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\noindent\parbox{2.95cm}{\includegraphics*[keepaspectratio=true,scale=0.125]{AFA.jpg}}
\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
\qquad Ann. Funct. Anal. 3 (2012), no. 1, 128--141\\
{\footnotesize \qquad \textsc{\textbf{$\mathscr{A}$}nnals of
\textbf{$\mathscr{F}$}unctional
\textbf{$\mathscr{A}$}nalysis}\\
\qquad ISSN: 2008-8752 (electronic)\\
\qquad URL:
\textcolor[rgb]{0.00,0.00,0.99}{www.emis.de/journals/AFA/}
}\\[.5in]}

\title[Wavelet characterization of Herz-type Hardy spaces]{The wavelet characterization of Herz-type Hardy spaces with variable exponent}

\author[H. Wang, Z. Liu]{Hongbin Wang$^1$$^{*}$ and Zongguang Liu$^2$}

\address{$^{1}$ Department of Mathematics, China University of Mining and
Technology(Beijing), Beijing 100083, P. R. China.}
\email{\textcolor[rgb]{0.00,0.00,0.84}{hbwang\_2006@163.com}}

\address{$^{2}$ Department of Mathematics, China University of Mining and
Technology(Beijing), Beijing 100083, P. R. China.}
\email{\textcolor[rgb]{0.00,0.00,0.84}{liuzg@cumtb.edu.cn}}

\dedicatory{{\rm Communicated by O. Christensen }}

\subjclass[2010]{Primary 46E30; Secondary 42B35, 42C40.}

\keywords{Herz-type Hardy space, variable exponent, tent space,
wavelet.}

\date{Received: 27 November 2011; Accepted: 2 March 2012.
\newline \indent $^{*}$ Corresponding author}

\begin{abstract}
In this paper, using the atomic theory of the Herz-type Hardy spaces
with variable exponent, we give their wavelet characterization by
means of some discrete tent spaces with variable exponent at the
origin.
\end{abstract} \maketitle

\section{Introduction and preliminaries}

The theory of function spaces with variable exponent has developed
since the paper \cite{K-R} of Kov\'{a}\v{c}ik and J.R\'{a}kosn\'{i}k
appeared in 1991. In \cite{HWY,L-Y}, Hern\'{a}ndez, Lu, Weiss and
Yang gave the $\varphi$-transform and wavelet characterizations of
Herz-type spaces. In addition, Kopaliani and Izuki introduced the
wavelets inequalities of Lebesgue spaces with variable exponent in
\cite{KOP2} and \cite{IZU1}, respectively. Recently, the authors
\cite{W-L} defined the Herz-type Hardy spaces with variable exponent
and gave their atomic characterizations.

Inspired by the aforementioned references, we give the wavelet
characterization of the Herz-type Hardy spaces with variable
exponent by using the atomic decomposition theory in Section 3. And
for this purpose, firstly in Section 2 we will introduce a kind of
discrete tent space with variable exponent.

To be precise, we first briefly recall some standard notations in
the remainder of this section. Given an open set $\Omega\subset
\mathbb{R}^{n}$, and a measurable function
$p(\cdot):\Omega\rightarrow[1,\infty),$ $L^{p(\cdot)}(\Omega)$
denotes the set of measurable functions $f$ on $\Omega$ such that
for some $\lambda>0,$
$$\int_\Omega\left(\frac{|f(x)|}{\lambda}\right)^{p(x)}dx < \infty.$$
This set becomes a Banach function space when equipped with the
Luxemburg-Nakano norm
$$\|f\|_{L^{p(\cdot)}(\Omega)}=\inf\left\{\lambda>0:\int_\Omega
\left(\frac{|f(x)|}{\lambda}\right)^{p(x)}dx \leq 1\right\}.$$ These
spaces are referred to as variable Lebesgue spaces or, more simply,
as variable $L^{p}$ spaces, since they generalized the standard
$L^{p}$ spaces: if $p(x)=p$ is a constant, then
$L^{p(\cdot)}(\Omega)$ is isometrically isomorphic to
$L^{p}(\Omega)$. The variable $L^{p}$ spaces are a special case of
Musielak-Orlicz spaces.

For all compact subsets $E\subset \Omega$, the space $L_{\rm
loc}^{p(\cdot)}(\Omega)$ is defined by $L_{\rm
loc}^{p(\cdot)}(\Omega):=\{f: f\in L^{p(\cdot)}(E)\}.$ Define
$\mathcal{P}(\Omega)$ to be the set of
$p(\cdot):\Omega\rightarrow[1,\infty)$ such that
$$1<p^{-}=\mathrm{ess} \inf\{p(x):x\in\Omega\}\leq\mathrm{ess} \sup\{p(x):x\in\Omega\}=p^{+}<\infty.$$
Denote $p'(x)=p(x)/(p(x)-1).$ Let $\mathcal{B}(\mathbb{R}^{n})$ be
the set of $p(\cdot)\in$ $\mathcal{P}(\mathbb{R}^{n})$
 such that the Hardy-Littlewood maximal operator
$M$ is bounded on $L^{p(\cdot)}(\mathbb{R}^{n})$.

