Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces

Abstract

In this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operators, on grand variable Herz spaces under some proper assumptions. To prove the boundedness results, we use Holder-type and Minkowski inequalities. In the proof of the main result, we use different techniques. We divide the summation into different terms and estimate each term under different conditions. Then, by combining the estimates, we prove that the rough Riesz potential operator of variable order and the fractional Hardy operators are bounded on grand variable Herz spaces. It is easy to show that the rough Riesz potential operator of variable order generalizes the Riesz potential operator and that the fractional Hardy operators are generalized versions of simple Hardy operators. So, our results extend some previous results to the more generalized setting of grand variable Herz spaces.

1. Introduction

Function spaces with variable exponents have drawn increasing attention in recent years because of their applications across a wide range of domains (see, for example, [1]). Lebesgue spaces with variable exponents, as well as Orlicz, Sobolev, and Lorentz spaces, have been the subject of substantial theoretical research over time. Our understanding of their possible applications has improved as a result of this effort, which has also helped define their properties and characteristics. A few significant sources in this field include publications like [2,3,4,5,6]. Herz spaces stand out in harmonic analysis because their norm takes into account both local and global information. It is well known that Beurling [7] and Herz [8] introduced some new spaces that characterize certain properties of functions. These spaces were called Herz spaces. The homogeneous space and non-homogeneous Herz space are denoted by ${\dot{K}}_q^{\alpha ,p}\left({\mathbb{R}}^n\right)$ and $K_q^{\alpha ,p}\left({\mathbb{R}}^n\right)$, respectively. The exponents $\alpha$, p, and q defining these spaces are constant. The first generalization of variable-exponent Herz spaces was introduced in [9]. The most general findings, which allowed the exponent $\alpha$ to vary, were presented in [10].

More recently, ongoing Herz spaces were defined using variable parameters in [11]. Furthermore, it was shown that sublinear operators on these continuous Herz spaces are bounded. Furthermore, [12,13] provided confirmation of the boundedness of other operators, including the Riesz potential operator and the Marcinkiewicz integral. Charles Morrey created Morrey spaces, a kind of function space, in 1938 as a means of examining the regularity characteristics of solutions to specific kinds of partial differential equations, especially those that arise in the calculus of variations. Morrey spaces provide a more refined notion of local regularity than Lebesgue spaces, which is useful in many different areas of mathematics, including harmonic analysis and PDE theory [14].

In [15], the authors explored some properties of generalized Morrey spaces and established some relations with Fefferman’s inequality, the Poisson equation, the p-Laplacian, and the unique continuation principle. The latter is fundamental and of independent interest in the theory of partial differential equations. Applications regarding the vanishing of solutions are often associated with, for instance, solvability, stability, the geometrical properties of solutions, and so on. For nonlinear diffusion equations driven by the $p(·)$-Laplacian with variable exponents in space, see [16].

The generalized forms of Herz spaces are called grand variable Herz spaces. Some papers that have examined the boundedness of certain operators on these spaces are [17,18]. Additionally, grand variable Herz spaces were generalized into grand variable Herz–Morrey spaces (see [19] for additional findings). The behavior of fractional integral operators on function spaces can be better understood by examining their higher-order commutators. It is easy to show that grand variable Herz spaces generalize classical Herz spaces and variable Herz spaces.

The past few decades have witnessed an increasing number of significant developments in the research of elliptic equations involving Hardy-type operators due to their applications across various scientific disciplines. The boundedness of operators on function spaces is an important class of problems that has gained significant attention around the globe. Among the Hardy-type operators are classical operators, the exploration of which began with Hardy’s proposal of the one-dimensional Hardy operator [20]. Over time, researchers have extended the definition of Hardy-type operators to higher dimensions. In [21], the authors defined the n-dimensional Hardy operator. In [22], the authors made contributions by presenting the n-dimensional fractional Hardy operator and studying its commutators. Fractional Hardy operators are a generalization of classical Hardy operators, allowing for more flexibility in the types of functions they can handle. In [23], the authors continued their investigation by studying Hardy operators with rough kernels and central $\mathrm{BMO}$ estimates for the commutators of the n-dimensional rough Hardy operator.

For a locally integrable function g on ${\mathbb{R}}^n$, the Hardy operators are given by

$$\mathcal{H}g\left(z\right):={\displaystyle \frac{1}{{\left|z\right|}^n}}{\int }_{\left|y\right|<\left|z\right|}g\left(y\right)dy,{\mathcal{H}}^{\ast }g\left(z\right):={\int }_{\left|y\right|\ge \left|z\right|}{\displaystyle \frac{g\left(y\right)}{{\left|y\right|}^n}}dy,$$

where $z\in {\mathbb{R}}^n\setminus \left\{0\right\}.$

Let g be a locally integrable function on ${\mathbb{R}}^n$. Then, the fractional Hardy operators are defined as

$${\mathcal{H}}_{\beta }g\left(z\right):={\displaystyle \frac{1}{{\left|z\right|}^{n-\beta }}}{\int }_{\left|x\right|<\left|z\right|}f\left(x\right)dx,{\mathcal{H}}_{\beta }^{\ast }g\left(z\right):={\int }_{\left|x\right|\ge \left|z\right|}{\displaystyle \frac{f\left(x\right)}{{\left|x\right|}^{n-\beta }}}dx,$$

where $z\in {\mathbb{R}}^n\setminus \left\{0\right\}.$

When $\beta =1$, a fractional Hardy operator becomes a Hardy operator.

Let g be a locally integrable function on ${\mathbb{R}}^n$ denoted by $L_{\mathrm{loc}}^1\left({\mathbb{R}}^n\right)$, and let $K_{\kappa }\left(y\right):={\left|y\right|}^{\kappa -n}$ denote the Riesz kernel. Let $n\in \mathbb{N}$ with $0<\kappa <n$. Then, the Riesz potential operator $I_{\kappa }$ for the convolution of a function g is given by

$$I_{\kappa }g\left(y\right):={\int }_{{\mathbb{R}}^n}{\displaystyle \frac{g\left(z\right)}{{|y-z|}^{n-\kappa }}}dm\left(z\right),y\in {\mathbb{R}}^n,$$

where m is the Lebesgue measure on ${\mathbb{R}}^n$ and $g\in L^p\left({\mathbb{R}}^n\right)$ with $1\le p<{\displaystyle \frac{n}{\kappa }}$.

It is easy to see that when $\kappa =2\ne n$, the operator $I_{\kappa }$ is referred to as a Newtonian potential, which is commonly used to describe the electrostatic potential generated by a charge distribution in physics or the potential energy distribution of a system of point masses in classical mechanics.

Many researchers have generalized Herz spaces to various other function spaces and settings to explore more complex structures and solve broader classes of problems. There are some common generalized versions of classical Herz spaces, including Herz spaces with variable exponents, Herz–Morrey spaces, grand variable Herz spaces, grand variable Herz–Morrey spaces, continual Herz–Morrey spaces, grand weighted Herz spaces, and grand weighted Herz–Morrey spaces (see [24,25,26,27,28,29,30]). Many researchers have worked in grand variable Herz spaces. Grand variable Herz spaces are more generalized than classical Herz spaces and Herz spaces with variable exponents. Under some conditions, grand variable Herz spaces become classical Herz spaces and Herz spaces with variable exponents. The idea of grand variable Herz spaces was introduced in [31], and the authors proved the boundedness of sublinear operators on these spaces under some assumptions on the exponents. For the boundedness of the Calderón–Zygmund singular integral operator on grand variable Herz spaces, see [32]. In this paper, we also use the variable $\alpha$ to prove the boundedness results. In this paper, we use the rough Riesz potential operator of variable order, which is a more generalized operator, as well as generalize the Riesz potential operator, the Riesz potential operator of variable order, and the rough Riesz potential operator. So, these results are more generalized than previously published results. Additionally, we prove the boundedness of fractional Hardy operators on grand variable Herz spaces, which are more generalized than previous results, because under some conditions, a fractional Hardy operator becomes a Hardy operator. Our findings not only consolidate and build on prior discoveries but also offer novel applications to the regularity solutions of some elliptic PDEs with smooth boundaries.

