在偏微分方程中, 为了更好的研究Possion方程, Sobolev^[1]引入了经典的分数次积分算子 (又称Riesz位势算子):

$I_\beta f(x)=\int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n-\beta}}dy, 0<\beta<n.$

并证明了$I_\beta(f)$是$(L^p(\mathbb{R}^n),L^q(\mathbb{R}^n))$型的.

1995年, Fan Dashan等^[2]给出了奇异积分算子在Morrey空间上的有界性. 2004年, Duong等^[4]给出了广义分数次积分算子$L^{-\frac{\beta}{2}}$在一定条件下从$(L^p(\mathbb{R}^n),L^q(\mathbb{R}^n))$是有界的. 2005年, Lu Shanzhen等^[5-6]在研究奇异积分算子时, 引入了一类与PDE相关的, 比Herz空间和Morrey空间更一般的齐次Morrey-Herz空间, 这类空间很快受到人们的重视, 随后, Morrey-Herz空间上的一些极具研究价值的结果不断出现. 2009年, Yasuo Komori等^[7]证明了分数次积分算子在加权Herz空间上的有界性.受此启示, 本文研究广义分数次积分算子在齐次加权Morrey-Herz空间上的有界性.

在叙述主要的结果之前, 首先给出一些必要的记号和定义, 设$B_k=\{x\in \mathbb{R}^n:|x|\leq2^k\}$, $A_k=B_k\backslash B_{k-1}$, $k\in Z$, $\chi_k=\chi_{A_k}$, 其中$\chi_{A_k}$表示$A_k$的特征函数.

对于$\mathbb{R}^n$上的可测函数$f$和非负的权函数$\omega(x)$, 记$\|f\|_{L^{p}(\omega)}=(\displaystyle\int_{\mathbb{R}^n}|f(x)|^p{\omega(x)}dx)^{1/p}$, 用$L^{p}(\omega)$表示$L^{p}(\mathbb{R}^n,\omega)$, 特别地$\omega=1$时, 记为$L^{p}(\mathbb{R}^n)$.

全文中, $C$表示一个不依赖于主要参数的常数, 但其值在不同的地方可能不尽相同; $p$与$p'$满足共轭关系, 即$1/p+1/p'=1$.

二阶散度型椭圆算子$Lf=-{\rm div}(A\nabla f), A=A(x)$是指一个定义在$\mathbb{R}^n$上的$L^{\infty}$系数的$n\times n$矩阵, 且满足一致性椭圆条件:存在$0<\lambda\leq\gamma<\infty$, 使得$\lambda|\xi|^2\leq ReA\xi\cdot\overline{\xi}, |A\xi\cdot\overline{\xi}|\leq\gamma|\xi|\zeta$, 其中$\xi, \zeta\in \textbf{C}^n$.

利用算子的谱理论, 算子$L$的广义分数次积分定义为

$L^{-\frac{\beta}{2}}f(x)=\frac{1}{\Gamma(\frac{\beta}{2})}\int\limits_0^\infty e^{-tL}(f)\frac{dt}{t^{1-\beta/2}}, 0<\beta<n.$

当$L=-\Delta$即$\mathbb{R}^n$上的Laplace算子时, 以上的广义分数次积分算子就是经典的分数次积分算子.

设$p_t(x,y)$是解析半群$e^{-tL}$的热核, 若满足$A$是实矩阵, 或者$A$是$n\leq2$的复矩阵, 或者当$n\geq3$时, 核是Hölder连续的, 那么$p_t(x,y)$具有Gaussian上界, 即

$|p_t(x,y)|\leq \frac{C}{t^\frac{n}{2}}e^{-C\frac{|x-y|^2}{t}}.$

(2.1)

容易验证:对于几乎处处$x\in \mathbb{R}^n$, 有

$|L^{-\frac{\beta}{2}}f(x)|\leq CI_\beta(|f|)(x).$

(2.2)

详见文献[4].

