对任意的$0<\beta\leq 1$, 设${\cal C}_\beta$是由函数$\phi: {\Bbb R}^n\rightarrow{\Bbb R}$构成的函数族, 其中$\phi$满足${\rm supp}\phi\subset\{x\in{\Bbb R}^n: |x|\leq 1\}$, $\int_{{\Bbb R}^n}\phi(x){\rm d}x=0$且对任意的$x_1, x_2\in{\Bbb R}^n$, 有

$ \begin{equation}\label{eq:a1} |\phi(x_1)-\phi(x_2)|\leq |x_1-x_2|^\beta. \end{equation} $

(1.1)

对于$(y, t)\in{\Bbb R}^{n+1}_+={\Bbb R}^n\times(0, \infty)$和$f\in L^1_{\rm loc}({\Bbb R}^n)$, 记

$ \begin{equation}\label{eq:a2} A_\beta(f)(y, t)=\sup\limits_{\phi\in{\cal C}_\beta}|f\ast\phi_t(y)|=\sup\limits_{\phi\in{\cal C}_\beta} \bigg|\int_{{\Bbb R}^n}\phi_t(y-z)f(z){\rm d}z\bigg|. \end{equation} $

(1.2)

本征平方函数$S_\beta$定义如下

$ \begin{equation}\label{eq:a3} S_\beta(f)(x)=\bigg(\int\!\!\!\int_{\Gamma(x)}(A_\beta(f)(y, t))^2\frac{ {\rm d}y{\rm d}t}{t^{n+1}}\bigg)^{\frac{1}{2}}, \end{equation} $

(1.3)

其中$\phi_t(x)=\frac{1}{t^n}\phi(\frac{x}{t})$, $\Gamma(x)=\{(y, t)\in{\Bbb R}^{n+1}_+: |x-y|<t\}$.

Wilson^[1]首先定义了这类新的本征平方函数$S_\beta$为了解决Fefferman和Stein在文献[2]中提出的关于经典Lusin面积函数$S$在加权Lebesgue空间上有界性的猜想.随后, 许多作者还研究了利用这类本征平方函数$S_\beta$刻画一般的函数空间, 如Hardy空间, Musielak-Orlicz Hardy空间等, 见文献[3-4].关于本征平方函数$S_\beta$的近期进展及广泛应用, 见文献[5-6]等.

另一方面, 由于Kováčik和Rákosník^[7]的基础性工作, 具有变指数Lebesgue空间及其它的一些函数空间得到了广泛的研究, 见文献[8-13]. 2012年, Almeida和Drihem^[14]定义了变指数Herz空间$\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$和$K_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$, 其中$\alpha$和$p$均为变指数.当指数$\alpha$和$p$满足一些自然的正则性条件如log-Hölder连续条件时, 他们建立了一类次线性算子(包含Hardy-Littlewood极大算子, Riesz位势算子和Calderón-Zygmund算子)在这些变指数Herz空间上的有界性. 2015年, Dong和Xu^[15]定义了变指数Herz-Hardy空间$H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$和$HK_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$, 并建立了它们的原子分解.利用这些分解, 他们证明了一些奇异积分算子, 如Calderón-Zygmund算子等的有界性.关于其它的变指数函数空间, 如变指数Morrey空间, 变指数Hardy空间等, 见文献[16-21].

变指数函数空间与经典的函数空间有许多本质的区别, 如变指数Lebesgue空间不具有平移不变性, 即若$p(\cdot)$在${\Bbb{ R}}^n$中不恒为常数, 则总是存在$f\in L^{p(\cdot)}({\Bbb{ R}}^n)$和$h\in {{\Bbb{ R}}^n}$, 使得$f(x+h)\notin L^{p(\cdot)}({\Bbb{ R}}^n)$.因而, 在经典Lebesgue空间中常用的good-$\lambda$方法及一些重排不等式在变指数Lebesgue空间中不再适用, 见文献[22-23].因为这个原因, 近二十年来将经典函数空间中一些重要结果推广到变指数情形吸引了越来越多学者的兴趣.最近, Wang^[24-25]证明了本征平方函数$S_\beta$在经典Herz及Herz-Hardy空间上的有界性, 受此启发, 该文将进一步在具有变指数的Herz及Herz-Hardy空间上考虑这些有界性质.

在该文中, 定义$B:=B(x, r)=\{y\in{\Bbb R}^n: |x-y|<r\}$. $f_B$表示$f$在$B$上的积分平均, 即$f_B=|B|^{-1}\int_Bf(x){\rm d}x$. $p'(\cdot)$为$p(\cdot)$的共轭指数, 定义为$1/p(\cdot)+1/p'(\cdot)=1$. ${\cal S}(\mathbb{R} ^n)$和${\cal S'}(\mathbb{R} ^n)$分别表示Schwartz函数空间和缓增广义函数空间.对于$x\in{\Bbb R}$, $[x]$表示不超过$x$的最大整数.字母$C$表示正常数, 但在不同的地方可能取值不同.表达式$f\lesssim g$意思为$f\leq Cg$, $f\thickapprox g$意思为$f\lesssim g\lesssim f$.