In addition, we denote the Lebesgue measure and the characteristic
function of a measurable set $A\subset \mathbb{R}^{n}$ by $|A|$ and
$\chi_A$, respectively.

\begin{lemma}(\cite{K-R})\label{1.1}
Let $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$. If $f\in
L^{p(\cdot)}(\mathbb{R}^{n})$ and $g\in
L^{p'(\cdot)}(\mathbb{R}^{n})$, then $fg$ is integrable on
$\mathbb{R}^{n}$ and
$$\int_{\mathbb{R}^{n}}|f(x)g(x)|dx \leq r_{p}\|f\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})},$$
where $$r_{p}=1+1/p^--1/p^+.$$

\end{lemma}

This inequality is named the generalized H\"{o}lder inequality with
respect to the variable $L^{p}$ spaces.

\begin{lemma}(\cite{IZU2})\label{1.2}
Let $q(\cdot)\in \mathcal{B}(\mathbb{R}^{n})$. Then there exists a
positive constant $C$ such that for all balls $B$ in
$\mathbb{R}^{n}$ and all measurable subsets $S\subset B$,

$$\displaystyle\frac{\|\chi_B\|_{L^{q(\cdot)}(\mathbb{R}^{n})}}{\|\chi_S\|_{L^{q(\cdot)}(\mathbb{R}^{n})}}\leq
C \frac{|B|}{|S|},$$
$$\frac{\|\chi_S\|_{L^{q(\cdot)}(\mathbb{R}^{n})}}{\|\chi_B\|_{L^{q(\cdot)}(\mathbb{R}^{n})}}\leq
C \left(\frac{|S|}{|B|}\right)^{\delta_1}$$ and
$$\displaystyle\frac{\|\chi_S\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}}{\|\chi_B\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}}\leq
C \left(\frac{|S|}{|B|}\right)^{\delta_2},$$

\noindent where $\delta_1, \delta_2$ are constants with $0<\delta_1,
\delta_2<1$.

\end{lemma}

\begin{remark}
The conclusions of Lemma \ref{1.2} are true if we replace the balls
$B$ by the cubes $Q$.

\end{remark}

\begin{remark}
Throughout this paper $\delta_2$ is the same as in Lemma \ref{1.2}.

\end{remark}

\begin{lemma}(\cite{DIE,KOP1,LER})\label{1.5}
Suppose $q(\cdot)\in \mathcal{B}(\mathbb{R}^{n})$. Then there exists
a constant $C>0$ such that for all cubes $Q$ in $\mathbb{R}^{n}$,
$$\frac{1}{|Q|}\|\chi_Q\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\|\chi_Q\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}\leq
C.$$

\end{lemma}

Next we give the definition of the Herz spaces with variable
exponent. Let $Q_k=\{x=(x_1,\cdot\cdot\cdot,x_n)\in \mathbb{R}^{n}:
|x_i|\leq 2^k\}$ and $A_k=Q_k\setminus Q_{k-1}$ for $k\in
\mathbb{Z}$. Denote $\mathbb{Z}_+$ as the set of positive integers,
$\chi_k=\chi_{A_k}$ for $k\in \mathbb{Z}$, $\tilde{\chi}_k=\chi_k$
if $k\in \mathbb{Z}_+$ and $\tilde{\chi}_0=\chi_{Q_0}$. Similar to
the definition of \cite{IZU2}, we have

\begin{definition}
Let $\alpha\in \mathbb{R}, 0<p<\infty$ and $q(\cdot)\in
\mathcal{P}(\mathbb{R}^{n})$. The homogeneous Herz space
$\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$ is defined by
$$\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})=\{f\in L_{\rm
loc}^{q(\cdot)}(\mathbb{R}^{n}\setminus \{0\}):
\|f\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}<\infty\},$$
where
$$\|f\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}=\left\{\sum_{k=-\infty}^\infty2^{k\alpha
p}\|f\chi_k\|^p_{L^{q(\cdot)}(\mathbb{R}^{n})}\right\}^{1/p}.$$ The
non-homogeneous Herz space $K^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$
is defined by $$K^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})=\{f\in
L_{\rm loc}^{q(\cdot)}(\mathbb{R}^{n}):
\|f\|_{K^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}<\infty\},$$ where
$$\|f\|_{K^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}=\left\{\sum_{k=0}^\infty2^{k\alpha
p}\|f\tilde{\chi}_k\|^p_{L^{q(\cdot)}(\mathbb{R}^{n})}\right\}^{1/p}.$$

\end{definition}

In \cite{W-L}, we gave the definitions of Herz-type Hardy spaces
with variable exponent and their atomic decomposition
characterizations. $\mathcal{S}(\mathbb{R}^{n})$ denotes the space
of Schwartz functions, and $\mathcal{S'}(\mathbb{R}^{n})$ denotes
the dual space of $\mathcal{S}(\mathbb{R}^{n})$. Let $G_Nf(x)$ be
the grand maximal function of $f(x)$ defined by
$$G_Nf(x)=\sup_{\phi\in\mathcal{A}_N}|\phi^*_\nabla(f)(x)|,$$
where
$\displaystyle\mathcal{A}_N=\{\phi\in\mathcal{S}(\mathbb{R}^{n}):\sup_{|\alpha|,|\beta|\leq
N}|x^\alpha D^\beta\phi(x)|\leq 1\}$ and $N>n+1$, $\phi^*_\nabla$ is
the nontangential maximal operator defined by
$$\phi^*_\nabla(f)(x)=\sup_{|y-x|<t}|\phi_t\ast f(y)|$$
with $\phi_t(x)=t^{-n}\phi(x/t)$.

\begin{definition}(\cite{W-L}) Let $\alpha\in\mathbb{R}, 0<p<\infty, q(\cdot)\in
\mathcal{P}(\mathbb{R}^{n})$ and $N>n+1$.

(i)The homogeneous Herz-type Hardy space
$H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$ is defined by
$$H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})=\{f\in \mathcal{S'}(\mathbb{R}^{n}):
G_Nf(x)\in\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})\}$$ and we
define
$\|f\|_{H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}=\|G_Nf\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}$.