定义2.1^[8] 设$\alpha\in R, \lambda\geq 0, 0<p,q<\infty$, 齐次Morrey-Herz空间定义如下

$M\dot{K}_{p,q}^{\alpha ,\lambda }({{\mathbb{R}}^{n}})=\{f\in L_{\text{loc}}^{q}({{\mathbb{R}}^{n}}\backslash \{0\}):\|f{{\|}_{M\dot{K}_{p,q}^{\alpha ,\lambda }({{\mathbb{R}}^{n}})}}<\infty \},$

其中$\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q}(\mathbb{R}^n)}=\sup\limits_{k_0\in Z}2^{-k_0\lambda}\{\sum\limits_{k=-\infty}^{k_0}2^{k\alpha p}\|f\chi_k\|_{L^q(\mathbb{R}^n)}^{p}\}^{1/p}.$

容易得出$M\dot{K}^{\alpha,\lambda}_{p,q}(\mathbb{R}^n)=\dot{K}^{\alpha,\lambda}_{q}(\mathbb{R}^n)$以及$M^{\lambda}_{q}(\mathbb{R}^n)\subseteq M\dot{K}^{0,\lambda}_{q,q}(\mathbb{R}^n)$.

定义2.2^[9] 设$\alpha \in R, \lambda\geq 0, 0<p,q<\infty$, 齐次加权Morrey-Herz空间定义如下

$M\dot{K}^{\alpha,\lambda}_{p,q}(\omega_1,\omega_2)=\{f\in L^{q}_{loc}(\mathbb{R}^n\backslash\{0\}, \omega_2):\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q}(\omega_1,\omega_2)}<\infty\},$

其中$\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q}(\omega_1,\omega_2)}=\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1}(B_k)]^\frac{\alpha p}{n}\|f\chi_k\|_{L^q(\omega_2)}^{p}\}^{1/p}.$

定义2.3^[7] 设$1<p<\infty$, 称$\omega\in A_{p}$, 是指如果对于任意球体Q, 有

$(\frac{1}{|Q|}\int_{Q}{\omega(x)}dx)(\frac{1}{|Q|}\int_{Q}{\omega(x)}^{-1/(p-1)}dx)^{p-1}\leq C.$

定义2.4^[7] 设$1<p_1,p_2<\infty$, 称$\omega\in A_{(p_1,p_2)}$, 是指

$(\frac{1}{|Q|}\int_{Q}{\omega(x)^{p_2}}dx)^{1/p_2}(\frac{1}{|Q|}\int_{Q}{\omega(x)}^{-p'_1}dx)^{1/p'_1}\leq C.$

定义2.5^[7] 设$\delta>0$, 称$\omega\in RD(\delta)$是指

$\frac{\omega(B_k)}{\omega(B_j)}\geq C2^{\delta(k-j)}, k>j.$

以下引理在本文证明中是必要的:

引理2.1^[3] 如果$\omega\in A_{p}$, 则$\frac{\omega(B_k)}{\omega(B_j)}\leq C2^{np(k-j)}, k>j$.

引理2.2^[3] 设$1<p<\infty$, 如果$\omega\in A_{p}$, 则存在$\bar{p}<p$, 使得$\omega\in A_{\bar{p}}$.

引理2.3^[3] 如果$\omega\in A_{(p_1,p_2)}, 1<p_1,p_2<\infty$, 则

${\omega^{-p'_1}(Q)^{1/{p'_1}}}{\omega^{p_2}(Q)^{1/{p_2}}}\leq C|Q|^{1/{p'_1}+1/{p_2}}.$

注  $\omega\in A_(p_1,p_2)$当且仅当$\omega^{p_2}\in A_{1+{p_2}/p'_1}$, 其中$1<p_1,p_2<\infty.$

引理2.4^[3] 设$0<\beta<n, 1<q_1<n/\beta$, 且${1/{q_2}}={1/{q_1}}-\beta/n$, 如果$\omega \in A_{(q_1,q_2)}$, 则$I_\beta$是$L^{q_1}(\omega^{q_1})$空间到$L^{q_2}(\omega^{q_2})$空间上的有界算子.