设$p(\cdot): {\Bbb{ R}}^n\rightarrow [1, \infty)$为可测函数.变指数Lebesgue空间$L^{p(\cdot)}({\Bbb{ R}}^n)$表示${\Bbb{ R}}^n$上所有满足以下条件的可测函数$f$的集合

$ I_{p(\cdot)}(f):=\int_{{\Bbb{ R}}^n}|f(x)|^{p(x)}{\rm d}x<\infty. $

当被赋予如下Luxemburg-Nakano范数时, $L^{p(\cdot)}({\Bbb{ R}}^n)$为Banach函数空间

$ \|f\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}:=\mbox{inf}\{\mu>0: I_{p(\cdot)}(f/\mu)\leq 1 \}. $

显然, 当$p(\cdot)\equiv p$时, $L^{p(\cdot)}({\Bbb{ R}}^n)=L^p({\Bbb{ R}}^n)$为经典的Lebesgue空间.给定一个开集$\Omega\subset {\Bbb{ R}}^n$, 局部$L^{p(\cdot)}_{\rm loc}(\Omega)$空间定义为

$ L^{p(\cdot)}_{\rm loc}(\Omega):=\{f: f\in L^{p(\cdot)}(K)\ {\mbox{对所有紧子集 }}K\subset \Omega\}. $

为了简便, 我们使用以下记号

$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p_-:=p_-({\Bbb{ R}}^n)=\mbox{ess inf}\{p(x): x\in {\Bbb{ R}}^n\}, \\ \ \ \ \ \ \ \ \ p_+:=p_+({\Bbb{ R}}^n)=\mbox{ess sup}\{p(x): x\in {\Bbb{ R}}^n\}, \\ \ \ \ \ \ \ \ \ \ \ {\cal P}({\Bbb{ R}}^n):=\{p(\cdot): \ p_->1 \ \mbox{且}\ p_+<\infty\}, \\ {\cal B}({\Bbb{ R}}^n):=\{p(\cdot)\in {\cal P}({\Bbb{ R}}^n): \ \ M \ {\mbox{在}} L^{p(\cdot)}({\Bbb{ R}}^n){\mbox{上有界}} \}, $

其中$M$表示Hardy-Littlewood极大算子, 定义为

$ Mf(x):=\sup\limits_{x\in {\Bbb{ R}}^n, r>0}r^{-n}\int_{B(x, r)}|f(y)|{\rm d}y. $

当$p(\cdot)\in{\cal P}({\Bbb{ R}}^n)$, $f\in L^{p(\cdot)}({\Bbb{ R}}^n)$且$g\in L^{p'(\cdot)}({\Bbb{ R}}^n)$, 则有以下广义Hölder不等式成立

$ \begin{equation}\label{eq:a4} \int_{{\Bbb R}^n}|f(x)g(x)|{\rm d}x\leq r_p\|f\|_{L^{p(\cdot)}({\Bbb R}^n)}\|g\|_{L^{p'(\cdot)}({\Bbb R}^n)}, \end{equation} $

(2.1)

其中$r_p :=1+1/p_--1/p_+$, 见文献[7, 定理2.1].

下面的引理2.1和2.2可见文献[17](或者[26]).

引理2.1  设$p(\cdot)\in{\cal B}({\Bbb{R}}^n)$, 则$ |B|^{-1}\|\chi_B\|_{L^{p(\cdot)}({\Bbb R}^n)}\|\chi_B\|_{L^{p'(\cdot)}({\Bbb R}^n)}\leq C. $

引理2.2  设$p(\cdot)\in{\cal B}({\Bbb{ R}}^n)$, 则对所有可测子集$E\subset B$, 有

$ \frac{\|\chi_E\|_{L^{p(\cdot)}({\Bbb R}^n)}}{\|\chi_B\|_{L^{p(\cdot)}({\Bbb R}^n)}} \leq C\bigg(\frac{|E|}{|B|}\bigg)^{\delta_1}, \quad \frac{\|\chi_E\|_{L^{p'(\cdot)}({\Bbb R}^n)}}{\|\chi_B\|_{L^{p'(\cdot)}({\Bbb R}^n)}}\leq C\bigg(\frac{|E|}{|B|}\bigg)^{\delta_2}, $

其中$\delta_1, \delta_2$为常数, 且$0<\delta_1, \delta_2<1$.

记$B_l := \{x \in {\mathbb{R} } ^n: |x| \leq 2^l\}$, $R_l := B_l\backslash B_{l-1}$, 这里$l\in {\Bbb Z}$, $\chi_l := \chi_{R_l}$表示$R_l$的特征函数.

定义2.1  设$0< q\leq\infty$, $p(\cdot)\in {\cal P}({\Bbb{ R}}^n)$及$\alpha(\cdot)\in L^\infty({\Bbb{ R}}^n)$.齐次Herz空间$\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$定义为

$ {\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}}({\Bbb{ R}}^n) := \bigg\{f\in L_{\rm {loc}}^{p(\cdot)}({{\Bbb{ R}}^n}\backslash\{0\}): \|f\|_{{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}}({\Bbb{ R}}^n)}<\infty\bigg\}, $

其中

$ \|f\|_{{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}}({\Bbb{ R}}^n)}:= \bigg(\sum\limits_{k\in{\Bbb Z}}\|2^{k\alpha(\cdot)} f\chi_k\|^q_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^{1/q}. $

当$q=\infty$时, 作通常意义下的修改.

我们称函数$\alpha(\cdot):{\Bbb{ R}}^n \rightarrow \Bbb R$在原点处log-Hölder连续, 如果存在常数$C_{\rm {log}}>0$使得

$ \begin{equation} |\alpha(x)-\alpha(0)|\leq \frac{C_{\rm {log}}}{\mbox{log}(e+1/|x|)}, \quad x\in {\Bbb{ R}}^n. \end{equation} $

(2.2)

如果存在常数$\alpha_\infty\in {\Bbb{R}}$和$C_{\rm {log}}>0$, 使得

$ \begin{equation} |\alpha(x)-\alpha_\infty|\leq \frac{C_{\rm {log}}}{\mbox{log}(e+|x|)}, \quad x\in {\Bbb{ R}}^n, \end{equation} $

(2.3)

则称$\alpha(\cdot)$在无穷处log-Hölder连续. Almeida和Drihem^[14]证明了以下结果.