The investigation of fractional Hardy operators belongs to one of the hot topics in the area of PDEs because of its wide-ranging interest to various fields in mathematics and physics. For instance, it is motivated by physical models related to the relativistic Schrödinger operator with Coulomb potential (see [33,34]) and by the study of Hardy inequalities and Hardy–Lieb–Thirring inequalities (see, e.g., [35,36,37]).

Motivated by the above results, the purpose of this study is to demonstrate that the Riesz potential operator of variable order on grand variable Herz spaces is bounded. In order to arrange and clearly explain the information, this research paper is divided into different sections. In addition to providing an introduction, the preliminaries section is crucial for establishing the definitions and background information that will be used throughout this paper. The boundedness of the Riesz potential operator of variable order on grand variable Herz spaces will be the main topic of the final section.

2. Preliminaries

Definition 1.

Consider a measurable set $\mathcal{A}$ in ${\mathbb{R}}^n$ and a measurable function $p(·):\mathcal{A}\to [1,\infty )$. We define the following spaces:

(a) 

The Lebesgue space with variable exponent $L^{p(·)}\left(\mathcal{A}\right)$ is defined as

$$L^{p(·)}\left(\mathcal{A}\right)=\left\{f\mathit{is}\mathit{measurable}:{\int }_{\mathcal{A}}{\left({\displaystyle \frac{\left|f\right(z\left)\right|}{\Gamma }}\right)}^{p\left(z\right)}dz<\infty \mathit{where}\Gamma \mathit{is}\mathit{a}\mathit{constant}\right\}.$$

The norm in $L^{p(·)}\left(\mathcal{A}\right)$ can be defined as

$${\parallel f\parallel }_{L^{p(·)}\left(\mathcal{A}\right)}=inf\left\{\Gamma >0:{\int }_H{\left({\displaystyle \frac{\left|f\right(z\left)\right|}{\Gamma }}\right)}^{p\left(z\right)}dz\le 1\right\}.$$

(b) 

The space $L_{\mathrm{loc}}^{p(·)}\left(\mathcal{A}\right)$ can be defined as

$$L_{\mathrm{loc}}^{p(·)}\left(\mathcal{A}\right):=\left\{f:f\in L^{p(·)}\left(K\right)\mathit{for}\mathit{all}\mathit{compact}\mathit{subsets}K\subset \mathcal{A}\right\}.$$

We use the following notations in this paper:

(i)

Suppose that $f\in L_{\mathrm{loc}}^1\left(\mathcal{A}\right)$. Then, the Hardy–Littlewood maximal operator $\mathfrak{M}$ can be given by

$$\mathfrak{M}f\left(z\right):=\underset{0<r}{sup}{\displaystyle \frac{1}{r^n}}\underset{B(i,r)}{\int }\left|f\left(z\right)\right|dz(z\in \mathcal{A}),$$

where $B(i,r):=\{x\in \mathcal{A}:|i-x|<r\}$.

(ii)

Let $p(·)$ be a measurable function and suppose that

$$1\le p_{-}\left(\mathcal{A}\right)\le p_{+}\left(\mathcal{A}\right)<\infty ,$$

(1)

where

$$p_{-}:=ess\underset{x\in \mathcal{A}}{inf}p\left(x\right),p_{+}:=ess\underset{x\in \mathcal{A}}{sup}p\left(x\right).$$

The set $\mathcal{P}\left(\mathcal{A}\right)$ consists of all $p(·)$ such that $p_{-}>1$ and $p_{+}<\infty$.

(iii)

Let $p\in \mathcal{P}\left(\mathcal{A}\right)$. Then, ${\mathfrak{B}}^{log}={\mathfrak{B}}^{log}\left(\mathcal{A}\right)$ is the class of functions satisfying (1) and the log-condition given by

$$|p\left(x_1\right)-p\left(x_2\right)|\le {\displaystyle \frac{C\left(p\right)}{-ln|x_1-x_2|}},|x_1-x_2|\le {\displaystyle \frac{1}{2}},x_1,x_2\in \mathcal{A}.$$

(2)

(iv)

For an unbounded set $\mathcal{A}$ in ${\mathbb{R}}^n$, ${\mathfrak{B}}_{\infty }\left(\mathcal{A}\right)$ is the subset of $\mathcal{P}\left(\mathcal{A}\right)$ with values in $[1,\infty )$ satisfying the following condition:

$$|p\left(x\right)-p_{\infty }|\le {\displaystyle \frac{C}{ln(e+|x\left|\right)}},$$

(3)

where $p_{\infty }\in (1,\infty )$. ${\mathfrak{B}}_{0,\infty }\left(\mathcal{A}\right)$ is the subset of $\mathcal{P}\left(\mathcal{A}\right)$ satisfying the condition (3) and the condition

$$|p\left(x\right)-p_0|\le {\displaystyle \frac{C}{ln\left|x\right|}},\left|x\right|\le {\displaystyle \frac{1}{2}},$$

(4)

for homogeneous Herz spaces.

(v)

$p(·)\in \mathcal{A}$. Then, $\mathcal{B}\left(\mathcal{A}\right)$ is the collection of $p(·)\in \mathcal{P}\left(\mathcal{A}\right)$ for which $\mathfrak{M}$ is bounded on $L^{p(·)}\left(\mathcal{A}\right)$.

(vi)

${\mathcal{S}}_t=\mathcal{S}(0,2^t)=\{x\in {\mathbb{R}}^n:\left|x\right|<2^t\}$ for all $l\in \mathbb{Z}.$ $F_t={\mathcal{S}}_t\setminus {\mathcal{S}}_{t-1}$, ${\mathbf{1}}_{F_t}={\mathbf{1}}_t$.

C is a constant; its value can be changed from line to line.

Lemma 1

([11]). Let $\mathcal{S}>1$, and let $p\in {\mathfrak{B}}_{0,\infty }\left({\mathbb{R}}^n\right)$. Then,

$${\displaystyle \frac{1}{t_0}}{s}^{{\displaystyle \frac{n}{p\left(0\right)}}}\le {\parallel {\mathbf{1}}_{R_{s,D_s}}\parallel }_{p(·)}\le t_0{s}^{{\displaystyle \frac{n}{p\left(0\right)}}},\mathit{for}0<s\le 1$$

(5)

and

$${\displaystyle \frac{1}{t_{\infty }}}{s}^{{\displaystyle \frac{n}{p_{\infty }}}}\le {\parallel {\mathbf{1}}_{R_{s,{\mathcal{S}}_s}}\parallel }_{p(·)}\le t_{\infty }{s}^{{\displaystyle \frac{n}{p_{\infty }}}},\mathit{for}s\ge 1,$$

(6)

respectively, where $t_0\ge 1$ and $t_{\infty }\ge 1$ depend on $\mathcal{S}$ but are independent of s.

Lemma 2

([38]). [Generalized Hölder inequality] Consider a measurable subset H such that $H\subseteq {\mathbb{R}}^n$, and $1\le p_{-}\left(\mathcal{A}\right)\le p_{+}\left(\mathcal{A}\right)\le \infty$. Then,

$${\parallel gf\parallel }_{L^{r(·)}\left(\mathcal{A}\right)}\le {\parallel g\parallel }_{L^{p(·)}\left(\mathcal{A}\right)}{\parallel f\parallel }_{L^{q(·)}\left(\mathcal{A}\right)}$$

holds, where $g\in L^{p(·)}\left(\mathcal{A}\right)$, $f\in L^{q(·)}\left(\mathcal{A}\right)$, and ${\displaystyle \frac{1}{r\left(z\right)}}={\displaystyle \frac{1}{p\left(z\right)}}+{\displaystyle \frac{1}{q\left(z\right)}}$ for every $z\in H$.

Definition 2.

