\def\papertitle{On the geometric-weighted-variable Hardy spaces on Lipschitz domains}

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\title{\papertitle}
\affiliation{Oussama Melkemi$^1$, Saibi Khedoudj$^{2,3}$, Zouhir Mokhtari$^{3}$}	% This command replaces \name{The DAFx Crew} 
{1-Laboratory of partial differential equations and applications\\
University of Batna 2\\
 Batna 05000, Algeria\\ 
2-School of Mathematics and Statistics\\
Central South University\\
Changsha 410083, China\\
3-Laboratory of applied mathematics,\\
University of Biskra, Biskra 07000, Algeria}

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	\maketitle

\begin{abstract}
In this paper, we introduce and explore the geometric-weighted Hardy spaces with variable exponent on bounded Lipschitz domain $\mathbb{D}$ of $\mathbb{R}^n.$
\end{abstract}

\section{Introduction}
The variable Lebesgue spaces is a well known generalization for the classical Lebesgue spaces and it has become an interesting field for scientists in recent decades due to its wide uses and applications in real world phenomena. The variable Lebesgue spaces $L^{p(\cdot)}(\Omega)$ was initiated first by Orlicz \cite{Orlicz}, then studied and investigated by many authors see for example \cite{S44,Cruz2006,Diening2011,S19}. 

The classical Hardy space on $\mathbb{R}^n$ was introduced and developed by Stein and Weiss \cite{Stein}. This kind of space was also extended to the variable setting. In particular, Nakai and Sawano \cite{Nakai2012} introduced and investigated the Hardy space $H^{p(\cdot)}(\RR^n)$ with variable exponent, where they established the atomic decomposition of this space. As in the results established in \cite{Nakai2012}, Liu \cite{Liu} extended the results obtained by Miyachi \cite{miyachi} from the real variable Hardy space on domains to the Hardy spaces with variable exponents. More precesily, Liu \cite{Liu} obtained the atomic decomposition of the variable Hardy spaces on domains and as an application, he studied the geometric Hardy spaces with variable exponent. On the other hand, the weighted variable Hardy spaces are considered as an extension for the variable Hardy spaces. K-P. Ho \cite{how} introduced and studied the variable weighted Hardy spaces $H^{p(\cdot)}_w(\RR^n),$ where he introduced a general class of weights compared to the ordinary Mukenhoupt class of weights. More recently, O. Melkemi et al. \cite{MSM} extended the results obtained by K-P. Ho \cite{how}, where the authors introduced and investigated the atomic characterization of the weighted variable Hardy spaces on general domains $\Omega$ of $\RR^n$. For more information and results concerning the variable Hardy spaces we refer the reader to \cite{yang,sawano,Khedoudj2,Izu} and the references therein. Motivated by the recent works in \cite{MSM} and \cite{Liu}, our main goal is to investigate the geometric-weighted-variable Hardy spaces $H^{p(\cdot)}_{w,\mathrm{r}}(\mathbb{D})$ on Lipschitz domain $\mathbb{D}$ of $\RR^n.$ We recall that the spaces $H^{p(\cdot)}_{w,\mathrm{r}}(\mathbb{D})$ are defined by means restricting arbitrary elements of $H^{p(\cdot)}_{w}(\RR^n)$ to a bounded domain $\mathbb{D}.$ By applying the atomic characterization of the space $H^{p(\cdot)}_w(\DD)$ obtained in \cite{MSM} for a general domains of $\RR^n$, the reflection technique for Lipschitz domains and borrowing some ideas from \cite{r4}, we prove the following identity $H^{p(\cdot)}_w(\mathbb{D}) = H^{p(\cdot)}_{w,r}(\mathbb{D})$ (see Theorem \ref{thm} below). The rest of this paper is arranged as follows. In the next section, we give some basic definitions and ingredients. In the last Section, we state and prove the main result obtained in this article.

As usual, throughout this paper, $C$ stands for a positive constant which may be different from line to line,
$ A \lesssim B $ means that there exists a positive constant $C$ such that $A \leq CB,$
the symbol  $ A \approx B $ means $A \lesssim B$  and  $ B \lesssim A $.

 
\section{Preliminaries}
A measurable function $p(\cdot):\mathbb{D}\rightarrow(0,\infty)$ is called a variable exponent. For a variable exponent
$p(\cdot)$, define
\beq
p_-={\rm{ess\,inf}}_{x\in \mathbb{D}} p(x) \quad \text{and} \quad p_+={\rm{ess\,sup}}_{x\in \mathbb{D}} p(x),\nonumber
\eeq
and let us denote by $\mathcal{P}(\mathbb{D})$ the collection of all variable exponents such that $0<p_-\leq p_+<\infty.$

Let $p(\cdot)\in\p(\mathbb{D})$. The Lebesgue space with variable exponents $L^{p(\cdot)}(\mathbb{D})$ consists of all measurable functions $f:\mathbb{D}\rightarrow\RR$ such that $\int_{\mathbb{D}}|f(x)|^{p(x)}dx<\infty,$
equipped with the Luxemburg quasi-norm
\beq
\|f\|_{L^{p(\cdot)}(\mathbb{D})}=\inf\Big\{\lambda>0:
\int_{\mathbb{D}}\bigg[\frac{|f(x)|}{\lambda}\bigg]^{p(x)}dx\leq1\Big\}.
\eeq