命题2.1  设$0< q\leq\infty$, $p(\cdot)\in {\cal P}({\Bbb{ R}}^n)$及$\alpha(\cdot)\in L^\infty({\Bbb{ R}}^n)$.若$\alpha(\cdot)$既在原点也在无穷处log-Hölder连续, 则

$ \|f\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}\approx\bigg(\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0) q}\| f\chi_k\|^q_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^{\frac{1}{q}}+\bigg(\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q}\| f \chi_k\|^q_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^{\frac{1}{q}}. $

设$G_Nf$为$f$的grand极大函数, 定义为

$ G_N(f)(x):=\sup\limits_{\phi\in{\cal A}_N}|\phi^{\ast}_\nabla(f)(x)|, \quad \forall x\in{\Bbb R}^n, $

其中

$ {\cal A}_N :=\Big\{\phi\in{\cal S}({\Bbb{ R}}^n): \sup\limits_{|\alpha|\leq N, |\beta|\leq N}|x^{\alpha}D^{\beta}\phi(x)|\leq1\Big\}, \quad N>n+1, $

且$\phi _\nabla ^ * (f)(x): = \mathop {\sup }\limits_{|y - x| < t} |{\phi _t} * f(y)|$, $\phi_t(x) :=t^{-n}\phi(\frac{x}{t})$.

极大算子$G_N$首先由Fefferman和Stein^[27]为了研究经典的Hardy空间而引入.关于经典的Hardy空间及其相应的变指数函数空间研究, 见文献[28-30]等.

定义2.2  设$0<q\leq\infty$, $p(\cdot)\in{\cal P}({\Bbb{ R}}^n)$, $\alpha(\cdot)\in L^\infty({\Bbb{ R}}^n)$且$N>n+1$.齐次Herz型Hardy空间$H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$定义为

$ H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n):=\bigg\{f\in{\cal S'}({\Bbb{ R}}^n): G_Nf\in\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)\bigg\} $

且$\|f\|_{H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}=\|G_Nf\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}$.

定义2.3  设$p(\cdot)\in{\cal P}({\Bbb{ R}}^n)$, $\alpha(\cdot)\in L^\infty({\Bbb{ R}}^n)$既在原点也在无穷处log-Hölder连续.若非负整数$s\geq [\alpha_r-n\delta_2]$, 其中$n\delta_2\leq \alpha_r<\infty$, $\delta_2$同引理2.2中所定义.这里$\alpha_r=\alpha(0)$若$0<r<1$, $\alpha_r=\alpha_\infty$若$r\geq 1$.称函数$a(\cdot)$为一个中心$(\alpha(\cdot), p(\cdot))$ -原子, 如果满足

$ \begin{eqnarray*} &&(1)\ \ \ {\rm supp}a\subset B(0, r):=\{x\in{\Bbb R}^n: |x|<r\}.\\ &&(2)\ \ \ \|a\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\leq|B(0, r)|^{-\frac{\alpha(0)}{n}}, \ 0<r<1.\\ &&(3)\ \ \ \|a\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\leq|B(0, r)|^{-\frac{\alpha_\infty}{n}}, \ r\geq 1.\\ &&(4)\ \ \ \int_{{\Bbb{ R}}^n}a(x)x^{\beta}{\rm d}x=0, \ |\beta|\leq s. \end{eqnarray*} $

显然, 若$p(\cdot)\equiv p$, $\alpha(\cdot)\equiv\alpha$为常数, 则取$\delta_2=1-\frac{1}{p}$我们即可得到经典的情形, 见文献[28]. Dong和Xu在文献[15, 定理2.1]中给出了$H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$空间的原子刻画.

命题2.2  设$0<q<\infty$, $p(\cdot)\in{\cal B}({\Bbb{ R}}^n)$, $\alpha(\cdot)\in L^\infty({\Bbb{ R}}^n)$既在原点也在无穷处log-H$\ddot{\rm o}$lder连续.若$n\delta_2\leq\alpha(0), \alpha_\infty<\infty$, 其中$\delta_2$同引理2.2中所定义.则$f\in H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$当且仅当在分布意义下它可以分解为$f=\sum\limits_{k=-\infty}^{\infty}\lambda_ka_k, $其中$a_k$是中心$(\alpha(\cdot), p(\cdot))$ -原子, $\mbox{supp}a_k\subset B_k$且$\sum\limits_{k=-\infty}^{\infty}|\lambda_k|^q<\infty$.进一步, $ \|f\|_{H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{R}}^n)} \approx \inf\Big(\sum\limits_{k=-\infty}^{\infty}|\lambda_k|^q \Big)^{1/q}, $其中下确界取遍$f$所有上述分解.

下面给出我们的主要结果.

定理2.1  设$0<\beta\leq 1$, $0<q\leq\infty$, $p(\cdot)\in{\cal B}({\Bbb{ R}}^n)$, $\alpha(\cdot)\in L^\infty({\Bbb{ R}}^n)$既在原点也在无穷处log-H$\ddot{\rm o}$lder连续, 使得$-n\delta_1<\alpha(0)\leq\alpha_{\infty}<n\delta_2$, 其中$0<\delta_1, \delta_2<1$同引理2.2中所定义.则对任意的$f\in\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$有$ \|S_\beta(f)\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)} \leq C\|f\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}. $

注意到在定理2.1中, 我们限制$\alpha(0), \alpha_{\infty}\in(-n\delta_1, n\delta_2)$, 下面的结果进一步将定理2.1推广到了$\alpha(0), \alpha_{\infty}\geq n\delta_2$时情形.