Let $a(·)\in L^{\infty }\left({\mathbb{R}}^n\right)$, $u\in [1,\infty )$, $v:{\mathbb{R}}^n\to [1,\infty )$, $\theta >0$. The grand variable Herz space ${\dot{K}}_{v(·)}^{r_0(·),u),\theta }$ is defined as

$${\dot{K}}_{v(·)}^{r_0(·),u),\theta }=\left\{g\in L_{\mathrm{loc}}^{v(·)}({\mathbb{R}}^n\setminus \left\{0\right\}):{\parallel g\parallel }_{{\dot{K}}_{v(·)}^{r_0(·),u),\theta }}<\infty \right\},$$

where

$$\begin{array}{cc}\hfill {\parallel g\parallel }_{{\dot{K}}_{v(·)}^{r_0(·),u),\theta }}& =\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{k\in \mathbb{Z}}{\parallel 2^{k{r}_0(·)}g{\mathbf{1}}_k\parallel }_{L^{v(·)}}^{u(1+p_0)}\right)}^{{\displaystyle \frac{1}{u(1+p_0)}}}\hfill \\ \hfill =& \underset{p_0>0}{sup}{p}_0^{{\displaystyle \frac{\theta }{u(1+p_0)}}}{\parallel g\parallel }_{{\dot{K}}_{v(·)}^{r_0(·),u(1+p_0)}}.\hfill \end{array}$$

Let ${\mathbb{S}}^{n-1}$ denote the unit sphere in ${\mathbb{R}}^n$ with the normalized Lebesgue measure. $\Phi \in L^r\left({\mathbb{S}}^{n-1}\right)$ is a function of degree zero, which is homogeneous, such that

$$\underset{{\mathbb{S}}^{n-1}}{\int }\Phi \left(z^{\prime }\right)d\Phi \left(z^{\prime }\right)=0,$$

(7)

where $z^{\prime }=z/\left|z\right|$ and z is not zero.

We consider the rough Riesz potential operator of variable order

$$I^{\zeta \left(x\right)}f\left(x\right)={\int }_{{\mathbb{R}}^n}{\displaystyle \frac{|\Phi (x-y\left)\right|f\left(y\right)}{{|x-y|}^{n-\zeta \left(x\right)}}}dy,0<\zeta \left(x\right)<n.$$

(8)

It is easy to see that when $\Phi (x-y)=1$, the rough Riesz potential operator of variable order becomes the Riesz potential operator of variable order. For $\Phi (x-y)=1$ and $\zeta \left(x\right)=$ constant, the rough Riesz potential operator of variable order is a simple Riesz potential operator.

Lemma 3

([39]). If $a>0$, $s\in [1,\infty ]$, $0<d\le s$, and $-m+(m-1)ds<u<\infty$, then

$${\left(\underset{\left|y\right|\le a\left|x\right|}{\int }{\left|y\right|}^u{|\Phi (x-y)|}^{d}dy\right)}^{1/d}\le {\left|x\right|}^{(u+m)/d}{\parallel \Phi \parallel }_{L^s\left({\mathbb{S}}^{m-1}\right)}.$$

3. Sobolev-Type Theorem for Grand Variable Herz Spaces

Theorem 1.

Let $1<ϵ<\infty$, $r_0,q_2\in {\mathfrak{P}}_{0,\infty }\left({\mathbb{R}}^n\right)$, $1/q_1\left(x\right)-1/q_2\left(x\right)=\zeta (·)/n$. If $0<u<min\{1/{\left(q_1\right)}_{+},1{\left(q_2^{\prime }\right)}_{+}\}$, $0<\zeta (·)<n$, and

$${\displaystyle \frac{-n}{q_{1\infty }}}-v-{\displaystyle \frac{n}{s}}<r_{0\infty }<{\displaystyle \frac{n}{q_{1\infty }^{\prime }}}-v-{\displaystyle \frac{n}{s}},{\displaystyle \frac{-n}{q_1\left(0\right)}}-v-{\displaystyle \frac{n}{s}}<r_0\left(0\right)<{\displaystyle \frac{n}{q_1^{\prime }\left(0\right)}}-v-{\displaystyle \frac{n}{s}}.$$

Then,

$$\parallel (1+{\left|x\right|)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}{\left(f\right)\parallel }_{{\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}\le C{\parallel f\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.$$

Proof. 

Let $f\in {\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)$, and let $f\left(x\right)={\sum }_{l=-\infty }^{\infty }f\left(x\right){\mathbf{1}}_l\left(x\right)={\sum }_{l=-\infty }^{\infty }{f}_l\left(x\right)$. Then, we have

$$\begin{array}{cc}\hfill \parallel (1+{\left|x\right|)}^{-\lambda \left(x\right)}& I^{\zeta (·)}{\left(f\right)\parallel }_{{\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ \hfill =& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t\in \mathbb{Z}}{2}^{t{r}_0(·)ϵ(1+p_0)}\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}{\left(f\right)\parallel }_{q_2(·)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t\in \mathbb{Z}}{2}^{t{r}_0(·)ϵ(1+p_0)}\left(\sum _{l=-\infty }^{\infty }\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}^{ϵ(1+p_0)}\right)\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t\in \mathbb{Z}}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=-\infty }^t\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill +& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t\in \mathbb{Z}}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=t+1}^{\infty }\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill =:& E_1+E_2.\hfill \end{array}$$

For $E_1$, let $k\in \mathbb{Z}$ with $l\le k$, $x\in F_t$, and $y\in F_l$, then $|x-y|\sim \left|x\right|\sim 2^k$, we obtain

$$\begin{array}{cc}\hfill |I^{\zeta (·)}\left(f{\mathbf{1}}_l\right)\left(x\right)& \le C{\int }_{F_l}{|x-y|}^{\zeta \left(x\right)-n}|\Phi (x-y)|\left|f\left(y\right)\right|dy\hfill \\ \hfill & \le C{2}^{-tn}{\int }_{F_l}{\left|x\right|}^{\zeta \left(x\right)}|\Phi (x-y)|\left|f\left(y\right)\right|dy\hfill \\ \hfill & \le C{2}^{-tn}{\left|x\right|}^{\zeta \left(x\right)}\parallel f{\mathbf{1}}_l{\parallel }_{L^{q_1(·)}}{\parallel \Phi (x-·){\mathbf{1}}_l(·)\parallel }_{L^{q_1^{\prime }(·)}}.\hfill \end{array}$$

It is known (see, e.g., [40]) that

$$\begin{array}{cc}\hfill I^{\zeta (·)}\left({\mathbf{1}}_{{\mathcal{S}}_t}\right)\left(x\right)& \ge I^{\zeta (·)}\left({\mathbf{1}}_{{\mathcal{S}}_t}\right)\left(x\right).\left({\mathbf{1}}_{{\mathcal{S}}_t}\right)\left(x\right)\hfill \\ \hfill & ={\int }_{{\mathcal{S}}_t}{\displaystyle \frac{1}{{|x-y|}^{\zeta \left(x\right)-n}}}dy.{\mathbf{1}}_{{\mathcal{S}}_t}\left(x\right)\hfill \\ \hfill & \ge {C\left|x\right|}^{\zeta \left(x\right)}.{\mathbf{1}}_{{\mathcal{S}}_t}\left(x\right)\hfill \\ \hfill & \ge {C\left|x\right|}^{\zeta \left(x\right)}.{\mathbf{1}}_k\left(x\right).\hfill \end{array}$$

Then, one can obtain

$$\begin{array}{cc}\hfill & \parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\hfill \\ \hfill & \le C{2}^{-tn}\parallel f{\mathbf{1}}_l{\parallel }_{L^{q_1(·)}}\parallel \Phi (x-·){\mathbf{1}}_l{(·)\parallel }_{L^{q_1^{\prime }(·)}}\parallel (1+{\left|x\right|)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left({\mathbf{1}}_{{\mathcal{S}}_t}\right){\parallel }_{q_2(·)}\hfill \\ \hfill & \le C{2}^{-tn}\parallel f{\mathbf{1}}_l{\parallel }_{L^{q_1(·)}}\parallel \Phi (x-·){\mathbf{1}}_l{(·)\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_{{\mathcal{S}}_t}\parallel }_{q_1(·)}.\hfill \end{array}$$