In the following lemma, we collect some useful properties of the variable exponent Lebesgue spaces. For the proofs and more details about these spaces, we refer to \cite{Diening2011}.
\begin{lemma}\label{vp}
Let $p(\cdot)\in\mathcal{P}(\mathbb{D})$ and $f,g\in L^{p(\cdot)}(\mathbb{D})$.
\begin{itemize}
  \item  [(1)] For $\lambda\in\CC$, we have $\|\lambda f\|_{L^{p(\cdot)}(\mathbb{D})}=|\lambda|\| f\|_{L^{p(\cdot)}(\mathbb{D})}.$
  \item  [(2)] For any $s\in(0,\infty),$ we have $\||f|^s\|_{L^{p(\cdot)}(\mathbb{D})}=\| f\|_{L^{sp(\cdot)}(\mathbb{D})}^s.$
  \item  [(3)] $\|f+g\|_{L^{p(\cdot)}(\mathbb{D})} ^{\underline{p}}\leq\|f\|_{L^{p(\cdot)}(\mathbb{D})}^{\underline{p}}+\|g\|_{L^{p(\cdot)}(\mathbb{D})}^{\underline{p}}$
      where $\underline{p}=\min\{p_-,1\}$.
\end{itemize}
\end{lemma}

We recall that for any $f\in L^{1}_{\text{loc}}(\mathbb{D})$, the Hardy-Littlewood maximal operator $M$ is defined for all $x\in \mathbb{D}$ by setting,
\beq
M(f)(x):=\sup_{B\ni x}\frac{1}{|B|}\int_{B}|f(y)|\,dy,\nonumber
\eeq
where the supremum is taken over all balls $B$ of $\mathbb{D}$ containing $x$.

\medskip

Next, we recall the definition and some properties of the weighted variable Lebesgue spaces. Let $w:\mathbb{D}\rightarrow (0,\infty)$ be a locally integrable function. The weighted variable
exponent Lebesgue space $L^{p(\cdot)}_{w}(\mathbb{D})$ is defined as the space of all measurable functions $f:\mathbb{D}\rightarrow\CC$
such that
 $$\|f\|_{L^{p(\cdot)}_{w}(\mathbb{D})}=\|fw\|_{L^{p(\cdot)}(\mathbb{D})}<\infty.$$

The weights used in this paper belongs to the following class.

\begin{definition}\label{dw}
Let $p(\cdot):\mathbb{D}\rightarrow (0,\infty)$ be a measurable function such that $0<p_-\leq p_+<\infty$ and $w:\mathbb{D}\rightarrow (0,\infty)$ be a Lebesgue measurable function. We denote by $W_{p(\cdot)}(\mathbb{D})$ the set of all Lebesgue measurable functions $w$ such that
\begin{enumerate}
  \item [(1)] $\|\chi_B\|_{L^{p(\cdot)/\underline{p}}_{w^{\underline{p}}}(\mathbb{D})}<\infty$ and $\|\chi_B\|_{L^{(\frac{p(\cdot)}{\underline{p}})'}_{w^{-\underline{p}}}(\mathbb{D})}<\infty$, for any ball $B$ of $\mathbb{D}$;
  \item [(2)] there exists $k>1$ and $s>1$ such that the Hardy-Littlewood maximal operator is bounded on $L^{(sp(\cdot))'/k}_{w^{-k/s}}(\mathbb{D})$.
\end{enumerate}
\end{definition} 
For any $w\in W_{p(\cdot)}(\mathbb{D})$, set
\beqn
s_{w}=\inf\{s\geq1: M\;\text{is bounded on}\; L^{(sp(\cdot))'}_{w^{-1/s}}(\mathbb{D})\},
\eeqn
\beqn
\mathbb{S}_{w}=\{s\geq1: M\;\text{is bounded on}\; L^{(sp(\cdot))'/k}_{w^{-k/s}}(\mathbb{D})\;\text{for some}\;k>1\}
\eeqn
and
\beq\label{k}
k_w^s=\sup\{k\geq1: M\;\text{is bounded on}\; L^{(sp(\cdot))'/k}_{w^{-k/s}}(\RR^n)\}.
\eeq
The following theorem gives the  Fefferman-Stein vector valued maximal inequalities on $L^{p(\cdot)}_{w}(\mathbb{D})$. For the proof, we refer to \cite{how}.
\begin{theorem}\label{theo1.6}
Let $p(\cdot):\mathbb{D}\rightarrow(0,\infty)$ be a measurable function with $0<p_-\leq p_+<\infty$ and $q\in(1,\infty)$. If $w\in W_{p(\cdot)}(\mathbb{D})$, then, for any $r>s_{w}$, we have
\beqn
\left\|\left(\sum_{i\in\NN}(Mf_i)^q\right)^{1/q}\right\|_{L^{rp(\cdot)}_{w^{1/r}}(\mathbb{D})}
\leq C \left\|\left(\sum_{i\in\NN}|f_i|^q\right)^{1/q}\right\|_{L^{rp(\cdot)}_{w^{1/r}}(\mathbb{D})}.
\eeqn
\end{theorem}