定理2.2  设$0<\beta\leq 1$, $0<q<\infty$, $p(\cdot)\in{\cal B}({\Bbb{ R}}^n)$, $\alpha(\cdot)\in L^\infty({\Bbb{ R}}^n)$既在原点也在无穷处log-Hölder连续.若$n\delta_2\leq\alpha(0), \alpha_\infty<n\delta_2+\beta$, 其中$\delta_2$同引理2.2中所定义.则对任意的$f\in H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$有$ \|S_\beta(f)\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)} \leq C\|f\|_{H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}. $

注2.1  正如文献[15]中指出, 若$p(\cdot)\in{\cal B}({\Bbb{ R}}^n)$且$\alpha(\cdot)\in L^\infty({\Bbb{ R}}^n)$既在原点也在无穷处log-Hölder连续, 则当$-n\delta_1<\alpha(0)\leq\alpha_{\infty}<n\delta_2$时, $H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)\bigcap L_{\rm {loc}}^{p(\cdot)}({{\Bbb{R}}^n}\backslash\{0\})=\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$.当$n\delta_2\leq\alpha(0), \alpha_{\infty}<\infty$时, $H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)\bigcap L_{\rm {loc}}^{p(\cdot)}({{\Bbb{ R}}^n}\backslash\{0\})\subsetneqq\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$.比较上述结果可以发现在定理2.2中, $H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$成为$\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$的合适替代.特别地, 当$\alpha(\cdot)\equiv\alpha$为常数时, 即在变指数Herz型Hardy空间$H\dot{K}_{p(\cdot)}^{\alpha, q}({\Bbb{ R}}^n)$中(定义见文献[20]), 定理2.2的结果也是新的.

以下我们仅证明$0<q<\infty$情形.当$q=\infty$时，证明过程只需作简单修改.

定理2.1的证明  记

$ \begin{eqnarray*} f(x)=\sum\limits_{i=-\infty}^\infty f(x)\chi_i(x)\equiv\sum\limits_{i=-\infty}^{\infty} f_i(x). \end{eqnarray*} $

由命题2.1及Minkowski不等式, 我们有

$ \begin{eqnarray*} \|S_\beta(f)\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}&\approx &\bigg\{\sum\limits_{k=-\infty}^{-1}2^{\alpha(0) kq} \|S_\beta(f)\chi_k\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\bigg\}^{1/q}\\ &&+\bigg\{\sum\limits_{k=0}^{\infty}2^{\alpha_\infty kq} \|S_\beta(f)\chi_k\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\bigg\}^{1/q}\\ &\leq& C(H_1+H_2+H_3+H_4+H_5+H_6), \end{eqnarray*} $

其中

$ H_1 :=\bigg\{\sum\limits_{k=-\infty}^{-1}2^{\alpha(0) kq}\bigg(\sum\limits_{i=-\infty}^{k-2} \|S_\beta(f_i)\chi_k\|_{L^{p(\cdot)}({\Bbb R}^n)}\bigg)^q\bigg\}^{1/q}, \\ H_2 :=\bigg\{\sum\limits_{k=-\infty}^{-1}2^{\alpha(0) kq}\bigg(\sum\limits_{i=k-1}^{k+1} \|S_\beta(f_i)\chi_k\|_{L^{p(\cdot)}({\Bbb R}^n)}\bigg)^q\Biggr\}^{1/q}, \\ H_3 :=\bigg\{\sum\limits_{k=-\infty}^{-1}2^{\alpha(0) kq}\bigg(\sum\limits_{i=k+2}^{\infty} \|S_\beta(f_i)\chi_k\|_{L^{p(\cdot)}({\Bbb R}^n)}\Biggr)^q\bigg\}^{1/q}, \\ \ \ \ H_4 :=\bigg\{\sum\limits_{k=0}^{\infty}2^{\alpha_\infty kq}\bigg(\sum\limits_{i=-\infty}^{k-2} \|S_\beta(f_i)\chi_k\|_{L^{p(\cdot)}({\Bbb R}^n)}\bigg)^q\bigg\}^{1/q}, \\ \ \ \ H_5 :=\bigg\{\sum\limits_{k=0}^{\infty}2^{\alpha_\infty kq}\bigg(\sum\limits_{i=k-1}^{k+1} \|S_\beta(f_i)\chi_k\|_{L^{p(\cdot)}({\Bbb R}^n)}\bigg)^q\Biggr\}^{1/q}, \\ \ \ \ H_6 :=\bigg\{\sum\limits_{k=0}^{\infty}2^{\alpha_\infty kq}\bigg(\sum\limits_{i=k+2}^{\infty} \|S_\beta(f_i)\chi_k\|_{L^{p(\cdot)}({\Bbb R}^n)}\Biggr)^q\bigg\}^{1/q}. $

首先估计$H_1$.注意到当$x\in R_k$, $(y, t)\in\Gamma(x)$, $z\in R_i\bigcap\{z: |y-z|\leq t\}$及$i\leq {k-2}$, 则有

$ \begin{equation}\label{eq:a5} t\geq \frac{1}{2}(|x-y|+|y-z|)\geq \frac{1}{2}|x-z|\geq \frac{1}{2}(|x|-|z|)\geq\frac{1}{4}|x|. \end{equation} $

(3.1)

因此, 由广义Hölder不等式(2.1), 我们有

$ \begin{eqnarray}\label{eq:a6} |S_\beta(f_i)(x)| &=&\bigg(\int\!\!\!\int_{\Gamma(x)}\sup\limits_{\phi\in{\cal C}_\beta} \bigg|\int_{{\Bbb{ R}}^n}\phi_t(y-z)f_i(z){\rm d}z\bigg|^2\frac{{\rm d}y{\rm d}t}{t^{n+1}}\bigg)^{\frac{1}{2}}\\ &\leq& C\bigg(\int_{\frac{|x|}{4}}^{\infty}\int_{|x-y|<t} \bigg|t^{-n}\int_{{R_i}\bigcap\{z: |y-z|\leq t\}}f_i(z){\rm d}z\bigg|^2\frac{{\rm d}y{\rm d}t}{t^{n+1}}\bigg)^{\frac{1}{2}} \\ &\leq &C\bigg(\int_{R_i}|f_i(z)|{\rm d}z\bigg)\bigg(\int_{\frac{|x|}{4}}^{\infty}\frac{{\rm d}t}{t^{2n+1}}\bigg)^{\frac{1}{2}} \\ &\leq &C2^{-kn}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)} \|\chi_{B_i}\|_{L^{p'(\cdot)}({\Bbb R}^n)}. \end{eqnarray} $