引理2.5^[4] 假设条件 (2.1) 成立, 设$0<\beta<n, 1<q_1<n/\beta$, 且${1/{q_2}}={1/{q_1}}-\beta/n$, 那么$\|L^{-\frac{\beta}{2}}(f)\|_{L^{q_2}(\mathbb{R}^n)}\leq C\|f\|_{L^{q_1}(\mathbb{R}^n)}$.

定理3.1 假设条件 (2.1) 成立, 设$1<q_1<n/\beta, {1/{q_2}}={1/{q_1}}-\beta/n$, 且$\omega \in A_{(q_1,q_2)}$, 则$L^{-\frac{\beta}{2}}$是$L^{q_1}(\omega^{q_1})$空间到$L^{q_2}(\omega^{q_2})$空间上的有界算子.

定理3.2 假设条件 (2.1) 成立, 设$n\geq2, 0\leq\lambda<\infty, 0<p<\infty, 1<q_1﹤n/\beta, \delta_1﹥0, \delta_2>0$且$1/q_2=1/q_1-\beta/n$, 如果

(1) $\omega_1\in A_m, \omega_1\in RD(\delta_1), 1\leq m<\infty$,

(2) $\omega^{q_2}_2\in A_r, r=1+{q_2/q'_1}$, 且$\omega^{q_2}_2\in RD(\delta_2)$,

(3) $-\delta_2/{\delta_1q_2}<\alpha<(1-\beta/n-\bar{r}/q_2)n/m$, 则$L^{-\frac{\beta}{2}}(f)$是$M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)$空间到$M\dot{K}^{\alpha,\lambda}_{p,q_2}(\omega_1,\omega^{q_2}_2)$空间的有界算子.

定理3.1的证明 当$\omega \in A_{(q_1,q_2)}$时, 根据 (2.2) 式, 引理2.4及引理2.5, 有

$\|I_\beta(|f|)\|_{L^{q_2}(\omega^{q_2})}\leq C\|f\|_{L^{q_1}(\omega^{q_1})},$

由$\|L^{-\frac{\beta}{2}}(f)\|_{L^{q_2}(\mathbb{R}^n)}\leq C\|I_\beta(|f|)\|_{L^{q_2}(\mathbb{R}^n)}$, 知

$\|L^{-\frac{\beta}{2}}(f)\|_{L^{q_2}(\omega^{q_2})}\leq C\|I_\beta(|f|)\|_{L^{q_2}(\omega^{q_2})}.$

综上所述, 得

$\|L^{-\frac{\beta}{2}}(f)\|_{L^{q_2}(\omega^{q_2})}\leq C\|f\|_{L^{q_1}(\omega^{q_1})}.$

定理3.1证明完毕.

定理3.2的证明 记$f(x)=\sum\limits_{j=-\infty}^{\infty}\chi_jf(x)=\sum\limits_{j=-\infty}^{\infty}f_j(x)$, 则

$\|L^{-\frac{\beta}{2}}(f)\|_{M\dot{K}^{\alpha,\lambda}_{p,q_2}(\omega_1,\omega^{q_2}_2)}= \sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n} \|L^{-\frac{\beta}{2}}(f)\chi_k\|^{p}_{L^{q_2}(\omega^{q_2}_2)}\}^{1/p}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n}({\sum\limits_{j=-\infty}^{k-2}\|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}})^{p}\}^{1/p}\\ + C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n}({\sum\limits_{j=k-1}^{k+1}\|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}})^{p}\}^{1/p}\\ + C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n}({\sum\limits_{j=k+2}^{\infty}\|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}})^{p}\}^{1/p}\\ \triangleq E+F+G.$

首先对$F$进行估计

$\begin{aligned} F\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n}({\sum\limits_{j=k-1}^{k+1}\|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}})^{p}\}^{1/p}. \end{aligned}$

由定理3.1知

$F\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n}\|f\chi_k\|^{p}_{L^{q_1}(\omega^{q_1}_2)}\}^{1/p}\\ \leq C\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}.$

其次, 对$E$进行估计, 当$j\leq k-2, x\in A_k, y\in B_j$时, 显然有$2^{k-2}\leq |x-y|\leq 2^{k+1}$, 根据Minkowski不等式及Hölder不等式, 有