Let $\phi\in\mathcal{D}(B(\mathbf{0}_n,1))$ such that $\int_{\RR^n} \phi(x)dx=1$. For any $t\in(0,\infty)$ and $x\in \mathbb{D}$, we set $\phi_t(x)=t^{-n}\phi(t^{-1}x)$. For any $f\in \mathcal{D}'(\mathbb{D})$, the radial maximal function $\M(f)$ is defined for any $x\in \mathbb{D}$ by
\begin{equation}\label{m}
\M(f)(x):=\sup_{t\in(0,{\rm{dist}}(x,\mathbb{D}^c))}|\langle f,\phi_t(x-\cdot)\rangle|,
\end{equation}
where $\mathbb{D}^c$ denotes the complementary set of $\mathbb{D}$ in $\RR^n$, ${\rm{dist}}(x,\mathbb{D}^c):=\inf\{|x-y|\;:\;y\in \mathbb{D}^c\}$ and $\langle\cdot,\cdot\rangle$ denotes the duality between $\mathcal{D}'(\mathbb{D})$ and $\mathcal{D}(\mathbb{D})$.

Now, we introduce weighted variable Hardy space on domains and we establish its atomic and maximal function  characterizations.

\begin{definition}
Let $\mathbb{D}$ be an open set of $\RR^n$ and $p(\cdot)\in \mathcal{P}(\mathbb{D})$. Then, the weighted variable Hardy space $H^{p(\cdot)}_{w}(\mathbb{D})$ is defined to be the set of all $f\in\mathcal{D}'(\mathbb{D})$ such that $\M(f)\in L^{p(\cdot)}_{w}(\mathbb{D})$, where $\M$ is as in \eqref{m}, equipped with the quasi-norm
\beq
\|f\|_{H^{p(\cdot)}_{w}(\mathbb{D})}=\|\M(f)\|_{L^{p(\cdot)}_{w}(\mathbb{D})}.\nonumber
\eeq
\end{definition}
Next, we give the definition of $(p(\cdot),r,w)$-atoms in $\DD$.
\begin{definition}\label{atom}
Let $\DD$ be an open set of $\RR^n$, $p(\cdot)\in \mathcal{P}(\DD)$, $w:\DD\rightarrow(0,\infty)$, $q\in(1,\infty]$ and
\beq\label{dp}
d_w = n(s_w - 1).
\eeq
\begin{enumerate} 
  \item A cube $Q\subset\RR^n$ is called of type (a) if $4Q\subset\DD$ and $\widetilde{Q}\subset\RR^n$ is called of type (b) if $2\widetilde{Q}\cap\DD^c= \varnothing$ and $4\widetilde{Q}\cap\DD^c\neq \varnothing$.
  \item  A measurable function $a$ on $\DD$ is called a type (a) $(p(\cdot),q,w)_{\DD}-$atom if there exists a cube $Q$  of type (a) such that
\begin{enumerate}
  \item[{(1)}] \text{supp} $a \subset Q$;
  \item[{(2)}] $\|a\|_{L^q(\DD)}\leq\frac{|Q|^{1/q}}{\|\chi_Q\|_{L^{p(\cdot)}_{w}(\DD)}}$;
  \item[{(3)}] there exist $s\geq d_w$ such that, $\int_{\RR^n}a(x)x^{\alpha}dx=0$ for all $\alpha\in\ZZ_+^n$ with $|\alpha|\leq s$.
\end{enumerate} 
  \item A measurable function $b$ on $\DD$ is called a type (b) $(p(\cdot),q,w)_{\DD}-$atom if there exists a cube $\widetilde{Q}$  of type (b) such that
      \begin{enumerate}
  \item[{(1)}] \text{supp} $b \subset \widetilde{Q}$;
  \item[{(2)}] $\|b\|_{L^q(\DD)}\leq\frac{|\widetilde{Q}|^{1/q}}{\|\chi_{\widetilde{Q}}\|_{L^{p(\cdot)}_{w}(\DD)}}$;
\end{enumerate}
\end{enumerate}
\end{definition}
Let $p(\cdot)\in \p(\DD)$, $\{\lambda_{i}\}_{i\in\NN}$ be a sequence of numbers in $\CC$, $\{Q_{i}\}_{i\in\NN}$ be a cube sequence of the supports of type (a) $(p(\cdot),r,w)_{\DD}-$ atoms,
$\{\kappa_{i}\}_{i\in\NN}$ be a sequence of numbers in $\CC$, $\{\widetilde{Q}_{i}\}_{i\in\NN}$ be a cube sequence of the supports of type (b) $(p(\cdot),r,w)_{\DD}-$ atoms. Define
\beq
\mathcal{A}(\{\lambda_{i}\}_{i\in\NN},\{Q_{i}\}_{i\in\NN}):=\bigg
\|\bigg\{\sum_{i\in\NN}\bigg[\frac{|\lambda_{i}|\chi_{Q_{i}}}{\|\chi_{Q_{i}}\|_{L^{p(\cdot)}_{w}(\DD)}}\bigg]^{\theta}
 \bigg\}^{\frac{1}{\theta}}\bigg\|_{L^{p(\cdot)}_{w}(\DD)},\nonumber
\eeq
and
\beq
\mathcal{B}(\{\kappa_{i}\}_{i\in\NN},\{\widetilde{Q}_{i}\}_{i\in\NN}):=
\bigg\|\bigg\{\sum_{i\in\NN}\bigg[\frac{|\kappa_{i}|\chi_{\widetilde{Q}_{i}}}{\|\chi_{\widetilde{Q}_{i}}\|_{L^{p(\cdot)}_{w}(\DD)}}\bigg]^{\theta}
 \bigg\}^{\frac{1}{\theta}}\bigg\|_{L^{p(\cdot)}_{w}(\DD)},\nonumber
\eeq

%The next result is the atomic decomposition of the weighted Hardy spaces with variable exponent on proper open sets $\Omega$ of $\RR^n$. This result established in \cite{MSM}.