(3.2)

结合(3.2)式, 引理2.1和引理2.2, 我们有

$ \begin{eqnarray}\label{eq:a7} \|S_\beta(f_i)(x)\chi_k\|_{L^{p(\cdot)}({\Bbb R}^n)} &\leq& C2^{-kn}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)} \|\chi_{B_i}\|_{L^{p'(\cdot)}({\Bbb R}^n)}\|\chi_{B_k}\|_{L^{p(\cdot)}({\Bbb R}^n)}\\ &\leq& C\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}\frac{\|\chi_{B_i}\|_{L^{p'(\cdot)}({\Bbb R}^n)}}{\|\chi_{B_k}\|_{L^{p'(\cdot)}({\Bbb R}^n)}}\\ &\leq &C2^{(i-k)n\delta_2}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}. \end{eqnarray} $

(3.3)

由(3.3)式可得

$ \begin{eqnarray*} H_1&\leq &C\bigg\{\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0) q} \bigg(\sum\limits_{i=-\infty}^{k-2}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}2^{(i-k)n\delta_2}\bigg)^q\bigg\}^{1/q}\\ &=& C\bigg\{\sum\limits_{k=-\infty}^{-1} \bigg(\sum\limits_{i=-\infty}^{k-2}2^{i\alpha(0)}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}2^{(i-k)(n\delta_2-\alpha(0))}\bigg)^q\bigg\}^{1/q}. \end{eqnarray*} $

我们分以下两种情况考虑.

情形Ⅰ  $0<q\leq 1$.由已知不等式

$ \begin{equation}\label{eq:a8} \bigg(\sum\limits_{j=1}^\infty \theta_j\bigg)^q\leq \sum\limits_{j=1}^\infty \theta_j^q, \quad (\theta_j>0, \ j\in{\Bbb N}) \end{equation} $

(3.4)

及$n\delta_2-\alpha(0)>0$, 我们得到以下估计

$ \begin{eqnarray*} H_1&\leq& C\bigg\{\sum\limits_{k=-\infty}^{-1} \sum\limits_{i=-\infty}^{k-2}2^{i\alpha(0) q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}2^{(i-k)(n\delta_2-\alpha(0))q}\bigg\}^{1/q}\\ &=&C\bigg\{\sum\limits_{i=-\infty}^{-3}2^{i\alpha(0) q} \|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\sum\limits_{k=i+2}^{-1} 2^{(i-k)(n\delta_2-\alpha(0))q}\bigg\}^{1/q}\\ &\leq& C\bigg\{\sum\limits_{i=-\infty}^{-3}2^{i\alpha(0) q} \|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\bigg\}^{1/q} \\ &\leq &C\|f\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}. \end{eqnarray*} $

情形Ⅱ  $1<q<\infty$.利用Hölder不等式, 我们有

$ \begin{eqnarray*} H_1&\leq &C\bigg\{\sum\limits_{k=-\infty}^{-1} \bigg(\sum\limits_{i=-\infty}^{k-2}2^{i\alpha(0) q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}2^{(i-k)(n\delta_2-\alpha(0))q/2}\bigg)\\ &&\times\bigg(\sum\limits_{i=-\infty}^{k-2} 2^{(i-k)(n\delta_2-\alpha(0))q'/2}\bigg)^{q/q'}\bigg\}^{1/q}\\ &\leq& C\bigg\{\sum\limits_{i=-\infty}^{-3}2^{i\alpha(0) q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\sum\limits_{k=i+2}^{-1} 2^{(i-k)(n\delta_2-\alpha(0))q/2}\bigg\}^{1/q}\\ &\leq &C\bigg\{\sum\limits_{i=-\infty}^{-3}2^{i\alpha(0) q} \|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\bigg\}^{1/q} \\ &\leq &C\|f\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}. \end{eqnarray*} $

对于$H_4$, 用$\alpha_\infty$代替$\alpha(0)$, 利用与$H_1$相同的估计方法可得

$ H_4\leq C\bigg\{\sum\limits_{k=0}^{\infty} \bigg(\sum\limits_{i=-\infty}^{k-2}2^{i\alpha_{\infty}}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}2^{(i-k)(n\delta_2-\alpha_{\infty})}\bigg)^q\bigg\}^{1/q}. $

若$0<q\leq 1$, 注意到$\alpha(0)\leq\alpha_{\infty}<n\delta_2$, 由Minkowski不等式和(3.4)式, 我们有

$\begin{eqnarray*} H_4&\leq& C\bigg\{\sum\limits_{k=0}^{\infty} \bigg(\sum\limits_{i=-\infty}^{-2}2^{i\alpha_{\infty}}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}2^{(i-k)(n\delta_2-\alpha_{\infty})}\bigg)^q\bigg\}^{1/q}\\ &&+C\bigg\{\sum\limits_{k=0}^{\infty} \bigg(\sum\limits_{i=-1}^{k-2}2^{i\alpha_{\infty}}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}2^{(i-k)(n\delta_2-\alpha_{\infty})}\bigg)^q\bigg\}^{1/q} \\ &\leq& C\bigg\{\sum\limits_{i=-\infty}^{-2}2^{i\alpha_{\infty} q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\sum\limits_{k=0}^{\infty}2^{(i-k)(n\delta_2-\alpha_{\infty})q}\bigg\}^{1/q}\\ &&+C\bigg\{\sum\limits_{i=-1}^{\infty}2^{i\alpha_{\infty} q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\sum\limits_{k=i+2}^{\infty}2^{(i-k)(n\delta_2-\alpha_{\infty})q} \bigg\}^{1/q}\\ &\leq& C\bigg\{\sum\limits_{i=-\infty}^{-2}2^{i\alpha(0) q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\bigg\}^{1/q} +C\bigg\{\sum\limits_{i=-1}^{\infty}2^{i\alpha_{\infty} q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)} \bigg\}^{1/q}\\ &\leq &C\|f\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}. \end{eqnarray*} $