(ii)The non-homogeneous Herz-type Hardy space
$HK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$ is defined by
$$HK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})=\{f\in \mathcal{S'}(\mathbb{R}^{n}):
G_Nf(x)\in K^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})\}$$ and we define
$\|f\|_{HK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}=\|G_Nf\|_{K^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}$.

\end{definition}

For $x\in\mathbb{R}$ we denote by $[x]$ the largest integer less
than or equal to $x$. Similar to the results of \cite{W-L}, we have
the following definition and lemma.

\begin{definition} Let
$n\delta_2\leq\alpha<\infty, q(\cdot)\in
\mathcal{B}(\mathbb{R}^{n})$ and non-negative integer $s\geq
[\alpha-n\delta_2]$.

(i) A function $a$ on $\mathbb{R}^{n}$ is said to be a central
$(\alpha, q(\cdot))$-atom, if it satisfies

\hspace{3mm}(1) supp\,$a\subset B(0,r)=\{x\in
\mathbb{R}^{n}:|x|<r\}$, for some $r>0$.

\hspace{3mm}(2) $\|a\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq
|B(0,r)|^{-\alpha/n}$.

\hspace{3mm}(3) $\int_{\mathbb{R}^{n}}a(x)x^\beta dx=0$, for any
multi-index $\beta$ with $|\beta|\leq s$.

(ii) A function $a$ on $\mathbb{R}^{n}$ is said to be a central
$(\alpha, q(\cdot))$-atom of restricted type, if it satisfies the
conditions (2), (3) above and

\hspace{3mm}(1)$'$ supp\,$a\subset B(0,r)$, for some $r\geq 1$.

\end{definition}

\begin{lemma}\label{1.9} Let
$n\delta_2\leq\alpha<\infty, 0<p<\infty$ and $q(\cdot)\in
\mathcal{B}(\mathbb{R}^{n})$. Then $f\in
H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$(or
$HK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$) if and only if
$$f=\sum_{k=-\infty}^\infty\lambda_ka_k\,\left(\mathrm{or}\,\sum_{k=0}^\infty\lambda_ka_k\right),\quad \mathrm{in\,\, the\,\, sense\,\, of\,\,} \mathcal{S'}(\mathbb{R}^{n}),$$ where each
$a_k$ is a central $(\alpha, q(\cdot))$-atom (or central $(\alpha,
q(\cdot))$-atom of restricted type) with support contained
in $Q_k$ and\\
$\displaystyle\sum_{k=-\infty}^\infty|\lambda_k|^p<\infty$(or
$\displaystyle\sum_{k=0}^\infty|\lambda_k|^p<\infty$). Moreover,
$$\|f\|_{H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}\approx\inf\left(\sum_{k=-\infty}^\infty|\lambda_k|^p\right)^{1/p}\,\left(\mathrm{or}\,\|f\|_{HK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}\approx\inf\left(\sum_{k=0}^\infty|\lambda_k|^p\right)^{1/p}\right),$$
where the infimum is taken over all above decompositions of $f$.

\end{lemma}

\section{The discrete Herz-type tent spaces with variable exponent}

In this section, we introduce a kind of discrete tent space with
variable exponent to establish the wavelet characterization of
Herz-type Hardy spaces with variable exponent. Let
$\nu\in\mathbb{Z}$ and $K\in\mathbb{Z}^n$. Define
$Q_{\nu,K}=\{x=(x_1,...,x_n)\in\mathbb{R}^n: 2^\nu x-K\in [0,1)^n\}$
and $\mathcal{D}=\{Q_{\nu,K}:\nu\in\mathbb{Z},K\in\mathbb{Z}^n\}$.
Moreover, we set $\displaystyle
s(\beta)(x)=\left(\sum_{Q\in\mathcal{D}}|\beta(Q)|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}$
and supp\,$\displaystyle\beta=\bigcup_{\{Q\in\mathcal{D}:
\beta(Q)\neq 0\}}Q$, where $\beta=\{\beta(Q)\}_{Q\in\mathcal{D}}$ is
a complex numerical series.

\begin{definition}\label{2.1} Let $0<\alpha<\infty, 0<p<
\infty, q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ and
$\beta=\{\beta(Q)\}_{Q\in\mathcal{D}}$. The tent space associated
with $\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$ is defined by
$$T\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})=\{\beta:s(\beta)\in \dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})\}$$
and we define
$\|\beta\|_{T\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}=\|s(\beta)\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}$.

\end{definition}

Similarly, we can define the space
$TK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$ replacing
$\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$ by
$K^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$ in Definition \ref{2.1}.

Firstly we establish the central $(\alpha, q(\cdot))$-atom-sequence
decomposition characterization of the space
$T\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$.

\begin{definition} Let $0<\alpha<\infty$ and
$q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$. If there is a cube $R$
with the center at the origin, such that $R\supset
\mathrm{supp}\,\beta$ and
$$\left\|\left(\sum_{Q\in \mathcal{D}}|\beta(Q)|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}\right\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq |R|^{-\alpha/n},$$
then $\beta=\{\beta(Q)\}_{Q\in\mathcal{D}}$ is said to be a central
$(\alpha, q(\cdot))$-atom-sequence, and the smallest cube $R$ with
above property is called the base of $\beta$.

\end{definition}

\begin{theorem}\label{2.3} Let $0<\alpha<\infty, 0<p<\infty$
and $q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$. The following two
statements are equivalent:

(i) $\beta\in T\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$.