By Lemma 3 and by virtue of the generalized Hölder inequality, we obtain

$$\begin{array}{cc}\hfill \parallel \Phi (x-·){\mathbf{1}}_l{(·)\parallel }_{L^{q_1^{\prime }(·)}}\le & \parallel \Phi (x-·){\mathbf{1}}_l{(·)\parallel }_{L^s\left({\mathbb{R}}^n\right)}{\parallel {\mathbf{1}}_l(·)\parallel }_{L^{q_1(·)}}\hfill \\ \hfill \le & 2^{-lv}{\left(\underset{2^{l-1}<\left|x\right|<2^l}{\int }{|\Phi (x-y)|}^s{\left|x\right|}^{sv}dx\right)}^{1/s}{\parallel {\mathbf{1}}_{{\mathcal{S}}_l}\parallel }_{L^{q_1(·)}}\hfill \\ \hfill \le & 2^{-lv}{2}^{t(v+{\displaystyle \frac{n}{s}})}{\parallel \Phi \parallel }_{L^s\left({\mathbb{S}}^{n-1}\right)}\parallel \parallel {\mathbf{1}}_{{\mathcal{S}}_l}{\parallel }_{L^{q_1(·)}}.\hfill \end{array}$$

By Lemma 1, we get

$$2^{-tn}\parallel {\mathbf{1}}_{{\mathcal{S}}_l}{\parallel }_{q_1^{\prime }(·)}{\parallel {\mathbf{1}}_{{\mathcal{S}}_t}\parallel }_{q_1(·)}\le C{2}^{-tn}{2}^{{\displaystyle \frac{tn}{q_1\left(0\right)}}}{2}^{{\displaystyle \frac{ln}{q_1^{\prime }\left(0\right)}}}\le C{2}^{{\displaystyle \frac{(l-t)n}{q_1^{\prime }\left(0\right)}}}.$$

(9)

Then, Minkowski’s inequality yields

$$\begin{array}{cc}\hfill E_1\le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t\in \mathbb{Z}}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=-\infty }^t\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=-\infty }^t\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill +& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=-\infty }^t\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill :=& E_{11}+E_{12}.\hfill \end{array}$$

By applying the above results to $E_{11}$, we obtain

$$\begin{array}{cc}\hfill & E_{11}\le \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=-\infty }^t\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le C& \underset{p_0>0}{sup}\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0\left(0\right)ϵ(1+p_0)}\right.\hfill \\ \hfill & {\left.\times {\left(\sum _{l=-\infty }^t{2}^{-tn}\parallel f{\mathbf{1}}_l{\parallel }_{L^{q_1(·)}}{2}^{-lv}{2}^{t(v+{\displaystyle \frac{n}{s}})}{\parallel \Phi \parallel }_{L^s\left({\mathbb{S}}^{n-1}\right)}\parallel \parallel {\mathbf{1}}_{{\mathcal{S}}_l}{\parallel }_{L^{q_1(·)}}{\parallel {\mathbf{1}}_{{\mathcal{S}}_t}\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le C& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0\left(0\right)ϵ(1+p_0)}{\left(\sum _{l=-\infty }^t{2}^{{\displaystyle \frac{(l-t)n}{q_1^{\prime }\left(0\right)}}}{2}^{-lv}{2}^{t(v+{\displaystyle \frac{n}{s}})}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le C& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0\left(0\right)ϵ(1+p_0)}{\left(\sum _{l=-\infty }^t{2}^{(l-t)({\displaystyle \frac{n}{q_1^{\prime }\left(0\right)}}-v-{\displaystyle \frac{n}{s}})}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}},\hfill \end{array}$$

where $b:={\displaystyle \frac{n}{q_1^{\prime }\left(0\right)}}-r_0\left(0\right)-v-{\displaystyle \frac{n}{s}}>0$. Applying Hölder’s inequality, Fubini’s theorem for series, and $2^{-ϵ(1+p_0)}<2^{-ϵ}$, we get

$$\begin{array}{cc}\hfill E_{11}& \le C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{\left(\sum _{l=-\infty }^t{2}^{r_0\left(0\right)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}{2}^{b(l-t)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & \le C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{\left(\sum _{l=-\infty }^t{2}^{r_0\left(0\right)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}{2}^{b(l-t)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & \le C\underset{p_0>0}{sup}\left[p_0^{\theta }\sum _{t=-\infty }^{-1}\left(\sum _{l=-\infty }^t{2}^{r_0\left(0\right)ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}{2}^{bϵ(1+p_0)(l-t)/2}\right)\right.\hfill \\ \hfill & {\left.\times {\left(\sum _{l=-\infty }^t{2}^{b{\left(ϵ(1+p_0)\right)}^{\prime }(l-t)/2}\right)}^{{\displaystyle \frac{ϵ(1+p_0)}{{\left(ϵ(1+p_0)\right)}^{\prime }}}}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left[p_0^{\theta }\sum _{t=-\infty }^{-1}\sum _{l=-\infty }^t{2}^{r_0\left(0\right)ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}{2}^{bϵ(1+p_0)(l-t)/2}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}[p_0^{\theta }\sum _{l=-\infty }^{-1}{2}^{r_0(·)ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\sum _{k=l}^{-1}{2}^{bϵ(1+p_0)(l-t)/2}{]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{l=-\infty }^{-1}{2}^{r_0\left(0\right)ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\sum _{k=l}^{-1}{2}^{bϵ(1+p_0)(l-t)/2}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{l=-\infty }^{-1}{2}^{r_0\left(0\right)ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill =& C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{l\in \mathbb{Z}}{2}^{r_0(·)ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & {C\parallel f\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Now, for $E_{12}$, using Minkowski’s inequality, we have

$$\begin{array}{cc}\hfill E_{12}\le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=-\infty }^t\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=-\infty }^{-1}\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill +& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=0}^k\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill :=& A_1+A_2.\hfill \end{array}$$

The estimate for $A_2$ follows in a similar manner to $E_{11}$, with $q^{\prime }\left(0\right)$ replaced with $q_{1\infty }^{\prime }$ and using the fact that $b:={\displaystyle \frac{n}{q_{1\infty }^{\prime }}}-r_{0\infty }-v-{\displaystyle \frac{n}{s}}>0$. For $A_1$, by using Lemma 1, we have

$$2^{-tn}\parallel {\mathbf{1}}_{{\mathcal{S}}_t}{\parallel }_{q_1(·)}{\parallel {\mathbf{1}}_{{\mathcal{S}}_l}\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbb{R}}^n\right)}\le C{2}^{-tn}{2}^{{\displaystyle \frac{tn}{q_{1_{\infty }}}}}{2}^{{\displaystyle \frac{ln}{q_1^{\prime }\left(0\right)}}}\le C{2}^{{\displaystyle \frac{-tn}{q_{1_{\infty }}}}}{2}^{{\displaystyle \frac{ln}{q_1^{\prime }\left(0\right)}}},$$

(10)

as $r_{0\infty }-{\displaystyle \frac{n}{q_{1\infty }^{\prime }}}<0$, and using the fact $2^{ka\left(z\right)}\approx 2^{ka(\infty )}$ for $k\ge 0$, we have

$$\begin{array}{cc}\hfill A_1\le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_{0\infty }ϵ(1+p_0)}{\left(\sum _{l=-\infty }^{-1}\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}[p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_{0\infty }ϵ(1+p_0)}{\left(\sum _{l=-\infty }^{-1}{2}^{l({\displaystyle \frac{n}{q_1\left(0\right)}}-v)}{2}^{t(v+{\displaystyle \frac{n}{s}}-{\displaystyle \frac{n}{q_{1\infty }}})}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}{]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}[p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t(v+{\displaystyle \frac{n}{s}}-{\displaystyle \frac{n}{q_{1\infty }}}+r_{0\infty })ϵ(1+p_0)}{\left(\sum _{l=-\infty }^{-1}{2}^{{\displaystyle \frac{ln}{q_1\left(0\right)}}-v}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}{]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }{\left(\sum _{l=-\infty }^{-1}{2}^{{\displaystyle \frac{ln}{q_1\left(0\right)-v}}}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }{\left(\sum _{l=-\infty }^{-1}{2}^{r_0\left(0\right)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}{2}^{l({\displaystyle \frac{n}{q_1\left(0\right)}}-v-{\displaystyle \frac{n}{s}}-r_0\left(0\right)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}.\hfill \end{array}$$