$ |L^{-\frac{\beta}{2}}(f_j)(x)|\leq C(\int_{B_j}(\int\limits_0^\infty t^{-\frac{n}{2}}|f_j(y)|e^{-C\frac{|x-y|^2}{t}}\frac{dtdy}{t^{1-\frac{\beta}{2}}})) \leq C\int_{B_j}|f_j(y)|(\int\limits_0^\infty t^{\frac{\beta-n}{2}-1}e^{-C\frac{|x-y|^2}{t}}dt)dy\\ \leq C\int_{B_j}|f_j(y)|(\int\limits_0^{(2^k)^2} t^{\frac{\beta-n}{2}-1}e^{-C\frac{(2^k)^2}{t}}dt+\int\limits_{(2^k)^2}^\infty t^{\frac{\beta-n}{2}-1}e^{-C\frac{(2^k)^2}{t}}dt)dy\\ \leq C2^{k(\beta-n)}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)}{\omega_2^{-{q'_1}}(B_j)^{1/{q'_1}}}.$

那么

$\|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}\leq C2^{k(\beta-n)}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)}{\omega_2^{-{q'_1}}(B_j)^{1/{q'_1}}}{\omega_2^{q_2}(B_k)^{1/{q_2}}}.$

结合引理2.1, 引理2.2及引理2.3, 得

$\|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}\leq C2^{k(\beta-n)}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)}{\omega_2^{-{q'_1}}(B_j)^{1/{q'_1}}}{\omega_2^{q_2}(B_j)^{1/{q_2}}} (\frac{\omega_2^{q_2}(B_k)}{\omega_2^{q_2}(B_j)})^{1/q_2}\\ \leq C2^{k(\beta-n)}2^{nj(1/{q'_1}+1/q_2)}2^{n\bar{r}(k-j)/q_2}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)}\\ \leq C2^{n(j-k)(1-\beta/n-\bar{r}/q_2)}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)}.$

将上述结果应用到$E$中, 有

$ E\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n}({\sum\limits_{j=-\infty}^{k-2}\|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}})^{p}\}^{1/p}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n}({\sum\limits_{j=-\infty}^{k-2}}2^{n(j-k)(1-\beta/n-\bar{r}/q_2)}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^{p}\}^{1/p}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=-\infty}^{k-2}} 2^{n(j-k)(1-\beta/n-\bar{r}/q_2)}(\frac{\omega_1(B_k)}{\omega_1(B_j)})^\frac{\alpha}{n}[{\omega_1(B_j)}]^\frac{\alpha}{n}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^{p}\}^{1/p}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=-\infty}^{k-2}} 2^{n(j-k)(1-\beta/n-\bar{r}/q_2-\alpha m/n)}[{\omega_1(B_j)}]^\frac{\alpha}{n}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^{p}\}^{1/p}.$

当$0<p\leq1$时, 根据不等式$(\sum\limits_{k=1}^{\infty}|a_k|)^p\leq \sum\limits_{k=1}^{\infty}|a_k|^p$, 以及$\alpha<{(1-\beta/n-\bar{r}/q_2)n}/m$, 有

$ E^p\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=-\infty}^{k-2}} 2^{n(j-k)(1-\beta/n-\bar{r}/q_2-\alpha m/n)p}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)})\}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{j=-\infty}^{k_0-2}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)}({\sum\limits_{k=j+2}^{k_0}}2^{n(j-k)(1-\beta/n-\bar{r}/q_2-\alpha m/n)p})\}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{j=-\infty}^{k_0}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)}\}\\ \leq C\|f\|^p_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}. $