%\begin{Theorem}\label{th2}
%Let $\Omega$ be a proper open subset of $\RR^n$ and $\phi\in\mathcal{D}(B(\mathbf{0}_n,1))$ such that $\int_{\RR^n}\phi(x)dx=1$, $p(\cdot)\in\mathcal{P}(\Omega)$, $w\in W_{p(\cdot)}(\Omega)$ and $q\in(\max\{1,p_+\},\infty]$. Then,  $H^{p(\cdot)}_{w}(\Omega)\approx H^{p(\cdot),q}_{w,{\rm{atom}}}(\Omega)$ with equivalent quasi-norms. In particular, if
%$n/(n+s)<p_-$ then
%\beq
%H^{p(\cdot)}_{w}(\Omega)\approx H^{p(\cdot),q}_{w,{\rm{atom}}}(\Omega)\approx H^{p(\cdot)}_{w,\max,s}(\Omega)\nonumber
%\eeq
%with equivalent quasi-norms.
%\end{Theorem}


\section{Main Result}

In this section, we present our main result. We start by introducing the geometric-weighted Hardy space with variable exponent on a proper open subset $\mathbb{D}$ of $\RR^n.$

\begin{definition}
Let $\mathbb{D} \subset \RR^n$ be a proper open subset and $p(\cdot) \in \mathcal{P}(\RR^n).$ Denote by $H^{p(\cdot)}_w(\RR^n)$ the weighted variable Hardy spaces on $\RR^n.$ Then, it is said that a distribution $f\in\mathcal{D}'(\mathbb{D})$ belongs to $H^{p(\cdot)}_{w,r}(\mathbb{D}),$ if $f$ is the restrection to $\mathbb{D}$ of $\mathrm{F} \in H^{p(\cdot)}_w(\RR^n).$ Namely, we have

$$ H^{p(\cdot)}_{w,r}(\mathbb{D}) := \Big\lbrace  f\in \mathcal{D}'(\mathbb{D}) : \text{ there exists } \mathrm{F} \in H^{p(\cdot)}_w(\RR^n) : \text{ such that } \mathrm{F}|_{\mathbb{D}} \equiv f \Big\rbrace.$$
\end{definition}


The definition of $(p(\cdot),q,w)-$atoms on $\RR^n$ is given as follows

\begin{definition}\label{atom}
Let $p(\cdot)\in \mathcal{P}(\DD)$, $w:\DD\rightarrow(0,\infty)$, $q\in(1,\infty]$ and

$$d_w = n(s_w - 1).$$
A measurable function $a$ on $\RR^n$ is called $(p(\cdot),q,w)_{\DD}-$atom if there exists a cube $Q$ such that

\begin{itemize} 
  \item[{(1)}] \text{supp} $a \subset Q$;
  \item[{(2)}] $\|a\|_{L^q(\RR^n)}\leq\frac{|Q|^{1/q}}{\|\chi_Q\|_{L^{p(\cdot)}_{w}(\RR^n)}}$;
  \item[{(3)}] there exist $s\geq d_w$ such that, $\int_{\RR^n}a(x)x^{\alpha}dx=0$ for all $\alpha\in\ZZ_+^n$ with $|\alpha|\leq s$.
\end{itemize}
\end{definition}

\begin{definition}\label{dd}
For sequences of $\{\lambda_{i}\}_{i\in\NN} \subset \CC$ and cubes $\{Q_j\}_{i\in\NN}$, define that
$$\mathcal{A}'(\{\lambda_{i}\}_{i\in\NN},\{Q_{i}\}_{i\in\NN}):=\bigg
\|\bigg\{\sum_{i\in\NN}\bigg[\frac{|\lambda_{i}|\chi_{Q_{i}}}{\|\chi_{Q_{i}}\|_{L^{p(\cdot)}_{w}(\mathbb{D})}}\bigg]^{\theta}
 \bigg\}^{\frac{1}{\theta}}\bigg\|_{L^{p(\cdot)/\theta}_{w^\theta}(\mathbb{D})},$$
 
 where $\theta\in\mathbb{S}_w.$
\end{definition}

The result of this section is given below

\begin{theorem}\label{thm}
Let $\mathbb{D} \subset \RR^n$ be a bounded Lipschitz domain, $p(\cdot) \in C^{\text{log}}(\RR^n)$ and $w\in W_{p(\cdot)}(\RR^n)$ with $ \frac{n}{1+n} \leq p_{-} \leq p_{+} \leq 1.$ Then we have the following identity:
$$ H^{p(\cdot)}_w(\mathbb{D}) = H^{p(\cdot)}_{w,r}(\mathbb{D}),$$

with equivalent quasi-norms. 
\end{theorem}

\begin{proof}[Proof of Theorem \ref{thm}]
 