where $d:={\displaystyle \frac{n}{q_{1\infty }}}+r_{0\infty }+v+{\displaystyle \frac{n}{s}}>0$. Then, we use Hölder’s inequality for series and $2^{-ϵ(1+p_0)}<2^{-ϵ}$ to get

$$\begin{array}{cc}\hfill A_1\le & C\underset{p_0>0}{sup}\left[p_0^{\theta }\left(\sum _{l=-\infty }^{-1}{2}^{r_0\left(0\right)lϵ(1+p_0)}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\right)\right.\hfill \\ \hfill & {\left.\times {\left(\sum _{l=-\infty }^{-1}{2}^{({\displaystyle \frac{ln}{q^{\prime }\left(0\right)}}-r_0\left(0\right)-v-{\displaystyle \frac{n}{s}}){\left(ϵ(1+p_0)\right)}^{\prime }}\right)}^{{\displaystyle \frac{ϵ(1+p_0)}{{\left(ϵ(1+p_0)\right)}^{\prime }}}}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\left(\sum _{l\in \mathbb{Z}}{2}^{r_0\left(0\right)lϵ(1+p_0)}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\right)\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & {C\parallel f\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

For $E_2$, let $k\in \mathbb{Z}$ with $l\ge t+2$ a.e. If $x\in F_t$ and $y\in F_l$, then, since $|x-y|\approx \left|y\right|\approx 2^l$, we get

$$\begin{array}{cc}\hfill |I^{\zeta (·)}\left(f{\mathbf{1}}_l\right)\left(x\right)& \le C{\int }_{F_l}{|x-y|}^{\zeta \left(x\right)-n}|\Phi (x-y)|\left|f\left(y\right)\right|dy\hfill \\ \hfill & \le C{2}^{-ln}{\int }_{F_l}{\left|x\right|}^{\zeta \left(x\right)}|\Phi (x-y)|\left|f\left(y\right)\right|dy\hfill \\ \hfill & \le C{2}^{-ln}{\left|x\right|}^{\zeta \left(x\right)}\parallel f{\mathbf{1}}_l{\parallel }_{L^{q_1(·)}}{\parallel \Phi (x-·){\mathbf{1}}_l(·)\parallel }_{L^{q_1^{\prime }(·)}}.\hfill \end{array}$$

It is known (see, e.g., [40]) that

$$\begin{array}{cc}\hfill I^{\zeta (·)}\left({\mathbf{1}}_{{\mathcal{S}}_t}\right)\left(x\right)& \ge I^{\zeta (·)}\left({\mathbf{1}}_{{\mathcal{S}}_t}\right)\left(x\right).\left({\mathbf{1}}_{{\mathcal{S}}_t}\right)\left(x\right)\hfill \\ \hfill & ={\int }_{{\mathcal{S}}_t}{\displaystyle \frac{1}{{|x-y|}^{\zeta \left(x\right)-n}}}dy.{\mathbf{1}}_{{\mathcal{S}}_t}\left(x\right)\hfill \\ \hfill & \ge {C\left|x\right|}^{\zeta \left(x\right)}.{\mathbf{1}}_{{\mathcal{S}}_t}\left(x\right)\hfill \\ \hfill & \ge {C\left|x\right|}^{\zeta \left(x\right)}.{\mathbf{1}}_k\left(x\right).\hfill \end{array}$$

Thus, we get

$$\begin{array}{cc}\hfill & \parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\hfill \\ \hfill & \le C{2}^{-ln}\parallel f{\mathbf{1}}_l{\parallel }_{L^{q_1(·)}}\parallel \Phi (x-·){\mathbf{1}}_l{(·)\parallel }_{L^{q_1^{\prime }(·)}}\parallel (1+{\left|x\right|)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left({\mathbf{1}}_{{\mathcal{S}}_t}\right){\parallel }_{q_2(·)}\hfill \\ \hfill & \le C{2}^{-ln}\parallel f{\mathbf{1}}_l{\parallel }_{L^{q_1(·)}}\parallel \Phi (x-·){\mathbf{1}}_l{(·)\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_{{\mathcal{S}}_t}\parallel }_{q_1(·)}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill E_2\le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t\in \mathbb{Z}}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=t+1}^{\infty }\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=t+1}^{\infty }\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill +& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=t+1}^{\infty }\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill :=& E_{21}+E_{22}.\hfill \end{array}$$

Here, we estimate $E_{22}$:

$$\begin{array}{cc}\hfill E_{32}& \le C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=t+1}^{\infty }\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & \le C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{\left(\sum _{l=t+1}^{\infty }{2}^{r_{0\infty }l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}{2}^{d(t-l)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}.\hfill \end{array}$$

Let ${\displaystyle \frac{n}{q_{1\infty }}}+r_{0\infty }=d>0$. Again, by applying Hölder’s inequality and $2^{-ϵ(1+p_0)}<2^{-ϵ}$, we get

$$\begin{array}{cc}\hfill & E_{22}\le C\underset{p_0>0}{sup}\left[p_0^{\theta }\sum _{t=0}^{\infty }\left(\sum _{l=t+1}^{\infty }{2}^{r_{0\infty }ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}{2}^{dϵ(1+p_0)(t-l)/2}\right.\right)\hfill \\ \hfill & {\left.\times {\left(\sum _{l=t+1}^{\infty }{2}^{d{\left(ϵ(1+p_0)\right)}^{\prime }(t-l)/2}\right)}^{{\displaystyle \frac{ϵ(1+p_0)}{{\left(ϵ(1+p_0)\right)}^{\prime }}}}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}[p_0^{\theta }\sum _{t=0}^{\infty }\sum _{l=t+1}^{\infty }{2}^{r_{0\infty }ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}{2}^{dϵ(1+p_0)(t-l)/2}{]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{l=0}^{\infty }{2}^{r_{0\infty }ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\sum _{t=0}^{l-1}{2}^{dϵ(1+p_0)(t-l)/2}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill <& C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{l\in \mathbb{Z}}{2}^{r_{0\infty }ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\sum _{t=-\infty }^{l-1}{2}^{dϵ(1+p_0)(t-l)/2}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill =& C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{l\in \mathbb{Z}}{2}^{r_0(·)ϵ(1+p_0)l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & {C\parallel f\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

For $E_{21}$, using Minkowski’s inequality,

$$\begin{array}{cc}\hfill E_{21}\le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=t+1}^{\infty }\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill E_{21}\le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=t+1}^{-1}\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill +& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{l=0}^{\infty }\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill :=& B_1+B_2.\hfill \end{array}$$

The estimate for $B_1$ is easy to find, so we omit the details here. For $B_2$, we have

$$2^{-ln}\parallel {\mathbf{1}}_{{\mathcal{S}}_t}{\parallel }_{q_1(·)}{\parallel {\mathbf{1}}_l\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbb{R}}^n\right)}\le C{2}^{-ln}{2}^{{\displaystyle \frac{tn}{q_1\left(0\right)}}}{2}^{{\displaystyle \frac{ln}{q^{\prime }{1}_{\infty }}}}\le C{2}^{{\displaystyle \frac{tn}{q_1\left(0\right)}}}{2}^{{\displaystyle \frac{-ln}{q_{1_{\infty }}}}},$$