当$1<p<\infty$时, 根据Hölder不等式, 得

$ E^p\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=-\infty}^{k-2}} 2^{n(j-k)(1-\beta/n-\bar{r}/q_2-\alpha m/n)}[{\omega_1(B_j)}]^\frac{\alpha}{n}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^p\}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=-\infty}^{k-2}} 2^{n(j-k)(1-\beta/n-\bar{r}/q_2-\alpha m/n)\frac{p}{2}}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)})\\ \times ({\sum\limits_{j=-\infty}^{k-2}}2^{n(j-k)(1-\beta/n-\bar{r}/q_2-\alpha m/n)\frac{p'}{2}})^{p/p'}\}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=-\infty}^{k-2}} 2^{n(j-k)(1-\beta/n-\bar{r}/q_2-\alpha m/n)\frac{p}{2}}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)})\}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{j=-\infty}^{k_0}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)}\}\\ \leq C\|f\|^p_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}.$

最后来估计$G$, 当$j\geq k-2, x\in A_k, y\in B_j$时, $2^{j-2}\leq |x-y|\leq 2^{j+1}$, 根据Minkowski不等式及Hölder不等式, 有

$|L^{-\frac{\beta}{2}}(f_j)(x)|\leq C2^{j(\beta-n)}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)}{\omega_2^{-{q'_1}}(B_j)^{1/{q'_1}}}$

以及

$\|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}\leq C2^{j(\beta-n)}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)}{\omega_2^{-{q'_1}}(B_j)^{1/{q'_1}}}{\omega_2^{q_2}(B_k)^{1/{q_2}}}.$

结合引理2.2及引理2.3得

$ \|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}\leq C2^{j(\beta-n)}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)}{\omega_2^{-{q'_1}}(B_j)^{1/{q'_1}}} {\omega_2^{q_2}(B_j)^{1/{q_2}}}(\frac{\omega_2^{q_2}(B_k)}{\omega_2^{q_2}(B_j)})^{1/q_2}\\ \leq C2^{j(\beta-n)}2^{nj(1/{q'_1}+1/q_2)}2^{(k-j)\delta_2/q_2}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)}\\ \leq C2^{(k-j)\delta_2/q_2}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)},$

有

$ G\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n}({\sum\limits_{j=k+2}^{\infty}\|(L^{-\frac{\beta}{2}}(f_j))\chi_k\|_{L^{q_2}(\omega^{q_2}_2)}})^{p}\}^{1/p}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_k)}]^\frac{\alpha p}{n}({\sum\limits_{j=k+2}^{\infty}2^{(k-j)\delta_2/q_2}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^p}\}^{1/p}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=k+2}^{\infty}} 2^{(k-j)(\alpha\delta_1+\delta_2/q_2)}[{\omega_1(B_j)}]^\frac{\alpha}{n}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^{p}\}^{1/p}. $

当$0<p\leq1$时, 根据不等式$(\sum\limits_{k=1}^{\infty}|a_k|)^p\leq \sum\limits_{k=1}^{\infty}|a_k|^p$以及$\alpha>-\delta_2/{\delta_1q_2}$, 有

$ G\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=k+2}^{k_0}} 2^{(k-j)(\alpha\delta_1+\delta_2/q_2)}[{\omega_1(B_j)}]^\frac{\alpha}{n}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^{p}\}^{1/p}\\ + C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=k_0+1}^{\infty}} 2^{(k-j)(\alpha\delta_1+\delta_2/q_2)}[{\omega_1(B_j)}]^\frac{\alpha}{n}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^{p}\}^{1/p}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{j=-\infty}^{k_0}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)}({\sum\limits_{k=-\infty}^{j-2}}2^{(k-j)(\alpha\delta_1+\delta_2/q_2)p})\}^{1/p}\\ + C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_j)}]^\frac{\alpha p}{n}{\sum\limits_{j=k_0+1}^{\infty}}2^{(k-j)(\alpha\delta_1+\delta_2/q_2)p}[{\omega_1(B_j)}]^{-\frac{\alpha p}{n}}\\ \times [{\omega_1(B_j)}]^\frac{\lambda p}{n}([{\omega_1(B_j)}]^{-\frac{\lambda}{n}}(\sum\limits_{l=-\infty}^{j} [{\omega_1(B_l)}]^\frac{\alpha p}{n}\|f_l\|^p_{L^{q_1}(\omega^{{q_1}}_2)})^{1/p})^p\}^{1/p}\\ \leq C\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}+ C\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1(B_j)}]^\frac{\alpha p}{n} $