(ii) There exist a sequence $\{\beta_j\}_{j=-\infty}^\infty$ of
central $(\alpha, q(\cdot))$-atom-sequences and a sequence
$\{\lambda_j\}_{j=-\infty}^\infty$ of numbers such that
$$\mathrm{supp}\,\beta_j\subset \bigcup_{Q\in\mathcal{D}_j}Q,\,\,\beta=\sum_{j=-\infty}^\infty\lambda_j\beta_j \,\,\mathrm{and} \,\left(\sum_{j=-\infty}^\infty|\lambda_j|^p\right)^{1/p}<\infty,$$
where $\mathcal{D}_j=\{Q\in\mathcal{D}:Q\subset Q_j\setminus
Q_{j-1}\}$. Furthermore, in this case, the following two norms are
mutually equivalent:
$$\|\beta\|_{T\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}\,\,\mathrm{and}\,\inf\left\{\left(\sum_{j=-\infty}^\infty|\lambda_j|^p\right)^{1/p}\right\},$$
where the infimum is taken over all the central $(\alpha,
q(\cdot))$-atom-sequence decompositions of $\beta$.

\end{theorem}

\begin{proof} We prove (i) implies (ii) firstly. Let
$$\lambda_j=|Q_j|^{\alpha/n}\left\|\left(\sum_{Q\in \mathcal{D}}|\beta(Q)|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}\chi_j(x)\right\|_{L^{q(\cdot)}(\mathbb{R}^{n})}.$$
Define $\beta_j=\{\beta_j(Q)\}_{Q\in \mathcal{D}}$ by
$$\displaystyle \beta_j(Q)=\left\{\begin{array}{ll}
\displaystyle \lambda_j^{-1}\beta(Q),\,\mathrm{if}\,Q\in \mathcal{D}_j;\\
\displaystyle 0,\quad\quad\quad\,\, \mathrm{otherwise}.
\end{array}\right.$$
Thus we have
$\displaystyle\beta=\sum_{j=-\infty}^\infty\lambda_j\beta_j$,
supp\,$\displaystyle\beta_j\subset \bigcup_{Q\in\mathcal{D}_j}Q$ and
$$\left(\sum_{j=-\infty}^\infty|\lambda_j|^p\right)^{1/p}\leq \|s(\beta)\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}=\|\beta\|_{T\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}.$$
Moreover, we claim that $\beta_j$ is a central $(\alpha,
q(\cdot))$-atom-sequence, since
$$\begin{array}{rl}
&\displaystyle\left\|\left(\sum_{Q\in
\mathcal{D}_j}|\beta_j(Q)|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}\right\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\\
&\displaystyle=\left\|\left(\sum_{Q\in
\mathcal{D}_j}\lambda_j^{-2}|\beta(Q)|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}\right\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\\
&\displaystyle=\lambda_j^{-1}\left\|\left(\sum_{Q\in
\mathcal{D}_j}|\beta(Q)|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}\right\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\\
&\displaystyle\leq\lambda_j^{-1}\left\|\left(\sum_{Q\in
\mathcal{D}}|\beta(Q)|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}\chi_j(x)\right\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\\
&\displaystyle=\lambda_j^{-1}\lambda_j|Q_j|^{-\alpha/n}\leq
\left|\bigcup_{Q\in\mathcal{D}_j}Q\right|^{-\alpha/n}.

\end{array}$$

Next we will prove (ii) implies (i). Suppose $\beta_j$ is a central
$(\alpha, q(\cdot))$-atom-sequence with
supp\,$\displaystyle\beta_j\subset
\bigcup_{Q\in\mathcal{D}_j}Q\subset Q_j$. We consider the two cases
$0<p\leq 1$ and $1<p<\infty$.

If $0<p\leq 1$, it suffices to prove that
$$\|s(\beta_j)\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}\leq C,$$
where $C$ is a positive constant independent of $\beta_j$. It is
easy to see that supp\,$s(\beta_j)\subset Q_j$, and thus we have
$$\begin{array}{rl}
\displaystyle\|s(\beta_j)\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}
&\displaystyle=\left\{\sum_{k=-\infty}^j 2^{k\alpha p}\|s(\beta_j)\|^p_{L^{q(\cdot)}(A_k)}\right\}^{1/p}\\
&\displaystyle\leq
\|s(\beta_j)\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\left\{\sum_{k=-\infty}^j
2^{k\alpha p}\right\}^{1/p}\\
&\displaystyle\leq C|Q_j|^{-\alpha/n}2^{j\alpha}\leq C.

\end{array}$$
If $1<p<\infty$, by the Minkowski inequality, we have
$$\begin{array}{rl}
\displaystyle s(\beta)(x)
&\displaystyle=\left(\sum_{Q\in\mathcal{D}}\left|\sum_{j=-\infty}^\infty\lambda_j\beta_j(Q)\right|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}\\
&\displaystyle\leq
\sum_{j=-\infty}^\infty|\lambda_j|\left(\sum_{Q\in\mathcal{D}}\left|\beta_j(Q)\right|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}\\
&\displaystyle=\sum_{j=-\infty}^\infty|\lambda_j|s(\beta_j)(x).