We first show that, for any $f\in H^{p(\cdot)}_{w,r}(\mathbb{D})$ we have $ \|f\|_{H^{p(\cdot)}_{w}(\mathbb{D})} \lesssim \|f\|_{H^{p(\cdot)}_{w,r}(\mathbb{D})}.$ Let $f\in H^{p(\cdot)}_{w,r}(\mathbb{D})$ then by definition there exists $\mathrm{F} \in H^{p(\cdot)}_{w}(\RR^n)$ such that $\mathrm{F}|_{\mathbb{D}} \equiv f.$ According to \cite[Theorem 5.3]{how} and \eqref{dd}, we infer that there exist a sequence $\{ \lambda_Q \}_{Q} \subset \CC,$ a sequence $\{a_Q\}_Q$ of $(p(\cdot),q)$-atoms and $\mathcal{A}'(\{\lambda_{Q}\}_{Q},\{Q\}_{Q})<\infty$ such that $\F = \sum_{Q} \lambda_Qa_Q $ in $\mathcal{S}'(\RR^n)$ and 
\begin{equation}\label{26}
\|\F\|_{H^{p(\cdot),q}_{\mathrm{atom},w}} \sim \mathcal{A}'(\{\lambda_{Q}\}_{Q},\{Q\}_{Q}), 
\end{equation} 
where, those atoms $a_Q$ supported in cubes $Q\in\DD$ with $(4Q)\cap \DD^c = \emptyset,$ are treated as type $(a)(p(\cdot),q)_\DD-$atoms and we have 

\begin{equation}\label{27}
\mathcal{A}(\{\lambda_{Q}\}_{Q_1},\{Q\}_{Q_1})=\mathcal{A}'(\{\lambda_{Q}\}_{Q_1},\{Q\}_{Q_1}) < \infty,
\end{equation}

where $Q_1 = \{ Q\subset \DD : (4Q) \cap \DD^c = \emptyset \}.$

We treat as type $(b)(p(\cdot),q)_\DD$-atoms and we have

\begin{equation}\label{28}
\mathcal{B}(\{\lambda_{Q}\}_{Q_2},\{Q\}_{Q_2})=\mathcal{A}'(\{\lambda_{Q}\}_{Q_2},\{Q\}_{Q_2}) < \infty,
\end{equation}

where $Q_2 = \{ Q\subset \DD : (4Q) \cap \DD^c \neq \emptyset \text{ and }  (2Q)\cap \DD^c = \emptyset\}.$ For those $a_Q$ atoms we can decompose them into type $(b)(p(\cdot),\infty)_\DD$-atoms via the Whitney decomposition. To do so, we consider atoms in the decomposition of $\F$ that intersect $\partial\DD.$ When we restrict the atom to $\DD$, we will lose part of its support. Without lose of generality we consider an atom $A$ fulfills $\|A\|_{L^q(\RR^n)}\leq 1,$ with $q\in(1,\infty],$ and its supported on the cube  

$$Q=\{(x',x_n) : |x_j|\leq 1/2 \text{ when } j=1,\cdots,n-1, 0<x_n\leq \alpha \},$$

for some $\alpha\in(0,1).$ From the Whitney decomposition on $\DD$ of $Q$ with respect to $\partial\DD$, we find that, the cube $Q$ is decomposed into a family of sub-cubes $\{ Q^k_j \}$ of distance $2^{-k}$ from $\partial\DD$ and there are $c_k \sim 2^{(n-1)k}$ of them. We can observe that $j$ varies from $1$ to $c_k$. From the Whitney decomposition, it follows that each cube $Q^k_j$ is a type $(b)$ cube and 

$$A(x',x_n) = \sum_{k=1}^{\infty} \sum_{j=1}^{c_k} \chi_{Q^k_j}A = \sum_{k=1}^{\infty} \sum_{j=1}^{c_k} \lambda_{Q^k_j}a_{Q^k_j},$$

where, for $j=1,\cdots,c_k$ and $k\in\NN,$
 
$$\lambda_{Q^k_j} := \|\chi_{Q^k_j}A\|_{L^\infty(\DD)}\|\chi_{Q^k_j}\|_{L^{p(\cdot)}_w(\DD)}$$ 
 
and 

$$ a_{Q^k_j} := \frac{\chi_{Q^k_j}A}{\|\chi_{Q^k_j}A\|_{L^\infty(\DD)\|\chi_{Q^k_j}\|_{L^{p(\cdot)}_w(\DD)}}}.$$
 
From the fact that $a_{Q^k_j}$ is supported in type $(b)$ cube $Q^k_j,$ we obtain 

$$ \|a_{Q^k_j}\|_{L^\infty(\DD)} \leq \frac{\chi_{Q^k_j}A}{\|\chi_{Q^k_j}A\|_{L^\infty(\DD)\|\chi_{Q^k_j}\|_{L^{p(\cdot)}_w(\DD)}}} \leq \frac{1}{\|\chi_{Q^k_j}\|_{L^{p(\cdot)}_w(\DD)}}.$$