(11)

$$\begin{array}{cc}\hfill B_2\le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0\left(0\right)ϵ(1+p_0)}{\left(\sum _{l=0}^{\infty }\parallel {\mathbf{1}}_t{(1+|x\left|\right)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}\left(f{\mathbf{1}}_l\right){\parallel }_{q_2(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0\left(0\right)ϵ(1+p_0)}{\left(\sum _{l=0}^{\infty }{2}^{-ln}{2}^{lv}{2}^{tv+{\displaystyle \frac{n}{s}}}{2}^{{\displaystyle \frac{tn}{q_1\left(0\right)}}}{2}^{{\displaystyle \frac{ln}{q_{1_{\infty }}^{\prime }}}}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0\left(0\right)ϵ(1+p_0)}{\left(\sum _{l=0}^{\infty }{2}^{-l({\displaystyle \frac{n}{q_{1\infty }}}+v+{\displaystyle \frac{n}{s}})}{2}^{t({\displaystyle \frac{n}{q_1\left(0\right)}}+v+{\displaystyle \frac{n}{s}})}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t({\displaystyle \frac{n}{q_1\left(0\right)}}+v+{\displaystyle \frac{n}{s}})ϵ(1+p_0)}{\left(\sum _{l=0}^{\infty }{2}^{-l({\displaystyle \frac{n}{q_{1\infty }}}+v+{\displaystyle \frac{n}{s}})}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }{\left(\sum _{l=0}^{\infty }{2}^{-l({\displaystyle \frac{n}{q_{1\infty }}}+v+{\displaystyle \frac{n}{s}})}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }{\left(\sum _{l=0}^{\infty }{2}^{r_{0\infty }l}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}{2}^{-l({\displaystyle \frac{n}{q_{1\infty }}}+v+{\displaystyle \frac{n}{s}}+r_{0\infty })}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }{\left(\sum _{l=0}^{\infty }{2}^{r_{0\infty }lϵ(1+p_0)}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\right)}^{ϵ(1+p_0)}{\left(\sum _{l=0}^{\infty }{2}^{-l({\displaystyle \frac{n}{q_{1\infty }}}+v+{\displaystyle \frac{n}{s}}){\left(ϵ(1+p_0)\right)}^{\prime }}\right)}^{{\displaystyle \frac{ϵ(1+p_0)}{{\left(ϵ(1+p_0)\right)}^{\prime }}}}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\left(\sum _{l\in \mathbb{Z}}{2}^{r_0(·)lϵ(1+p_0)}{\parallel f{\mathbf{1}}_l\parallel }_{q_1(·)}^{ϵ(1+p_0)}\right)\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & {C\parallel f\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Combining the estimates for $E_1$ and $E_2$ yields

$$\parallel (1+{\left|x\right|)}^{-\lambda \left(x\right)}{I}^{\zeta (·)}{\left(f\right)\parallel }_{{\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}\le C{\parallel f\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.$$

which ends the proof. □

4. Boundedness of Fractional Hardy Operators on Grand Variable Herz Spaces

Theorem 2.

Let $1<ϵ<\infty$, $\alpha ,q_2\in {\mathfrak{P}}_{0,\infty }\left({\mathbb{R}}^n\right)$, where α satisfies ${\displaystyle \frac{-n}{q_{1\infty }}}<r_{0\infty }<{\displaystyle \frac{n}{q_{1\infty }^{\prime }}}$ and ${\displaystyle \frac{-n}{q_1\left(0\right)}}<r_0\left(0\right)<{\displaystyle \frac{n}{q_1^{\prime }\left(0\right)}}$. Define a variable exponent $q_2(·)$ by the relation

$${\displaystyle \frac{1}{q_2(·)}}={\displaystyle \frac{1}{q_1(·)}}-{\displaystyle \frac{\beta }{n}}.$$

Then, the Hardy operator ${\mathcal{H}}_{]}beta$ is bounded from ${\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)$ to ${\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)$.

Proof. 

Let $g\in {\dot{K}}_{q(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)$, and let $g\left(z\right)={\sum }_{j=-\infty }^{\infty }g\left(z\right){\mathbf{1}}_l\left(z\right)={\sum }_{j=-\infty }^{\infty }{g}_j\left(z\right)$. Then, we have

$$\begin{array}{cc}\hfill |{\mathcal{H}}_{\beta }\left(g\right)\left(z\right).{\mathbf{1}}_t\left(z\right)|& \le {\displaystyle \frac{1}{{\left|z\right|}^{n-\beta }}}{\int }_{{\mathcal{S}}_t}\left|g\left(x\right)\right|dx.{\mathbf{1}}_t\left(z\right)\hfill \\ \hfill & \le C{2}^{t\beta -tn}\sum _{j=-\infty }^t\parallel g_j{\parallel }_{L^{q_1(·)}}{\parallel {\mathbf{1}}_j\parallel }_{L^{q_1^{\prime }(·)}}.{\mathbf{1}}_t\left(z\right).\hfill \end{array}$$

Next, we have

$$\begin{array}{c}\hfill \parallel {\mathcal{H}}_{\beta }\left(g\right)\left(z\right).{\mathbf{1}}_t{\left(z\right)\parallel }_{L^{q_2(·)}}\le 2^{t\beta -tn}\sum _{j=-\infty }^t\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}.\end{array}$$

$$\begin{array}{cc}\hfill & \parallel {\mathcal{H}}_{\beta }{\left(g\right)\parallel }_{{\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}=\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t\in \mathbb{Z}}{2}^{t{r}_0(·)ϵ(1+p_0)}{\parallel {\mathbf{1}}_t{\mathcal{H}}_{\beta }\left(g\right)\parallel }_{q_2(·)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t\in \mathbb{Z}}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=-\infty }^t{2}^{t\beta -tn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}.\hfill \end{array}$$

Consequently, we get

$$\begin{array}{cc}\hfill & \parallel {\mathcal{H}}_{\beta }{\left(g\right)\parallel }_{{\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=-\infty }^t{2}^{t\beta -tn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill +& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=-\infty }^t{2}^{t\beta -tn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill :=& F_1+F_2.\hfill \end{array}$$

Since fractional operators are known to be bounded on Lebesgue spaces (see e.g., [41]), we can easily find that

$$\parallel {\mathbf{1}}_t{\parallel }_{L^{q_2(·)}}\le C{2}^{-t\beta }{\parallel {\mathbf{1}}_t\parallel }_{L^{q_1(·)}}.$$

For $F_1$, applying Lemma 1 and the fact $2^{k\alpha \left(z\right)}\approx 2^{k\alpha \left(0\right)}$ for $k<0$, we get

$$2^{-tn}\parallel {\mathbf{1}}_j{\parallel }_{q_1^{\prime }(·)}{\parallel {\mathbf{1}}_t\parallel }_{q_1(·)}\le C{2}^{-tn}{2}^{{\displaystyle \frac{tn}{q_1\left(0\right)}}}{2}^{{\displaystyle \frac{jn}{q_1^{\prime }\left(0\right)}}}\le C{2}^{{\displaystyle \frac{(j-t)n}{q_1^{\prime }\left(0\right)}}}.$$

(12)

Now, we apply these estimates to $E_1$ to obtain

$$\begin{array}{cc}\hfill F_1\le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=-\infty }^t{2}^{t\beta -tn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0\left(0\right)ϵ(1+p_0)}{\left(\sum _{j=-\infty }^t{2}^{{\displaystyle \frac{(j-t)n}{q_1^{\prime }\left(0\right)}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{\left(\sum _{j=-\infty }^t{2}^{r_0\left(0\right)j}{2}^{{\displaystyle \frac{(j-t)n}{q_1^{\prime }\left(0\right)}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}.\hfill \end{array}$$