若$1<q<\infty$, 由Hölder不等式可得

$ \begin{eqnarray*} H_4&\leq& C\bigg\{\sum\limits_{k=0}^{\infty} \bigg(\sum\limits_{i=-\infty}^{-2}2^{i\alpha_{\infty} q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}2^{(i-k)(n\delta_2-\alpha_{\infty})q/2}\bigg)\\ &\quad&\times\bigg(\sum\limits_{i=-\infty}^{-2} 2^{(i-k)(n\delta_2-\alpha_{\infty})q'/2}\bigg)^{q/q'}\bigg\}^{1/q}\\ &\quad&+C\bigg\{\sum\limits_{k=0}^{\infty} \bigg(\sum\limits_{i=-1}^{k-2}2^{i\alpha_{\infty} q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}2^{(i-k)(n\delta_2-\alpha_{\infty})q/2}\bigg)\\ &\quad&\times\bigg(\sum\limits_{i=-1}^{k-2} 2^{(i-k)(n\delta_2-\alpha_{\infty})q'/2}\bigg)^{q/q'}\bigg\}^{1/q}\\ &\leq& C\bigg\{\sum\limits_{i=-\infty}^{-2}2^{i\alpha_{\infty} q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\sum\limits_{k=0}^{\infty}2^{(i-k)(n\delta_2-\alpha_{\infty})q/2}\bigg\}^{1/q}\\ &\quad&+C\bigg\{\sum\limits_{i=-1}^{\infty}2^{i\alpha_{\infty} q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\sum\limits_{k=i+2}^{\infty}2^{(i-k)(n\delta_2-\alpha_{\infty})q/2} \bigg\}^{1/q}\\ &\leq& C\bigg\{\sum\limits_{i=-\infty}^{-2}2^{i\alpha(0) q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)}\bigg\}^{1/q} +C\bigg\{\sum\limits_{i=-1}^{\infty}2^{i\alpha_{\infty} q}\|f_i\|^q_{L^{p(\cdot)}({\Bbb R}^n)} \bigg\}^{1/q}\\ &\leq& C\|f\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}. \end{eqnarray*} $

对于$H_2$和$H_5$, 由$S_\beta$的$L^{p(\cdot)}({\Bbb R}^n)$有界性(见文献[5, 定理3.4])可得

$ H_2\leq C\bigg\{\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0) q} \|f_k\|^q_{L^{p(\cdot)}({\Bbb R}^n)} \bigg\}^{1/q}\leq C\|f\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}.\\ \ \ \ H_5\leq C\bigg\{\sum\limits_{k=0}^{\infty}2^{k\alpha_{\infty} q} \|f_k\|^q_{L^{p(\cdot)}({\Bbb R}^n)} \bigg\}^{1/q} \leq C\|f\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}. $

下面我们估计$H_3$.注意到$x\in R_k$, $(y, t)\in\Gamma(x)$, $z\in R_i\bigcap\{z: |y-z|\leq t\}$及$i\geq {k+2}$, 我们有

$ \begin{equation}\label{eq:a9} t\geq \frac{1}{2}(|x-y|+|y-z|)\geq \frac{1}{2}|x-z|\geq \frac{1}{2}(|z|-|x|)\geq\frac{1}{4}|z|. \end{equation} $

(3.5)

由(3.5)式和广义Hölder不等式(2.1)可得

$\begin{eqnarray}\label{eq:a10} |S_\beta(f_i)(x)| &=&\bigg(\int\!\!\!\int_{\Gamma(x)}\sup\limits_{\phi\in{\cal C}_\beta} \bigg|\int_{{\Bbb{ R}}^n}\phi_t(y-z)f_i(z){\rm d}z\bigg|^2\frac{{\rm d}y{\rm d}t}{t^{n+1}}\bigg)^{\frac{1}{2}} \\ &\leq &C\bigg(\int_{\frac{|z|}{4}}^{\infty}\int_{|x-y|<t} \bigg|t^{-n}\int_{{R_i}\bigcap\{z: |y-z|\leq t\}}f_i(z){\rm d}z\bigg|^2\frac{{\rm d}y{\rm d}t}{t^{n+1}}\bigg)^{\frac{1}{2}} \\ &\leq &C\bigg(\int_{R_i}|f_i(z)|{\rm d}z\bigg)\bigg(\int_{\frac{|z|}{4}}^{\infty}\frac{{\rm d}t}{t^{2n+1}}\bigg)^{\frac{1}{2}}\\ &\leq& C2^{-in}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)} \|\chi_{B_i}\|_{L^{p'(\cdot)}({\Bbb R}^n)}. \end{eqnarray} $

(3.6)

结合(3.6)式, 引理2.1和引理2.2, 我们有

$ \begin{eqnarray}\label{eq:a11} \|S_\beta(f_i)(x)\chi_k\|_{L^{p(\cdot)}({\Bbb R}^n)} &\leq& C2^{-in}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)} \|\chi_{B_i}\|_{L^{p'(\cdot)}({\Bbb R}^n)}\|\chi_{B_k}\|_{L^{p(\cdot)}({\Bbb R}^n)}\\ &\leq& C\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}\frac{\|\chi_{B_k}\|_{L^{p(\cdot)}({\Bbb R}^n)}}{\|\chi_{B_i}\|_{L^{p(\cdot)}({\Bbb R}^n)}}\\ &\leq &C2^{(k-i)n\delta_1}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}. \end{eqnarray} $

(3.7)

由(3.7)式可得

$ H_3\leq C\bigg\{\sum\limits_{k=-\infty}^{-1} \bigg(\sum\limits_{i=k+2}^{\infty}2^{i\alpha(0)}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}2^{(k-i)(n\delta_1+\alpha(0))}\bigg)^q\bigg\}^{1/q}. $

用$\alpha_\infty$代替$\alpha(0)$, 同理可得

$ H_6\leq C\bigg\{\sum\limits_{k=0}^{\infty} \bigg(\sum\limits_{i=k+2}^{\infty}2^{i\alpha_{\infty}}\|f_i\|_{L^{p(\cdot)}({\Bbb R}^n)}2^{(k-i)(n\delta_1+\alpha_{\infty})}\bigg)^q\bigg\}^{1/q}. $

我们略去$H_3$和$H_6$的估计, 因为它们本质上分别与$H_4$和$H_1$估计相似.从而定理2.1证明完毕.

