自交换子的概念被引进以来, 学者们对各种交换子的有界性研究产生了浓厚的兴趣, 尤其是1971年Conifman和Weiss在文献[1]中提出在更一般的空间上研究算子的有界性, 也就是后来的Conifman-weiss意义下的齐型空间之后.在文献[2]中, Bramanti和Cerctti介绍了奇异积分算子交换子及分数次积分算子交换子在齐型空间上的有界性.在文献[3]中, Gao Wenhua和Gao Yonghui研究了次线性算子在齐型空间中的Morrey-Herz空间上的弱有界性. Jorge在文献[4]中讨论了分数次积分算子交换子在齐型空间上的有界性. 2010年, 胡国恩等在文献[5]中给出了齐型空间上奇异积分算子极大交换子的加权估计.陶双平, 曹薇在文献[6]中给出了一类次线性算子和BMO函数生成的交换子在齐型空间中的Morrey-Herz空间上的弱有界性.我们知道, 抛物空间, 双曲空间, 球, Heisenberg群以及欧式空间都是齐型空间的特例.因此, 齐型空间上交换子的研究在抛物和双曲偏微分方程的研究中有着广泛的应用.

受上述有研究结果的启发, 本文主要研究了满足如下两个尺寸条件:

(ⅰ) $|T_{l}f(x)|\leq C\mu(B(x_{0}, d(x_{0}, x)))^{l-1}\|f\|_{L^{1}(X)}, $supp$f\subseteq C_{j}, d(x_{0}, x)\geq k_{d}a^{j+1}, j\in\mathbb{Z};$

(ⅱ) $|T_{l}f(x)|\leq C\mu(B_{j})^{l-1}\|f\|_{L^{1}(X)}, $supp$f\subseteq C_{j}, d(x_{0}, x)\leq a^{j-2}/k_{d}, j\in\mathbb{Z}$的次线性算子及其交换子在弱Morrey-Herz空间上的有界性.

在本文中, $C$表示大于0的常数, 且在不同的地方可能不同.

令$x_{0}\in X, a\geq 2, B_{j}=\{x\in X:d(x_{0}, x)\leq a^{j}\}, C_{j}=B_{j}\backslash B_{j-1}, j\in \mathbb{Z}, \chi_{j}=\chi_{C_{j}}.$首先, 我们介绍本文的相关定义.

定义 1.1  ^[7] 设$X $是一个集合, 在这个集合上赋予一个正则的Borel测度$\mu$和一个拟距离$d$.所谓$d$是一个拟距离, 意味着存在一个常数$k_{d}\geqslant 1$, 使得$\forall x, y, z\in X$有下式成立:

$ d(x, y)\leqslant k_{d}(d(x, z)+d(z, y)). $

假如$\mu$还满足双倍条件, 即存在常数$C\geqslant 1$, 使得对于任意的$x\in $X和$r>0$有

$ \mu(B(x, 2r))\leqslant C\mu(B(x, r))<\infty, $

其中$B(x, r)$表示以$x$为中心, $r$为半径的球, 则称$(X, d, \mu)$是一个Coifman-Weiss意义下的齐型空间.

齐型空间上的测度$\mu$还存在下面的性质(见参考文献[8]).

对于$a\in\mathbb{R}, a\geq2, $存在常数$A>1$, 使得对$x\in X, 0<r<\infty, $有

$ \mu(B(x, ar))\geq A\mu(B(x, r)). $

定义 1.2  ^[9] 称齐型空间$(X, \mu, d)$是正规的, 如果存在常数$C_{1}, C_{2}>0$, 使得对任一$x\in X, r:\mu(x)<r<\mu(X)$有

$ C_{1}r\leq \mu(B(x, r))\leq C_{2}r . $

定义 1.3  ^[3] 设$(X, d, \mu)$为齐型空间, $\alpha\in \mathbb{R}, 0\leqslant \lambda<\infty, 0<p<\infty, 1\leqslant q<\infty, $