\end{array}$$
This implies that
$$\begin{array}{rl}
\displaystyle\|s(\beta)\|^p_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}
&\displaystyle=\sum_{k=-\infty}^\infty 2^{k\alpha p}\|s(\beta)\chi_k\|^p_{L^{q(\cdot)}(\mathbb{R}^{n})}\\
&\displaystyle\leq
\sum_{k=-\infty}^\infty 2^{k\alpha p}\left(\sum_{j=-\infty}^\infty|\lambda_j|\|s(\beta_j)\chi_k\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\right)^p\\
&\displaystyle\leq \sum_{k=-\infty}^\infty 2^{k\alpha p}\left(\sum_{j=k-1}^\infty|\lambda_j|\|s(\beta_j)\chi_k\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\right)^p\\
&\displaystyle\leq \sum_{k=-\infty}^\infty
\left(\sum_{j=k-1}^\infty|\lambda_j|2^{k\alpha
}|Q_j|^{-\alpha/n}\right)^p\\
&\displaystyle\leq C\sum_{k=-\infty}^\infty
\left(\sum_{j=k-1}^\infty|\lambda_j|2^{(k-j)\alpha }\right)^p\\
&\displaystyle\leq C\sum_{k=-\infty}^\infty
\left(\sum_{j=k-1}^\infty|\lambda_j|^p2^{(k-j)p\alpha/2 }\right)
\left(\sum_{j=k-1}^\infty2^{(k-j)p'\alpha/2 }\right)^{p/p'}\\
&\displaystyle\leq C\sum_{j=-\infty}^\infty
|\lambda_j|^p\left(\sum_{k=-\infty}^{j+1}2^{(k-j)p\alpha/2
}\right)\\
&\displaystyle\leq C\sum_{j=-\infty}^\infty |\lambda_j|^p,

\end{array}$$
where $1/p+1/p'=1$.

Hence, (i) holds, and the proof of Theorem \ref{2.3} is completed.

\end{proof}

Similarly, we also introduce the definition of central $(\alpha,
q(\cdot))$-atom-sequence of restricted type.

\begin{definition} Let $0<\alpha<\infty$ and
$q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$. If there is a cube $R$
with the center at the origin and having side length no less than 2,
such that $R\supset \mathrm{supp}\,\beta$ and
$$\left\|\left(\sum_{Q\in \mathcal{D}}|\beta(Q)|^2|Q|^{-1}\chi_Q(x)\right)^{1/2}\right\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq |R|^{-\alpha/n},$$
then $\beta=\{\beta(Q)\}_{Q\in\mathcal{D}}$ is said to be a central
$(\alpha, q(\cdot))$-atom-sequence of restricted type, and the
smallest cube $R$ with above property is called the base of $\beta$.

\end{definition}

If the spaces
$T\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n}),\,\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$
and the central $(\alpha, q(\cdot))$-atom-sequence were replaced by
the spaces
$TK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n}),\,K^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$
and the central $(\alpha, q(\cdot))$-atom-sequence of restricted
type, then there is a similar result. The proof is similar to
Theorem \ref{2.3}, so we omit it.

\begin{theorem} Let $0<\alpha<\infty, 0<p< \infty$
and $q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$. The following two
statements are equivalent:

(i) $\beta\in TK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$.

(ii) There are a sequence $\{\beta_j\}_{j=0}^\infty$ of central
$(\alpha, q(\cdot))$-atom-sequences of restricted type and a
sequence $\{\lambda_j\}_{j=0}^\infty$ of numbers such that
$$\mathrm{supp}\,\beta_j\subset \bigcup_{Q\in\mathcal{D}_j}Q,\,\,\beta=\sum_{j=0}^\infty\lambda_j\beta_j \,\mathrm{and} \,\left(\sum_{j=0}^\infty|\lambda_j|^p\right)^{1/p}<\infty,$$
where $\mathcal{D}_j=\{Q\in\mathcal{D}:Q\subset Q_j\setminus
Q_{j-1}\}$. Moreover, in this case, the following norms are mutually
equivalent:
$$\|\beta\|_{TK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}\,\mathrm{and}\,\inf\left\{\left(\sum_{j=0}^\infty|\lambda_j|^p\right)^{1/p}\right\},$$
where the infimum is taken over all the above central $(\alpha,
q(\cdot))$-atom-sequence of restricted type decompositions of
$\beta$.

\end{theorem}

\section{ The wavelet characterization of Herz-type Hardy spaces with variable exponent}

In this section, we will use the $\gamma$-regular compactly support
wavelets obtained by Daubechies in \cite{DAU} (or see \cite{MEY}) to
give the characterization of the elements in Herz-type Hardy spaces
with variable exponent.

Set $Q_{j,K}$ and $\mathcal{D}$ be as before. Let
$E=\{0,1\}^n\setminus\{0,\cdots,0\}$, $\varphi$ and $\psi$ be
$\gamma$-regular compactly supported functions obtained by the
multiresolution approximation in \cite{DAU}. For any $\varepsilon\in
E$ and $Q\in \mathcal{D}$, set
$$\psi_Q^\varepsilon(x)=2^{nj/2}\psi^{\varepsilon_1}(2^jx_1-k_1)\cdots\psi^{\varepsilon_n}(2^jx_n-k_n),$$
where $\psi^0=\varphi$ and $\psi^1=\psi$. Let $mQ$ be the cube with
the same center as $Q$ and whose sides are $m$ times as long. It is
well known that
$\{\psi_Q^\varepsilon\}_{Q\in\mathcal{D},\varepsilon\in E}$ have the
following properties (see \cite{DAU,HWY,MEY}):

(A) $\{\psi_Q^\varepsilon\}_{Q\in\mathcal{D},\varepsilon\in E}$ is
an orthonormal basis of $L^2(\mathbb{R}^{n})$.

(B) supp\,$\psi_Q^\varepsilon\subset mQ$ with $m\geq 1$ for all
$Q\in\mathcal{D}$.

(C) For any index $\alpha\in \mathbb{N}^n$ and $|\alpha|\leq
\gamma$,
$$|\partial^\alpha \psi_Q^\varepsilon(x)|\leq C2^{nj/2}2^{j|\alpha|}.$$

(D) $\int_{\mathbb{R}^n}x^\alpha
\psi_Q^\varepsilon(x)dx=0,\,\,|\alpha|\leq \gamma$.