Hence, each $a_{Q^k_j}$ is a type $(b)(p(\cdot),\infty)_\DD-$atom, Moreover, by combining the H\"older inequality, $\|A\|_{L^q(\RR^n)} \leq 1, \sum_{k=1}^{\infty}\sum_{j=1}^{c_k} |Q^k_j|= |Q|$ and $\|\chi_{Q}\|_{L^{p(\cdot)}_w(\RR^n)} \lesssim |Q|^{1/p_+},$ we find out that 

\begin{eqnarray}\label{29}
\nonumber\mathcal{B}(\{\lambda_{Q^k_j}\}_{j,k},\{Q^k_j\}_{j,k})&=&\mathcal{A}'(\{\lambda_{Q^k_j}\}_{j,k},\{Q^k_j\}_{j,k})\\
&\leq & \Bigg\| \Big(\sum_{k=1}^{\infty}\sum_{j=1}^{c_k} \|\chi_{Q^k_j}A\|_{L^\infty(\DD)}\Big)\Big(\sum_{k=1}^{\infty}\sum_{j=1}^{c_k} \chi_{Q^k_j}\Big)^{\frac{{1-\theta}}{\theta}}        \Bigg\|_{L^{p(\cdot)/\theta}_{w^\theta}(\RR^n)}\\ \nonumber
&\leq & \|A\|_{L^\infty(\RR^n)}\|\chi_{Q}\|_{L^{p(\cdot)}_w(\RR^n)}\\ \nonumber
&\lesssim & |Q|^{1/p_+}<\infty, 
\end{eqnarray}

which means that the atomic decomposition converges in $H^{p(\cdot)}_w(\RR^n).$ Furthermore, in view of \cite[Theorem 3.7]{MSM}, \eqref{27}, \eqref{28} and \eqref{29}, we get

\begin{align*}
\|f\|_{H^{p(\cdot)}(\DD)} &\sim \|f\|_{H^{p(\cdot),q}_{q,\mathrm{atom}}(\DD)}\\
&\leq  \mathcal{A}(\{\lambda_{Q}\}_{Q1},\{Q\}_{Q_1})+ \mathcal{B}(\{\lambda_{Q}\}_{Q_2},\{Q\}_{Q_2}) +\mathcal{B}(\{\lambda_{Q^k_j}\}_{j,k},\{Q^k_j\}_{j,k})\\
&\sim \mathcal{A}'(\{\lambda_{Q}\}_{Q1},\{Q\}_{Q_1})+ \mathcal{A}'(\{\lambda_{Q}\}_{Q_2},\{Q\}_{Q_2}) +\mathcal{A}'(\{\lambda_{Q^k_j}\}_{j,k},\{Q^k_j\}_{j,k}) < \infty,
\end{align*}
from this, \eqref{26}, the definition of the variable geometric-weighted-Hardy space and \cite[Theorem 5.3]{how}, we deduce that

$$\|f\|_{H^{p(\cdot)}_{w}(\DD)} \leq \|\F\|_{H^{p(\cdot),q}_{w,\mathrm{atom}}(\RR^n)} \lesssim \|f\|_{H^{p(\cdot)}_{r,w}(\DD)} < \infty,$$

which implies that $f\in H^{p(\cdot)}(\DD).$

Let us now turn out to show that, for $f\in H^{p(\cdot)}_w(\DD)$ then we have, $ \|f\|_{H^{p(\cdot)}_{w,r}(\DD)} \leq \|f\|_{H^{p(\cdot)}_{w}(\DD)}.$ Let $f\in H^{p(\cdot)}_{w}(\DD)$, then by \cite[Theorem 3.7]{MSM} and Definition \ref{atom}, we know that there exist two sequences ${\lambda_j}_{j\in\NN} \subset \CC$ and ${\kappa_j}_{j\in\NN}\subset \CC,$  a sequence ${a_j}_{j\in\NN}$ of type $(a)(p(\cdot),q)_\DD$atoms and  a sequence ${b_j}_{j\in\NN}$ of type $(b)(p(\cdot),q)_\DD$atoms such that 

\begin{equation}\label{30}
f = \sum_{j\in\NN} \lambda_j a_j + \sum_{j\in\NN} \kappa_j b_j \quad \text{in  } \mathcal{D}'(\DD),
\end{equation}
   
and
   
$$ \mathcal{A}(\{\lambda_{j}\}_{j\in\NN},\{Q_j\}_{j\in\NN})+ \mathcal{B}(\{\kappa_{j}\}_{j\in\NN},\{\tilde{Q}_j\}_{j\in\NN})<\infty.$$

The type $(a)(p(\cdot),q)_\DD$atoms $a_j$ in \eqref{30} are already $(p(\cdot),q)$atoms and we have 

$$ \mathcal{A}(\{\lambda_{j}\}_{j\in\NN},\{Q_j\}_{j\in\NN}) = \mathcal{A}'(\{\lambda_{j}\}_{j\in\NN},\{Q_j\}_{j\in\NN}) < \infty.$$
   
While, we treat the $(b)(p(\cdot),q)_\DD$atoms $b_j$ in \eqref{30} in two different cases.