定理2.2的证明  设$f\in H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)$, 由命题2.2, 在分布意义下我们有$f=\sum\limits_{j=-\infty}^{\infty}\lambda_ja_j, $其中$\lambda_j\geq 0$, $a_j$是中心$(\alpha(\cdot), p(\cdot))$ -原子, $\mbox{supp}a_j\subset B_j$且

$ \|f\|_{H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)}\approx \inf \bigg(\sum\limits_{j=-\infty}^{\infty}|\lambda_j|^q\bigg)^{\frac{1}{q}}. $

因此, 我们有

$ \begin{eqnarray*} \|S_\beta (f)\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\Bbb{ R}}^n)} &\approx&\bigg(\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0) q}\| S_\beta(f)\chi_k\|^q_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^{\frac{1}{q}}\\ &\quad& +\bigg(\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q}\| S_\beta(f)\chi_k\|^q_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^{\frac{1}{q}}\\ \\ &\leq& C\bigg\{\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q} \bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j|\| S_\beta(a_j)\chi_k\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^q\bigg\}^{\frac{1}{q}}\\ &\quad& +C\bigg\{\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q} \bigg(\sum\limits_{j=k-1}^{\infty}|\lambda_j|\| S_\beta(a_j)\chi_k\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^q\bigg\}^{\frac{1}{q}}\\ &\quad &+C\bigg\{\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q} \bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j|\| S_\beta(a_j)\chi_k\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^q\bigg\}^{\frac{1}{q}}\\ &\quad &+C\bigg\{\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q} \bigg(\sum\limits_{j=k-1}^{\infty}|\lambda_j|\| S_\beta(a_j)\chi_k\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^q\bigg\}^{\frac{1}{q}}\\ &:=&J_1+J_2+J_3+J_4. \end{eqnarray*} $

对于$J_1$, 注意到若$x\in R_k$, $(y, t)\in\Gamma(x)$, $z\in B_j\bigcap\{z: |y-z|\leq t\}$及$j\leq {k-2}$, 则有$t\geq\frac{1}{4}|x|$.利用$a_j$的消失性条件, 我们有

$ \begin{eqnarray}\label{eq:a12} |S_\beta(a_j)(x)| &=&\bigg(\int\!\!\!\int_{\Gamma(x)}\sup\limits_{\phi\in{\cal C}_\beta} \bigg|\int_{{\Bbb{ R}}^n}\phi_t(y-z)a_j(z){\rm d}z\bigg|^2\frac{{\rm d}y{\rm d}t}{t^{n+1}}\bigg)^{\frac{1}{2}}\\ &=&\bigg(\int\!\!\!\int_{\Gamma(x)}\sup\limits_{\phi\in{\cal C}_\beta} \bigg|\int_{{\Bbb{ R}}^n}(\phi_t(y-z)-\phi_t(y))a_j(z){\rm d}z\bigg|^2\frac{{\rm d}y{\rm d}t}{t^{n+1}}\bigg)^{\frac{1}{2}}\\ &\leq& C2^{j\beta}\bigg(\int_{B_j}|a_j(z)|{\rm d}z\bigg)\bigg(\int_{\frac{|x|}{4}}^{\infty}\int_{|x-y|<t} \frac{{\rm d}y{\rm d}t}{t^{3n+1+2\beta}}\bigg)^{\frac{1}{2}}\\ &\leq &C2^{(j-k)\beta-kn}\|\chi_{B_j}\|_{L^{p'(\cdot)}({\Bbb{R}}^n)}\|a_j\|_{L^{p(\cdot)}({\Bbb{R}}^n)}. \end{eqnarray} $

(3.8)

由(3.8)式, 引理2.1和2.2, 可得

$ \begin{eqnarray}\label{eq:a13} \|S_\beta(a_j)\chi_k\|_{L^{p(\cdot)}({\Bbb{R}}^n)} &\leq& C2^{(j-k)\beta-kn}\|a_j\|_{L^{p(\cdot)}({\Bbb{R}}^n)} \|\chi_{B_k}\|_{L^{p(\cdot)}({\Bbb{R}}^n)}\|\chi_{B_j}\|_{L^{p'(\cdot)}({\Bbb{R}}^n)}\\ &\leq &C2^{(j-k)\beta}\|a_j\|_{L^{p(\cdot)}({\Bbb{R}}^n)} \frac{\|\chi_{B_j}\|_{L^{p'(\cdot)}({\Bbb{R}}^n)}}{\|\chi_{B_k} \|_{L^{p'(\cdot)}({\Bbb{R}}^n)}}\\ &\leq &C2^{(j-k)(\beta+n\delta_2)}\|a_j\|_{L^{p(\cdot)}({\Bbb{R}}^n)}. \end{eqnarray} $

(3.9)

为了简洁, 以下我们记$\xi=\beta+n\delta_2-\alpha(0)>0$, $\eta=\beta+n\delta_2-\alpha_\infty>0$.若$0<q\leq 1$, 注意到$j\leq k-2$且$k<0$时, 有$\|a_j\|_{L^{p(\cdot)}({\Bbb{R}}^n)}\leq 2^{-j\alpha(0)}$, 由(3.9)式和$n\delta_2\leq \alpha(0)<n\delta_2+\beta$, 我们有