Let $\Lambda$ denote the set of indices $\lambda=K2^{-j}+\varepsilon
2^{-j-1}$ corresponding to $\psi_Q^\varepsilon$, where $Q=Q_{j,K}$.
For simplicity, we write $\psi_Q^\varepsilon=\psi_\lambda$ with
$\lambda\in \Lambda$ and $\langle
f,\,g\rangle=\int_{\mathbb{R}^{n}}f(x)\overline{g(x)}dx$.

\begin{theorem}\label{3.1}
Let $n\delta_2\leq\alpha<\infty, \,0<p<\infty$ and $q(\cdot)\in
\mathcal{B}(\mathbb{R}^{n})$. Suppose $f\in
\mathcal{S}(\mathbb{R}^{n}),\,\alpha(\lambda)=\langle
f,\,\psi_\lambda\rangle$ and $\displaystyle
f(x)=\sum_{\lambda\in\Lambda}\alpha(\lambda)\psi_\lambda(x)$, where
$\psi$ is a $\gamma$-regular compactly supported wavelet and
$\gamma\geq \alpha-n\delta_2+1$. Then the following two statements
are equivalent:

(1) $\displaystyle
S(f)(x)\equiv\left(\sum_{\lambda\in\Lambda}|\alpha(\lambda)|^2|Q(\lambda)|^{-1}\chi_{Q(\lambda)}(x)\right)^{1/2}
\in \dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$.

(2) $f\in H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$.

Moreover, the norms
$$\|f\|_{H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}\,\mathrm{and}\,
\|S(f)\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}$$ are
equivalent.

\end{theorem}

\begin{remark}
This result is true for the spaces
$HK^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$ and
$K^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$.

\end{remark}

To prove Theorem \ref{3.1}, we need the following lemma, which is a
kind of wavelet characterization of $L^{q(\cdot)}(\mathbb{R}^{n})$.


\begin{lemma}\label{3.3}
Let $q(\cdot)\in \mathcal{B}(\mathbb{R}^{n})$. Then there exists a
constant $C\geq 1$ such that for all $f\in
L^{q(\cdot)}(\mathbb{R}^{n})$,
$$C^{-1}\|f\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq\|S(f)\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C\|f\|_{L^{q(\cdot)}(\mathbb{R}^{n})}.$$

\end{lemma}

\begin{proof} The method of proof is similar to \cite{IZU1} or \cite{KOP2}, here we omit it. Actually, Lemma \ref{3.3} is one of
two cases in \cite{IZU1}.

\end{proof}

\noindent{\it Proof of Theorem \ref{3.1}}\quad We first show (1)
implies (2). It is easy to see that for any given $\varepsilon\in
E,\,\{\alpha^\varepsilon(Q)\}_{Q\in\mathcal{D}}\in
T\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$. Therefore, by
Theorem \ref{2.3}, there are a sequence
$\{\alpha_j^\varepsilon\}_{j=-\infty}^\infty$ of central $(\alpha,
q(\cdot))$-atom-sequences and a sequence $\{\lambda_j\}$ of numbers
such that supp\,$\displaystyle\alpha_j^\varepsilon\subset
Q_j,\,\alpha^\varepsilon=\sum_{j=-\infty}^\infty\lambda_j^\varepsilon\alpha_j^\varepsilon$
and
$$\left(\sum_{j=-\infty}^\infty|\lambda_j^\varepsilon|^p\right)^{1/p}\leq \|S(f)\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}.$$
So we have

$$\begin{array}{rl}
\displaystyle f(x)&\displaystyle=\sum_{\lambda\in\Lambda}\alpha(\lambda)\psi_\lambda(x)\\
&\displaystyle=\sum_{\varepsilon\in E}\sum_{Q\in\mathcal{D}}\alpha^\varepsilon(Q)\psi_Q^\varepsilon(x)\\
&\displaystyle=\sum_{\varepsilon\in E}\sum_{Q\in\mathcal{D}}\sum_{j=-\infty}^\infty\lambda_j^\varepsilon\alpha_j^\varepsilon(Q)\psi_Q^\varepsilon(x)\\
&\displaystyle=\sum_{\varepsilon\in
E}\sum_{j=-\infty}^\infty\lambda_j^\varepsilon\left(\sum_{Q\in\mathcal{D}}\alpha_j^\varepsilon(Q)\psi_Q^\varepsilon(x)\right).

\end{array}$$
Set $\displaystyle
a_j^\varepsilon=\sum_{Q\in\mathcal{D}}\alpha_j^\varepsilon(Q)\psi_Q^\varepsilon(x)$,
then supp\,$a_j^\varepsilon\subset mQ_j$ and by Lemma \ref{3.3} we
have

$$\begin{array}{rl}
\displaystyle \|a_j^\varepsilon\|_{L^{q(\cdot)}(\mathbb{R}^{n})}&\displaystyle\leq C\left\|\left(\sum_{Q\in\mathcal{D}}|\alpha_j^\varepsilon(Q)|^2|Q|^{-1}\chi_Q\right)^{1/2}\right\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\\
&\displaystyle\leq C|Q_j|^{-\alpha/n}.

\end{array}$$
Since $\psi$ satisfies property (D), it is easy to see that
$a_j^\varepsilon$ is a central $(\alpha, q(\cdot))$-atom up to an
absolute constant with support in $mQ_j$.