\textbf{Case 1.} If $b_j$ is of type $(b)(p(\cdot),q)_\DD$ atoms as in \eqref{30} and is supported on a type $(b)$ cube $Q$ with $\ell(Q) \leq \epsilon$ for some $\epsilon$ sufficiently small $\epsilon>0$, then we find a cube $\tilde{Q} \subset (\bar{\partial\DD})^c$ which has the same size of $Q$. Now, we set the extension $(b_j)_{*}$ of function $b_j$ as follows:

$$(b_j)_{*}(x):=\begin{cases}
      b_j(x)\quad\text{for}\quad x\in Q;\\
      -\frac{1}{|Q|}\int_{Q}b_j(y)dy\quad\text{for}\quad x\in\tilde{Q},
      \end{cases}$$
from this, we infer that the function $(b_j)_{*}$ is supported on $Q\cup \tilde{Q}$. As the distance between $Q$ and $\tilde{Q}$ to $\partial\DD$ is comparable to $\ell(Q)$, we may find another cube denoted $\hat{Q}$ such that $Q\cup \tilde{Q}$ is a subset of $\hat{Q}$ and $|Q| \leq |\hat{Q}| \lesssim |Q|.$ From this and the H\"older inequality, we obtain   
   
\begin{align*}
\|(b_j)_{*}\|_{L^q(\RR^n)} &\leq \|b_j\|_{L^q(\RR^n)} + \Big\| \bigg( -\frac{1}{|Q|}\int_Q b_j(y)dy \bigg)\chi_{\tilde{Q}}  \Big\|_{L^q(\RR^n)}\\
&\leq \|b_j\|_{L^q(\RR^n)} + |\tilde{Q}||Q|^{-1}\int_Q b_j(y)dy\\
&\leq \|b_j\|_{L^q(\RR^n)} + |\tilde{Q}|^{1/q}|Q|^{-1+1/q'}\|b_j\|_{L^q(\RR^n)} \lesssim \frac{|\hat{Q}|^{1/q}}{\|\chi_{\hat{Q}}\|_{L^{p(\cdot)}_w(\RR^n)}}
\end{align*}   
   
and 

\begin{align*}
\int_{\RR^n} (b_j)_{*}(x)dx &= \int_Q b_j(x)dx - \int_{\tilde{Q}} \big(\frac{1}{|Q|}\int_Q b_j(y)dy\big)\chi_{\tilde{Q}}(x)dx\\
&= \int_Q b_j(x)dx - |\tilde{Q}||Q|^{-1}\int_{Q}b_j(x)dx = 0.
\end{align*}   
   
Thus, we find out

\begin{equation}\label{32}
\mathcal{A}'(\{\kappa_{Q}\}_{Q_3},\{Q\}_{Q_3}) = \mathcal{B}(\{\kappa_{Q}\}_{Q_3},\{Q\}_{Q_3}) < \infty,
\end{equation}   
   
where $Q_3 := \{ Q \subset \DD : Q \text{ be a type } (b) \text{ cube with} \ell(Q) \leq \epsilon \}.$

\textbf{Case 2.}  If $b_j$ is a type $(b)(p(\cdot,q)_\DD$atom
in \eqref{30} and is supported on a type $(b)$ cube $Q$ with $\ell(Q) > \epsilon$, then from the fact that $\DD \subset \RR^n$ is a bounded Lipschitz domain, we find out that there exists a cube $\tilde{Q} \subset (\bar{\partial\DD})^c$, such that $\ell(\tilde{Q}) = \ell(Q)$ and $\text{dist}(Q,\tilde{Q}) \lesssim \ell(Q).$ Then we can find another cube denoted $\hat{Q}$, such that $Q\cup \tilde{Q} \subset \hat{Q}$ and $\ell(\hat{Q}) = \ell(Q).$ As in \textbf{Case 1.} we define the function   

$$(b_j)_{\sharp}(x):=\begin{cases}
      b_j(x)\quad\text{for}\quad x\in Q;\\
      -\frac{1}{|Q|}\int_{Q}b_j(y)dy\quad\text{for}\quad x\in\tilde{Q},
      \end{cases}$$

which implies that $(b_j)_{\sharp}$ is supported on $Q\cup \tilde{Q},$ such that


$$ \|(b_j)_{\sharp}\|_{L^q(\RR^n)} \leq \|b_j\|_{L^q(\RR^n)} + |Q|^{-1}\|(\int_Qb_j(y)dy)\chi_{\tilde{Q}}\|_{L^q(\RR^n)} \lesssim \frac{|Q|^{1/q}}{\|\chi_{\tilde{Q}}\|_{L^{p(\cdot)}_w(\RR^n)}}$$


and 


\begin{align*}
\int_{\RR^n} (b_j)_{\sharp}(x)dx &= \int_Q b_j(x)dx - \int_{\tilde{Q}} \big(\frac{1}{|Q|}\int_Q b_j(y)dy\big)\chi_{\tilde{Q}}(x)dx\\
&= \int_Q b_j(x)dx - |\tilde{Q}||Q|^{-1}\int_{Q}b_j(x)dx = 0.
\end{align*}