$ \times {\sum\limits_{j=k_0+1}^{\infty}}2^{(k-j)(\alpha\delta_1+\delta_2/q_2)p}[{\omega_1(B_j)}]^{-\frac{\alpha p}{n}} [{\omega_1(B_j)}]^\frac{\lambda p}{n}\}^{1/p}\\ \leq C\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}.$

当$1<p<\infty$时, 根据Hölder不等式得

$ G\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=k+2}^{k_0}} 2^{(k-j)(\alpha\delta_1+\delta_2/q_2)}[{\omega_1(B_j)}]^\frac{\alpha}{n}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^{p}\}^{1/p}\\ + C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=k_0+1}^{\infty}} 2^{(k-j)(\alpha\delta_1+\delta_2/q_2)}[{\omega_1(B_j)}]^\frac{\alpha}{n}\|f_j\|_{L^{q_1}(\omega^{{q_1}}_2)})^{p}\}^{1/p}\\ \triangleq G_1+G_2, \\ G_1\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=k+2}^{k_0}} 2^{(k-j)(\alpha\delta_1+\delta_2/q_2)\frac{p}{2}}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)})\\ \times ({\sum\limits_{j=k+2}^{k_0}}2^{(k-j)(\alpha\delta_1+\delta_2/q_2)\frac{p'}{2}})^{p/p'}\}^{1/p}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda}{n}}\{\sum\limits_{j=-\infty}^{k_0}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)}({\sum\limits_{k=-\infty}^{j-2}}2^{(k-j)(\alpha\delta_1+\delta_2/q_2)\frac{p}{2}})\}^{1/p}\\ \leq C\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}, \\ G_2^p\leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=k_0+1}^{\infty}} 2^{(k-j)(\alpha\delta_1+\delta_2/q_2)\frac{p}{2}}[{\omega_1(B_j)}]^\frac{\alpha p}{n}\|f_j\|^p_{L^{q_1}(\omega^{{q_1}}_2)})\\ \times ({\sum\limits_{j=k_0+1}^{\infty}}2^{(k-j)(\alpha\delta_1+\delta_2/q_2)\frac{p'}{2}})^{p/p'}\}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{k=-\infty}^{k_0}({\sum\limits_{j=k_0+1}^{\infty}} 2^{(k-j)(\alpha\delta_1+\delta_2/q_2)\frac{p}{2}}\sum\limits_{l=-\infty}^{j}[{\omega_1(B_l)}]^\frac{\alpha p}{n}\|f_l\|^p_{L^{q_1}(\omega^{{q_1}}_2)})\}\\ \leq C\sup\limits_{k_0\in Z}[{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}} \{\sum\limits_{k=-\infty}^{k_0}{\sum\limits_{j=k_0+1}^{\infty}}2^{(k-j)(\alpha\delta_1+\delta_2/q_2)\frac{p}{2}} [{\omega_1(B_j)}]^{\frac{\lambda p}{n}}([{\omega_1(B_j)}]^{-{\frac{\lambda}{n}}}\\ \times (\sum\limits_{l=-\infty}^{j}[{\omega_1(B_l)}]^\frac{\alpha p}{n}\|f_l\|^p_{L^{q_1}(\omega^{{q_1}}_2)})^{1/p})^p\}\\ \leq C\|f\|^p_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}\sup\limits_{k_0\in Z} [{\omega_1}(B_{k_0})]^{-\frac{\lambda p}{n}}\{\sum\limits_{k=-\infty}^{k_0}[{\omega_1}(B_k)]^{\frac{\lambda p}{n}}\}\\ \leq C\|f\|^p_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}.$

故有$G\leq G_1+G_2\leq C\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}$.

综上所述, 得$\|L^{-\frac{\beta}{2}}(f)\|_{M\dot{K}^{\alpha,\lambda}_{p,q_2}(\omega_1,\omega^{q_2}_2)}\leq C\|f\|_{M\dot{K}^{\alpha,\lambda}_{p,q_1}(\omega_1,\omega^{q_1}_2)}$, 证明完毕.