Because we only have a finite number of $\varepsilon'$ s, by Lemma
\ref{1.9} we know that $f\in
H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$. Moreover,
$$\|f\|_{H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}\leq C\|S(f)\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}.$$
This finishes the proof of the fact that (1) implies (2).

Next we need to prove that (2) implies (1) to complete the proof of
Theorem \ref{3.1}. We will consider two cases $0<p\leq 1$ and
$1<p<\infty$.

When $0<p\leq 1$, we only need to prove that
$\|S(a)\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}\leq C$ for
each central $(\alpha, q(\cdot))$-atom $a$ with support in $Q_k$,
where $C$ is independent of $k$. Let $m$ satisfy $2^{m_0}\leq
m<2^{m_0+1}$ with $m_0\in \mathbb{N}$ (see (B) at the beginning of
this section). Write

$$\begin{array}{rl}
\displaystyle
\|S(a)\|^p_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}&\displaystyle=\sum_{l=-\infty}^{k+2}2^{l\alpha
p}\|S(a)\chi_l\|^p_{L^{q(\cdot)}(\mathbb{R}^{n})}
+\sum_{l=k+3}^\infty2^{l\alpha p}\|S(a)\chi_l\|^p_{L^{q(\cdot)}(\mathbb{R}^{n})}\\
&\displaystyle\equiv I_1+I_2.

\end{array}$$
For $I_1$, by Lemma \ref{3.3} we obtain
$$I_1\leq C\sum_{l=-\infty}^{k+2}2^{l\alpha p}\|a\|^p_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C\sum_{l=-\infty}^{k+2}2^{(l-k)\alpha p}\leq C<\infty.$$
To deal with $I_2$, we first estimate $S(a)(x)$ when $x\in A_l$ with
$l\geq k+3$. Let $Q(\lambda)$ denote the cube associated with
$\lambda\in\Lambda$ and $\Lambda^k=\{\lambda\in\Lambda:
mQ(\lambda)\cap Q_k\neq\emptyset \,\mathrm{and}\,\psi_\lambda(x)\neq
0\}$. If $\lambda\in \Lambda^k$, we write the side length
$L(Q(\lambda))$ of $Q(\lambda)$ is equal to $2^{-j}$, so that
$m2^{-j}>2^{l-1}-2^{l-2}=2^{l-2}$, where $l\geq k+3$. So $j\leq
m_0+3-l$. It is observed that $a$ is a central $(\alpha,
q(\cdot))$-atom and $\psi$ is $\gamma$-regular with $\gamma\geq
\alpha-n\delta_2+1$. Let $\gamma_0= \alpha-n\delta_2$ and
$P_{\gamma_0}(x)$ be the $\gamma_0$-order Taylor expansion for
$\psi_\lambda(x)$ at 0. Then we have

$$\begin{array}{rl}
\displaystyle |\langle a,
\psi_\lambda\rangle|&\displaystyle\leq \int_{\mathbb{R}^{n}} |a(x)||\psi_\lambda(x)-P_{\gamma_0}(x)|dx\\
&\displaystyle\leq C2^{nj/2}2^{j(\gamma_0+1)}\int_{\mathbb{R}^{n}}
|a(x)||x|^{\gamma_0+1}dx\\
&\displaystyle\leq
C2^{nj/2+(j+k)(\gamma_0+1)-k\alpha}\|\chi_{Q_k}\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}.

\end{array}$$
On the other hand, it is easy to show that for any given $j$, the
number of $\lambda$ in $\Lambda^k$ is less than an absolute constant
$C_0$. Therefore, if $x\in A_l$ with $l\geq k+3$, then by the
generalized H\"{o}lder inequality we have

$$\begin{array}{rl}
\displaystyle
[S(a)(x)]^2&\displaystyle=\sum_{\lambda\in\Lambda}|\langle a,
\psi_\lambda\rangle|^2|Q(\lambda)|^{-1}\chi_{Q(\lambda)}(x)\\
&\displaystyle=\sum_{\lambda\in\Lambda^k}|\langle a,
\psi_\lambda\rangle|^2|Q(\lambda)|^{-1}\chi_{Q(\lambda)}(x)\\
&\displaystyle\leq
C_0\sum_{j=-\infty}^{m_0+3-l}2^{2j(n/2+\gamma_0+1)+2k(\gamma_0+1-\alpha)+nj}\|\chi_{Q_k}\|^2_{L^{q'(\cdot)}(\mathbb{R}^{n})}\\
&\displaystyle\leq
C_02^{2k(\gamma_0+1-\alpha)-2l(n+\gamma_0+1)}\|\chi_{Q_k}\|^2_{L^{q'(\cdot)}(\mathbb{R}^{n})}.

\end{array}$$
That is, $$S(a)(x)\leq
C_02^{k(\gamma_0+1-\alpha)-l(n+\gamma_0+1)}\|\chi_{Q_k}\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}.$$
So by Lemma \ref{1.2} and Lemma \ref{1.5} we have
$$\begin{array}{rl}
\displaystyle I_2&\displaystyle\leq C \sum_{l=k+3}^\infty
2^{(l-k)(\alpha-\gamma_0-1)p}2^{-lnp}\|\chi_{Q_k}\|^p_{L^{q'(\cdot)}(\mathbb{R}^{n})}\|\chi_{Q_l}\|^p_{L^{q(\cdot)}(\mathbb{R}^{n})}\\
&\displaystyle\leq C \sum_{l=k+3}^\infty
2^{(l-k)(\alpha-\gamma_0-1)p}2^{-lnp}\|\chi_{Q_k}\|^p_{L^{q'(\cdot)}(\mathbb{R}^{n})}(|Q_l|\|\chi_{Q_l}\|^{-1}_{L^{q'(\cdot)}(\mathbb{R}^{n})})^p\\
&\displaystyle\leq C \sum_{l=k+3}^\infty
2^{(l-k)(\alpha-\gamma_0-1)p}\left(\frac{\|\chi_{Q_k}\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{Q_l}\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}}\right)^p\\
&\displaystyle\leq C \sum_{l=k+3}^\infty
2^{(l-k)(\alpha-n\delta_2-\gamma_0-1)p}=C<\infty.