If ${\displaystyle \frac{n}{q_1^{\prime }\left(0\right)}}-r_0\left(0\right)=b>0$, then, since $2^{-ϵ(1+p_0)}<2^{-ϵ}$, applying the well-known Fubini theorem and Hölder inequality yields

$$\begin{array}{cc}\hfill F_1& \le C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{\left(\sum _{j=-\infty }^t{2}^{r_0\left(0\right)l}{\parallel g_j\parallel }_{L^{q_1(·)}}{2}^{b(l-t)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & \le C\underset{p_0>0}{sup}[p_0^{\theta }\sum _{t=-\infty }^{-1}\left(\sum _{j=-\infty }^t{2}^{r_0\left(0\right)ϵ(1+p_0)l}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}{2}^{bϵ(1+p_0)(j-t)/2}\right)\hfill \\ \hfill & \times {\left(\sum _{j=-\infty }^t{2}^{b{\left(ϵ(1+p_0)\right)}^{\prime }(j-t)/2}\right)}^{{\displaystyle \frac{ϵ(1+p_0)}{{\left(ϵ(1+p_0)\right)}^{\prime }}}}{]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & \le C\underset{p_0>0}{sup}{\left[p_0^{\theta }\sum _{t=-\infty }^{-1}\left(\sum _{j=-\infty }^t{2}^{r_0\left(0\right)ϵ(1+p_0)l}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}{2}^{bϵ(1+p_0)(j-t)/2}\right)\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill =& C\underset{p_0>0}{sup}{\left[p_0^{\theta }\sum _{j=-\infty }^{-1}{2}^{r_0\left(0\right)ϵ(1+p_0)j}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}\sum _{k=j}^{-1}{2}^{bϵ(1+p_0)(j-t)/2}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{j=-\infty }^{-1}{2}^{r_0\left(0\right)ϵ(1+p_0)j}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill =& C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{j\in \mathbb{Z}}{2}^{r_0(·)ϵ(1+p_0)j}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & {C\parallel g\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Now, for $F_2$, using Minkowski’s inequality, we have

$$\begin{array}{cc}\hfill F_2\le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=-\infty }^{-1}{2}^{t\beta -tn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill +& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=0}^k{2}^{t\beta -tn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill :=& C_1+C_2.\hfill \end{array}$$

It can be easily seen that the estimate for $C_2$ can be checked similarly. For that estimate, we simply replace $q^{\prime }\left(0\right)$ with $q_{\infty }^{\prime }$ and use the fact that $b:={\displaystyle \frac{n}{q_{\infty }^{\prime }}}-\alpha >0$. For $C_2$, we have

$$2^{t\beta -tn}\parallel {\mathbf{1}}_t{\parallel }_{L^{q_2(·)}}{\parallel {\mathbf{1}}_j\parallel }_{L^{q_1^{\prime }(·)}}\le C{2}^{-tn}{2}^{{\displaystyle \frac{tn}{q_{1\infty }}}}{2}^{{\displaystyle \frac{jn}{q^{\prime }\left(0\right)}}}\le C{2}^{{\displaystyle \frac{-tn}{q_{1\infty }^{\prime }}}}{2}^{{\displaystyle \frac{jn}{q_1^{\prime }\left(0\right)}}},$$

(13)

as $r_{0\infty }-{\displaystyle \frac{n}{q_{1\infty }^{\prime }}}<0$, and using the fact $2^{ta\left(z\right)}\approx 2^{ta(\infty )}$ for $t\ge 0$, we have

$$\begin{array}{cc}\hfill C_1\le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=-\infty }^{-1}{2}^{t\beta -tn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left[p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_{0\infty }ϵ(1+p_0)}{\left(\sum _{j=-\infty }^{-1}{2}^{-tn}{2}^{{\displaystyle \frac{tn}{q_{1\infty }}}}{2}^{{\displaystyle \frac{jn}{q_1^{\prime }\left(0\right)}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left[p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_{0\infty }ϵ(1+p_0)}{\left(\sum _{j=-\infty }^{-1}{2}^{{\displaystyle \frac{-tn}{q_{1\infty }^{\prime }}}}{2}^{{\displaystyle \frac{jn}{q_1^{\prime }\left(0\right)}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}[p_0^{\theta }\sum _{t=0}^{\infty }{2}^{{\displaystyle \frac{k\alpha -tn}{q_{1\infty }^{\prime }}}ϵ(1+p_0)}{\left(\sum _{j=-\infty }^{-1}{2}^{{\displaystyle \frac{jn}{q_1^{\prime }\left(0\right)}}}{\parallel g_j\parallel }_{L^{q(·)}}\right)}^{ϵ(1+p_0)}{]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }{\left(\sum _{j=-\infty }^{-1}{2}^{{\displaystyle \frac{jn}{q_1^{\prime }\left(0\right)}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }{\left(\sum _{j=-\infty }^{-1}{2}^{r_0\left(0\right)j}{\parallel g_j\parallel }_{L^{q(·)}}{2}^{{\displaystyle \frac{jn}{q_1^{\prime }\left(0\right)}}-r_0\left(0\right)j}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}.\hfill \end{array}$$

As we can check that ${\displaystyle \frac{n}{q^{\prime }\left(0\right)}}-\alpha \left(0\right)>0$, using Hölder’s inequality we have

$$\begin{array}{cc}\hfill C_1\le & C\underset{p_0>0}{sup}{\left[p_0^{\theta }\left(\sum _{j=-\infty }^{-1}{2}^{r_0\left(0\right)jϵ(1+p_0)}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}\right){\left(\sum _{j=-\infty }^{-1}{2}^{({\displaystyle \frac{jn}{q_1^{\prime }\left(0\right)}}-r_0\left(0\right)l)ϵ{(1+p_0)}^{\prime }}\right)}^{{\displaystyle \frac{ϵ(1+p_0)}{ϵ{(1+p_0)}^{\prime }}}}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\left(\sum _{j\in \mathbb{Z}}{2}^{r_0(·)ϵ(1+p_0)}{\parallel g_j\parallel }_{L^{q(·)}}^{ϵ(1+p_0)}\right)\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & {C\parallel g\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Combining these estimates, we get

$$\parallel {\mathcal{H}}_{\beta }{\left(g\right)\parallel }_{{\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}\le C{\parallel g\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)},$$

which completes the proof.

□

Theorem 3.

Let $1<ϵ<\infty$, $\alpha ,q\in {\mathfrak{P}}_{0,\infty }\left({\mathbb{R}}^n\right)$, where α satisfies ${\displaystyle \frac{-n}{q_{1\infty }}}<r_{0\infty }<{\displaystyle \frac{n}{q_{1\infty }^{\prime }}}$ and ${\displaystyle \frac{-n}{q_1\left(0\right)}}<r_0\left(0\right)<{\displaystyle \frac{n}{q^{\prime }\left(0\right)}}$. Define a variable exponent $q_2(·)$ by the relation

$${\displaystyle \frac{1}{q_2(·)}}={\displaystyle \frac{1}{q_1(·)}}-{\displaystyle \frac{\beta }{n}}.$$

Then, the Hardy operator ${\mathcal{H}}_{\beta }^{\ast }$ is bounded from ${\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)$ to ${\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)$.

Proof. 

This proof is similar to our previous result; therefore, we only give a simple proof. Let $g\in {\dot{K}}_{q(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)$, and let $g\left(z\right)={\sum }_{j=-\infty }^{\infty }g\left(z\right){\mathbf{1}}_l\left(z\right)={\sum }_{j=-\infty }^{\infty }{g}_j\left(z\right)$. Then, we have

$$\begin{array}{cc}\hfill |{\mathcal{H}}_{\beta }^{\ast }\left(g\right)\left(z\right).{\mathbf{1}}_t\left(z\right)|& \le {\int }_{{\mathbb{R}}^n\setminus {\mathcal{S}}_t}{\displaystyle \frac{\left|g\right(x\left)\right|}{{\left|z\right|}^{n-\beta }}}dx.{\mathbf{1}}_t\left(z\right)\hfill \\ \hfill & \le C\sum _{j=t+1}^{\infty }{\left|z\right|}^{\beta -n}\parallel g_j{\parallel }_{L^{q_1(·)}}{\parallel {\mathbf{1}}_j\parallel }_{L^{q_1^{\prime }(·)}}.{\mathbf{1}}_t\left(z\right).\hfill \end{array}$$

Next, we have

$$\begin{array}{c}\hfill \parallel {\mathcal{H}}_{\beta }^{\ast }\left(g\right)\left(z\right).{\mathbf{1}}_t{\left(z\right)\parallel }_{L^{q_2(·)}}\le C\sum _{j=t+1}^{\infty }{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}.\end{array}$$

$$\begin{array}{cc}\hfill & \parallel {\mathcal{H}}_{\beta }^{\ast }{\left(g\right)\parallel }_{{\dot{K}}_{q_2(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}\le \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t\in \mathbb{Z}}{2}^{t{r}_0(·)ϵ(1+p_0)}{\parallel {\mathcal{H}}_{\beta }^{\ast }\left(g\right){\mathbf{1}}_t\parallel }_{L^{q_2(·)}}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=t+1}^{\infty }{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}.\hfill \end{array}$$