Consequently, we obtain

\begin{equation}\label{33}
\mathcal{A}'(\{\kappa_{Q}\}_{Q_4},\{Q\}_{Q_4}) = \mathcal{B}(\{\kappa_{Q}\}_{Q_4},\{Q\}_{Q_4}) < \infty,
\end{equation}   

where $Q_4 := \{ Q\subset \DD : Q \text{ be a type (b) cube with} \ell(Q) > \epsilon   \}.$ Thus By Definition of the space $H^{p(\cdot)}_{w,r}(\DD)$, \eqref{26}, \eqref{32}, \eqref{33} and \cite[Theorem 5.3]{how}, we infer that

\begin{align*}
 \| f \|_{H^{p(\cdot)}_{w,r}(\DD)} &\leq \|\F\|_{H^{p(\cdot)}_w(\RR^n)} \sim \|\F\|_{H^{p(\cdot),q}_{w,\rm{atom}}(\RR^n)}\\
&\leq  \mathcal{A}'(\{\kappa_{\lambda_j}\}_{j\in\NN},\{Q_j\}_{j\in\NN}) + \mathcal{A}'(\{\kappa_{Q}\}_{Q_3},\{Q\}_{Q_3}) + \mathcal{A}'(\{\kappa_{Q}\}_{Q_4},\{Q\}_{Q_4}) < \infty. 
\end{align*}

In view of \cite[Theorem 3.7]{MSM}, we obtain that

$$ \|f\|_{H^{p(\cdot)}_{w,r}(\DD)} \leq \|f\|_{H^{p(\cdot),q}_{w,\rm{atom}}(\DD)} \sim \|f\|_{H^{p(\cdot)}_w(\DD)} < \infty,$$
which is the required result.
\end{proof}



\begin{thebibliography}{9999}

\bibitem{r4}
Chang, D.-C.: {The dual of Hardy spaces on a bounded domain in $\RR^n$.} Forum Math. 6, 65–81 (1994)

\bibitem{Cruz2006}
Cruz-Uribe, D., Fiorenza, A., Martell J., and P\'{e}rez C.: \emph{The
  boundedness of classical operators on variable $L^p$ spaces.} Ann. Acad. Sci. Fenn. Math. \textbf{31}, 239--264 (2006)

\bibitem{Diening2011}
Diening, L., Harjulehto, P., {H\"{a}st\"{o}} P. and {Ru\v{z}i\v{c}ka}, M.: \emph{Lebesgue and Sobolev spaces with variable exponents.} Lecture Notes
  in Mathematics, vol. 2017, Springer, Heidelberg (2011)

\bibitem{S19}
Fan, X. and Zhao, D.: \emph{On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$.} J. Math. Anal. Appl. \textbf{263}, 424--446 (2001)

\bibitem{how}
K.-P. Ho, \emph{Atomic decompositions of weighted Hardy spaces with variable exponents.} Tohoku Math. J. (2)
69 (2017), no. 3, 383–413.

\bibitem{Izu}
Izuki, M., Nakai, E. and Sawano, Y.: Function spaces with variable exponents-an introduction. Sci. Math. Jpn. \textbf{77}, 187--315 (2014)

\bibitem{S44}
Kova\'{a}\v{c}k, O. and R\'{a}kosn\'{\i}k, J.: \emph{On spaces $L^{p(x)}$ and $W^{k,p(x)}$}. Czechoslovak Math. J. \textbf{41} , 592--618 (1991)

\bibitem{Orlicz}
Orlicz, W.: \emph{Uber konjugierte Expoentenfolgen.} Studia Math. \textbf{3}, 200--211 (1931)

\bibitem{Liu}
Liu, X.:
Atomic characterizations of variable Hardy spaces on domains and their applications. Banach J. Math. Anal. \textbf{15}, 26 (2021)


\bibitem {MSM} 
Melkemi, O., Saibi, K. and Mokhtari, Z. \emph{Weighted variable Hardy spaces on domains.} Adv. Oper. Theory 6, 56 (2021). https/ doi.org/10.1007/s43036-021-00151-4


\bibitem{miyachi}
Miyachi, A.: Maximal functions for distributions on open sets.
  Hitotsubashi J. Arts Sci.  \textbf{28}, 45--58 (1987)


\bibitem{Nakai2012}
Nakai, E. and Sawano, Y.: Hardy spaces with variable exponents and
  generalized {C}ampanato spaces. J. Funct. Anal. \textbf{262},
  3665--3748 (2012)

\bibitem{Khedoudj2}
Saibi, K.: Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces Journal of Function Spaces. vol. 2020, Article ID 2681719, 9 pages (2020)

\bibitem{sawano}
Sawano, Y. : \emph{ Atomic Decompositions of Hardy Spaces with Variables Exponents and its Applications to Bounded Linear Operators.} Integr. Equ. Oper. Theory 77, 123--148 (2013) 

\bibitem{Stein}
Stein, E. M. and Weiss G.: \emph{On the theory of harmonic functions of several variables: I. The theory of Hp-spaces.} Acta Mathematica. vol. 103, 25--62(1960)

\bibitem{yang}
Yang, D., Zhuo, C. and Nakai, E.: \emph{Characterizations of variable exponent Hardy spaces via Riesz transforms,} Rev. Mat. Complut. \textbf{29}, no. 2, 245-270 (2016)

\end{thebibliography}

\end{document}