$ \begin{eqnarray*} J_1^q&=&C\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q} \bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j|\| S_\beta(a_j)\chi_k\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^q\\ &\leq &C\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q} \bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j| 2^{(j-k)(\beta+n\delta_2)-j\alpha(0)}\bigg)^q\\ &\leq &C\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q} \bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j|^q 2^{(j-k)(\beta+n\delta_2)q-j\alpha(0) q}\bigg)\\ &=&C\sum\limits_{j=-\infty}^{-3}|\lambda_j|^q\sum\limits_{k=j+2}^{-1}2^{(j-k)\xi q} \leq C\sum\limits_{j=-\infty}^{-1}|\lambda_j|^q. \end{eqnarray*} $

若$1<q<\infty$, 由Hölder不等式可得

$ \begin{eqnarray*} J_1^q&=&C\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q} \bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j|\| S_\beta(a_j)\chi_k\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^q\\ &\leq &C\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q} \bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j| 2^{(j-k)(\beta+n\delta_2)-j\alpha(0)}\bigg)^q\\ &\leq &C\sum\limits_{k=-\infty}^{-1}\bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j|^q2^{(j-k)\xi q/2}\bigg) \bigg(\sum\limits_{j=-\infty}^{k-2}2^{(j-k)\xi q'/2}\bigg)^{q/q'}\\ &\leq& C\sum\limits_{j=-\infty}^{-3}|\lambda_j|^q\sum\limits_{k=j+2}^{-1}2^{(j-k)\xi q/2} \leq C\sum\limits_{j=-\infty}^{-1}|\lambda_j|^q. \end{eqnarray*} $

对于$J_3$, 若$0<q\leq 1$, 利用(3.9)式以及$a_j$的尺寸条件, 我们有

$ \begin{eqnarray*} J_3^q&=&C\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q} \bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j|\| S_\beta(a_j)\chi_k\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^q\\ &\leq &C\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q} \bigg(\sum\limits_{j=-\infty}^{-1}|\lambda_j|^q 2^{(j-k)(\beta+n\delta_2)q-j\alpha(0)q}\bigg)\\ &\quad&+C\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q} \bigg(\sum\limits_{j=0}^{k-2}|\lambda_j|^q 2^{(j-k)(\beta+n\delta_2)q-j\alpha_\infty q}\bigg)\\ &\leq &C\sum\limits_{k=0}^{\infty}2^{k(\alpha_\infty-\beta-n\delta_2)q}\sum\limits_{j=-\infty}^{-1}|\lambda_j|^q 2^{j\xi q} +C\sum\limits_{j=0}^{\infty}|\lambda_j|^q\sum\limits_{k=j+2}^{\infty}2^{(j-k)\eta q}\\ &\leq& C\bigg(\sum\limits_{j=-\infty}^{-1}|\lambda_j|^q+\sum\limits_{j=0}^{\infty}|\lambda_j|^q\bigg) =C\sum\limits_{j=-\infty}^{\infty}|\lambda_j|^q. \end{eqnarray*} $

若$1<q<\infty$, 由Hölder不等式可得

$ \begin{eqnarray*} J_3^q&=&C\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q} \bigg(\sum\limits_{j=-\infty}^{k-2}|\lambda_j|\| S_\beta(a_j)\chi_k\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^q\\ &\leq &C\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q} \bigg(\sum\limits_{j=-\infty}^{-1}|\lambda_j| 2^{(j-k)(\beta+n\delta_2)-j\alpha(0)}\bigg)^q\\ &\quad&+C\sum\limits_{k=0}^{\infty}2^{k\alpha_\infty q} \bigg(\sum\limits_{j=0}^{k-2}|\lambda_j| 2^{(j-k)(\beta+n\delta_2)-j\alpha_\infty}\bigg)^q\\ &\leq &C\sum\limits_{k=0}^{\infty}2^{k(\alpha_\infty-\beta-n\delta_2)q} \bigg(\sum\limits_{j=-\infty}^{-1}|\lambda_j| 2^{j\xi}\bigg)^q +C\sum\limits_{k=0}^{\infty}\bigg(\sum\limits_{j=0}^{k-2}|\lambda_j| 2^{(j-k)\eta}\bigg)^q\\ &\leq &C\bigg(\sum\limits_{j=-\infty}^{-1}|\lambda_j|^q 2^{j\xi q/2}\bigg) \bigg(\sum\limits_{j=-\infty}^{-1} 2^{j\xi q'/2}\bigg)^{q/q'}\\ &\quad&+C\sum\limits_{k=0}^{\infty}\bigg(\sum\limits_{j=0}^{k-2}|\lambda_j|^q 2^{(j-k)\eta q/2}\bigg)^q \bigg(\sum\limits_{j=0}^{k-2} 2^{(j-k)\eta q'/2}\bigg)^{q/q'} \\ &\leq& C\sum\limits_{j=-\infty}^{-1}|\lambda_j|^q 2^{j\xi q/2} +C\sum\limits_{j=0}^{\infty}|\lambda_j|^q\sum\limits_{k=j+2}^{\infty}2^{(j-k)\eta q/2} \\ &\leq& C\sum\limits_{j=-\infty}^{\infty}|\lambda_j|^q. \end{eqnarray*} $

对于$J_2$和$J_4$, 利用$S_\beta$的$L^{p(\cdot)}({\Bbb R}^n)$有界性以及文献[15, p338]中对$I_1$和$I_3$相同的估计方法, 我们容易得到

$ J_2^q\leq C\sum\limits_{j=-\infty}^{\infty}|\lambda_j|^q \quad \mbox{及}\quad J_4^q\leq C\sum\limits_{j=0}^{\infty}|\lambda_j|^q. $

因此, 结合以上估计, 定理2.2证明完毕.