Then, Minkowski’s inequality leads to

$$\begin{array}{cc}\hfill & \parallel {\mathcal{H}}_{\beta }^{\ast }{\left(g\right)\parallel }_{{\dot{K}}_{q(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=t+1}^{\infty }{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=t+1}^{\infty }{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill +& \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=t+1}^{\infty }{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill :=& G_1+G_2.\hfill \end{array}$$

For $G_2$, Lemma 1 yields

$$2^{j\beta -jn}\parallel {\mathbf{1}}_t{\parallel }_{L^{q_2(·)}}{\parallel {\mathbf{1}}_j\parallel }_{L^{q_1^{\prime }(·)}\left({\mathbb{R}}^n\right)}\le C{2}^{-jn}{2}^{{\displaystyle \frac{tn}{q_{1\infty }}}}{2}^{{\displaystyle \frac{jn}{q_{1\infty }^{\prime }}}}\le C{2}^{{\displaystyle \frac{(t-j)n}{q_{1\infty }}}}.$$

(14)

We get

$$\begin{array}{cc}\hfill G_2& \le C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=t+1}^{\infty }{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & \le C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=t+1}^{\infty }{2}^{{\displaystyle \frac{(t-j)n}{q_{1\infty }}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & \le C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=0}^{\infty }{\left(\sum _{j=t+1}^{\infty }{2}^{r_{0\infty }j}{\parallel g_j\parallel }_{L^{q_1(·)}}{2}^{d(t-j)}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}.\hfill \end{array}$$

If ${\displaystyle \frac{n}{q_{1\infty }}}+r_{0\infty }=d>0$, then Hölder’s inequality and $2^{-ϵ(1+p_0)}<2^{-ϵ}$ yields

$$\begin{array}{cc}\hfill & G_2\le C\underset{p_0>0}{sup}\left[p_0^{\theta }\sum _{t=0}^{\infty }\left(\sum _{j=t+1}^{\infty }{2}^{r_{0\infty }ϵ(1+p_0)j}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}{2}^{dϵ(1+p_0)(t-j)/2}\right)\right.\hfill \\ \hfill & \times {\left.{\left(\sum _{j=t+1}^{\infty }{2}^{d{\left(ϵ(1+p_0)\right)}^{\prime }(t-j)/2}\right)}^{{\displaystyle \frac{ϵ(1+p_0)}{{\left(ϵ(1+p_0)\right)}^{\prime }}}}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left[p_0^{\theta }\sum _{t=0}^{\infty }\sum _{j=t+1}^{\infty }{2}^{r_{0\infty }ϵ(1+p_0)j}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}{2}^{dϵ(1+p_0)(t-j)/2}\right]}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{j=0}^{\infty }{2}^{r_{0\infty }ϵ(1+p_0)j}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}\sum _{t=0}^{j-1}{2}^{dϵ(1+p_0)(t-j)/2}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill <& C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{j\in \mathbb{Z}}{2}^{r_{0\infty }ϵ(1+p_0)j}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}\sum _{t=-\infty }^{j-1}{2}^{dp(t-j)/2}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill =& C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{j\in \mathbb{Z}}{2}^{r_0(·)ϵ(1+p_0)j}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & {C\parallel g\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Now, we find the estimate for $E_1$. By using Minkowski’s inequality, we obtain

$$\begin{array}{cc}\hfill G_1& \le \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=t+1}^{\infty }{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & \le \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=t+1}^{-1}{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & +\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=0}^{\infty }{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill & :D_1+D_2.\hfill \end{array}$$

The estimate for $B_1$ is easy to calculate, so we omit the details here. For $B_2$, we get

$$2^{j\beta -jn}\parallel {\mathbf{1}}_t{\parallel }_{L^{q_2(·)}}{\parallel {\mathbf{1}}_j\parallel }_{L^{q_1^{\prime }(·)}}\le C{2}^{-jn}{2}^{{\displaystyle \frac{tn}{q_1\left(0\right)}}}{2}^{{\displaystyle \frac{jn}{q_{1\infty }^{\prime }}}}\le C{2}^{{\displaystyle \frac{tn}{q_1\left(0\right)}}}{2}^{{\displaystyle \frac{-jn}{q_{1\infty }}}}.$$

(15)

$$\begin{array}{cc}\hfill D_2\le & \underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0(·)ϵ(1+p_0)}{\left(\sum _{j=0}^{\infty }{2}^{j\beta -jn}\parallel g_j{\parallel }_{L^{q_1(·)}}\parallel {\mathbf{1}}_j{\parallel }_{L^{q_1^{\prime }(·)}}{\parallel {\mathbf{1}}_t\parallel }_{L^{q_2(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0\left(0\right)ϵ(1+p_0)}{\left(\sum _{j=0}^{\infty }{2}^{-jn}{2}^{{\displaystyle \frac{tn}{q_1\left(0\right)}}}{2}^{{\displaystyle \frac{jn}{q_{1\infty }^{\prime }}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t{r}_0\left(0\right)ϵ(1+p_0)}{\left(\sum _{j=0}^{\infty }{2}^{{\displaystyle \frac{tn}{q_1\left(0\right)}}}{2}^{{\displaystyle \frac{-jn}{q_{1\infty }}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}\left(p_0^{\theta }\sum _{t=-\infty }^{-1}{2}^{t(r_0\left(0\right)+n)/q_1\left(0\right)ϵ(1+p_0)}\right.\hfill \\ \hfill & \times {\left.{\left(\sum _{j=0}^{\infty }{2}^{{\displaystyle \frac{-jn}{q_{1\infty }}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }{\left(\sum _{j=0}^{\infty }{2}^{{\displaystyle \frac{-jn}{q_{1\infty }}}}{\parallel g_j\parallel }_{L^{q_1(·)}}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }{\left(\sum _{j=0}^{\infty }{2}^{r_{0\infty }j}{\parallel g_j\parallel }_{L^{q_1(·)}}{2}^{j(n{q}_{1\infty }+r_{0\infty })}\right)}^{ϵ(1+p_0)}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}\left(p_0^{\theta }{\left(\sum _{j=0}^{\infty }{2}^{r_{0\infty }jϵ(1+p_0)}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}\right)}^{ϵ(1+p_0)}\right.\hfill \\ \hfill & \times {\left.{\left(\sum _{j=0}^{\infty }{2}^{j(n{q}_{1\infty }+r_{0\infty }){\left(ϵ(1+p_0)\right)}^{\prime }}\right)}^{{\displaystyle \frac{ϵ(1+p_0)}{{\left(ϵ(1+p_0)\right)}^{\prime }}}}\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & C\underset{p_0>0}{sup}{\left(p_0^{\theta }\left(\sum _{j\in \mathbb{Z}}{2}^{\alpha (·)jϵ(1+p_0)}{\parallel g_j\parallel }_{L^{q_1(·)}}^{ϵ(1+p_0)}\right)\right)}^{{\displaystyle \frac{1}{ϵ(1+p_0)}}}\hfill \\ \hfill \le & {C\parallel g\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Combining the estimates yields

$$\parallel {\mathcal{H}}_{\beta }^{\ast }{\left(g\right)\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)}\le C{\parallel f\parallel }_{{\dot{K}}_{q_1(·)}^{r_0(·),ϵ),\theta }\left({\mathbb{R}}^n\right)},$$

which ends the proof. □

5. Conclusions

This manuscript contributes significantly to the field of mathematical analysis by introducing new results within the framework of grand variable Herz spaces. This study deepens our understanding of the boundedness of the rough Riesz potential operator of variable order and fractional Hardy operators on grand variable Herz spaces by building on previous findings. The application of these results to establishing the existence of regularity solutions of some elliptic PDEs with smooth boundaries in these spaces is demonstrated. There are a number of exciting avenues for further study in grand variable weighted Herz–Morrey spaces in the future. Extending these results to two-weighted grand Herz–Morrey spaces with variable exponents for solving PDEs within such spaces could be particularly valuable.