\end{array}$$

When $1<p<\infty$, take $1/p+1/p'=1$. Let $f\in
H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})$. Then $\displaystyle
f=\sum_{k=-\infty}^\infty\lambda_ka_k$ and
$$\left\{\sum_{k=-\infty}^\infty|\lambda_k|^p\right\}^{1/p}\leq
C\|f\|_{H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})},$$ where
$a_k$ is a central $(\alpha, q(\cdot))$-atom with support in $Q_k$.
It follows from the Minkowski inequality that $$S(f)(x)\leq
\sum_{k=-\infty}^\infty|\lambda_k| S(a_k)(x).$$ Write

$$\begin{array}{rl}
\displaystyle
\|S(f)\|_{\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}
&\displaystyle\leq
\left\{\sum_{k=-\infty}^\infty2^{k\alpha p}\left(\sum_{j=-\infty}^{k-3}|\lambda_j|\|S(a_j)\chi_k\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\right)^p\right\}^{1/p}\\
&\displaystyle\hspace{3mm}+\left\{\sum_{k=-\infty}^\infty2^{k\alpha p}\left(\sum_{j=k-2}^\infty|\lambda_j|\|S(a_j)\chi_k\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\right)^p\right\}^{1/p}\\
&\displaystyle\equiv II_1+II_2.

\end{array}$$
Similarly to the estimate for $I_1$, we obtain

$$\begin{array}{rl}
\displaystyle II_2^p &\displaystyle\leq
C\sum_{k=-\infty}^\infty2^{k\alpha p}\left(\sum_{j=k-2}^\infty|\lambda_j|\|a_j\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\right)^p\\
&\displaystyle\leq C\sum_{k=-\infty}^\infty2^{k\alpha p}\left(\sum_{j=k-2}^\infty|\lambda_j|2^{-j\alpha}\right)^p\\
&\displaystyle\leq C\sum_{k=-\infty}^\infty\left(\sum_{j=k-2}^\infty|\lambda_j|^p2^{(k-j)\alpha p/2}\right)\left(\sum_{j=k-2}^\infty2^{(k-j)\alpha p'/2}\right)^{p/p'}\\
&\displaystyle\leq C\sum_{j=-\infty}^\infty|\lambda_j|^p.

\end{array}$$
That is, $$II_2\leq
C\left(\sum_{j=-\infty}^\infty|\lambda_j|^p\right)^{1/p}\leq
C\|f\|_{H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}.$$ Similar
to the estimate for $I_2$, we obtain that if $x\in A_k$ with $k\geq
j+3$, then $$S(a_j)(x)\leq
C2^{j(\gamma_0+1-\alpha)-k(n+\gamma_0+1)}\|\chi_{Q_j}\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}.$$
So by Lemma \ref{1.2} and Lemma \ref{1.5} we have

$$\begin{array}{rl}
\displaystyle II_1^p &\displaystyle\leq
C\sum_{k=-\infty}^\infty2^{k\alpha p}\left(\sum_{j=-\infty}^{k-3}|\lambda_j|2^{j(\gamma_0+1-\alpha)-k(n+\gamma_0+1)}\|\chi_{Q_j}\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}\|\chi_{Q_k}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\right)^p\\
&\displaystyle\leq
C\sum_{k=-\infty}^\infty2^{k\alpha p}\\
&\displaystyle\hspace{3mm}\times\left(\sum_{j=-\infty}^{k-3}|\lambda_j|2^{j(\gamma_0+1-\alpha)-k(n+\gamma_0+1)}\|\chi_{Q_j}\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}\left(|Q_k|\|\chi_{Q_k}\|^{-1}_{L^{q'(\cdot)}(\mathbb{R}^{n})}\right)\right)^p\\
&\displaystyle\leq
C\sum_{k=-\infty}^\infty2^{k\alpha p}\left(\sum_{j=-\infty}^{k-3}|\lambda_j|2^{j(\gamma_0+1-\alpha)-k(\gamma_0+1)}\left(\frac{\|\chi_{Q_j}\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{Q_k}\|_{L^{q'(\cdot)}(\mathbb{R}^{n})}}\right)\right)^p\\
&\displaystyle\leq C\sum_{k=-\infty}^\infty\left(\sum_{j=-\infty}^{k-3}|\lambda_j|2^{(j-k)(\gamma_0+1+n\delta_2-\alpha)}\right)^p\\
&\displaystyle\leq C\sum_{j=-\infty}^\infty|\lambda_j|^p.

\end{array}$$
That is, $$II_1\leq
C\left(\sum_{j=-\infty}^\infty|\lambda_j|^p\right)^{1/p}\leq
C\|f\|_{H\dot{K}^{\alpha,p}_{q(\cdot)}(\mathbb{R}^{n})}.$$ This
finishes the proof of the fact that (2) implies (1).

Thus we finish the proof of Theorem \ref{3.1}.

\qed


\textbf{Acknowledgement.} The authors are very grateful to the
referees for their valuable comments. This work was supported by the
National Natural Science Foundation of China (Grant Nos. 11001266
and 11171345) and the Fundamental Research Funds for the Central
Universities (Grant No. 2010YS01).


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\end{document}

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