3.14), we get

The desired result is obtained by combining the estimates of $J_{1}$, $J_{2}$ and $J_{3}$.

$\mathbf{Theorem\; 3.2.}$ Let $0 < q_{1}\leq q_{2} < \infty$, $p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\cap \mathcal{C}^{\mathrm{log}}(\mathbb{R}^{n})$ and $p_{1}(\cdot)$ be such that $\frac{1}{p_{2}(\cdot)} = \frac{1}{p_{1}(\cdot)}-\frac{\beta}{n}$. Also, let $\omega^{p_{2}(\cdot)}\in A_{1}$, $b\in \mathrm{BMO}(\mathbb{R}^{n})$, $\lambda > 0$ and $\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\cap \mathcal{C}^{\mathrm{log}}(\mathbb{R}^{n})$ be log-Hölder continuous at the origin, with $\lambda-n\delta_{1} < \alpha(0)\leq\alpha(\infty)$, where $\delta_{1}\in(0, 1)$ is the constant appearing in (2.12). Then,

Proof. From an application of the inequality of $C_{p}$, it is not difficult to see that

For $F_{1}$, by the generalized Hölder inequality, we have

By taking the $(L^{p_{2}(\cdot)}(\omega^{p_{2}(\cdot)}))$-norm on both sides of (3.18) and using (2.15) of Lemma 2.9, we get

For $F_{2}$, by the generalized Hölder inequality, we have

By taking the $(L^{p_{2}(\cdot)}(\omega^{p_{2}(\cdot)}))$-norm on both sides of (3.20) and using (2.15) of Lemma 2.9, we get

Hence, from inequalities (3.17), (3.19) and (3.21), we get

On the other hand, by (2.7) and (2.8), combining (2.12) and (3.10), we have

Hence combining (3.22) and (3.23), we obtain

Next, by virtue of the condition $q_{1}\leq q_{2}$ and the definition of weighted variable exponent Morrey-Herz space along with the use of Proposition 3.1, we get

where

First, we estimate $Y_{1}$. Since $\lambda-n\delta_{1} < \alpha(0)\leq \alpha(\infty)$, combining (3.24) and (3.13), we get

The estimate of $Y_{2}$ is similar to that of $Y_{1}$.

Lastly, we estimate $Y_{3}$. Since $\lambda-n\delta_{1} < \alpha(0)\leq \alpha(\infty)$, combining (3.24) and (3.14), we get

The desired result is obtained by combining the estimates of $Y_{1}$, $Y_{2}$ and $Y_{3}$.

4.   Conclusions

This paper considers the boundedness for $m$th order commutators of $n-$dimensional fractional Hardy operators $\mathcal{H}^{m}_{_{\beta, b}}$ and adjoint operators $\mathcal{H}_{\beta, b}^{\ast m}$ on weighted variable exponent Morrey-Herz spaces $\mathrm{M\dot{K}}_{q, p(\cdot)}^{\alpha(\cdot), \lambda}(\omega)$. When $m = 0$, our main result holds on weighted variable exponent Morrey-Herz space for fractional Hardy operators and generalizes the result of Asim et al. in [1, Theorems 4.2 and 4.3]. When $m = 1$, our main result holds on weighted variable exponent Morrey-Herz space for commutators of the fractional Hardy operators and generalizes the result of Hussain et al. in [12, Theorems 18 and 19].

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Grant No. 12161071), Qinghai Minzu University campus level project (Nos. 23GH29, 23GCC10).

Conflict of interest

All authors declare that they have no conflicts of interest.