(1) Morrey-Herz空间定义为

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

其中

$ \|f\|_{M\dot{K}_{p, q}^{\alpha, \lambda}(X)}=\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda }(\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p}\|f\chi_{k}\|_{L^{q}(X)}^{p})^{\frac{1}{p}}. $

(2) 弱Morrey-Herz空间定义为

$ WM\dot{K}_{p, q}^{\alpha, \lambda}(X)=\{f:f是X上的可测函数, 且\|f\|_{WM\dot{K}_{p, q}^{\alpha, \lambda}(X)}<\infty\}, $

其中

$ \|f\|_{WM\dot{K}_{p, q}^{\alpha, \lambda}(X)}=\sup\limits_{\gamma>0}\gamma\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda} \left\{\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p}\mu\{x\in C_{k}:|f(x)|>\gamma\}^{p/q}\right\}^{\frac{1}{p}}. $

当$p=\infty$时, 作通常的修改.

定义 1.4  ^[10] 设$(X, d, \mu)$是齐型空间, $0<l<1$, 齐型空间上的分数次积分算子定义为

$ I_{l}(f)(x)=\int_{X}K_{l}(x, y)f(y)d\mu(y), $

其中

$ K_{l}(x, y)= \begin{cases} \mu(B(x, d(x, y)))^{l-1}, &x\in X\backslash\{y\};\\ \mu(\{x\})^{l-1}, &x=y, \mu(\{x\})>0. \end{cases} $

定义1.5  ^[11] 设$(X, d, \mu)$为齐型空间,

$ Mf(x)=\sup\limits_{x\in B}\mu(B)^{-1}\int_{B}|f(y)|d\mu(y), \\ f^{\sharp}(x)=\sup\limits_{x\in B}\mu(B)^{-1}\int_{B}|f(y)-f_{B}|d\mu(y), $

齐型空间上的BMO空间定义为

$ {\hbox{BMO}}(X)=\{f\in L_{{\hbox{loc}}}^{1}(X):\|f\|_{\ast}<\infty\}, $

其中$f_{B}=\mu(B)^{-1}\displaystyle\int_{B}|f(x)|d\mu(x)$, $\|f\|_{\ast}=\sup\limits_{x}f^{\sharp}(x)$.

定义1.6  ^[12] 设$(X, d, \mu)$是齐型空间, $0<\beta<1$, 齐型空间上的Lipschitz空间定义为

$ {\hbox{Lip}}_{\beta}(X)=\{f:\|f\|_{{\hbox{Lip}}_{\beta}}=\sup\limits_{x, y\in X, x\neq y}\frac{|f(x)-f(y)|}{d(x, y)^{\beta}}<\infty\}. $

定义 1.7  ^[13] 设$T$是次线性算子, $b\in L_{{\hbox{loc}}}(X)$, 定义由$b$和$T$生成的交换子为

$ [b, T](f)(x)=T[(b(x)-b)f](x). $

为证我们的结果, 需要以下一些结论.

引理2.1  ^[11] (1) 对每个$p:1\leq p<\infty$, 存在常数$C$使得对每个$f\in {\hbox{BMO}}(X)$及球$B$有

$ \left(\mu(B)^{-1}\int_{B}|f(y)-f_{B}|^{p}d\mu(y)\right)^{1/p}\leq C\|f\|_{\ast}. $

(2) 令$b\in {\hbox{BMO}}(X), k, j\in \mathbb{Z}, k>j$, 则存在常数$C$使得

$ |b_{k}-b_{j}|\leq C(k-j)\|b\|_{\ast}, $

其中$b_{k}, b_{j}$分别是$b$在球$B_{k}, B_{j}$上的平均值.

引理 2.2  ^[14] 设$(X, d, \mu)$是齐型空间, $0<\beta<1, 1\leq p\leq\infty, f\in {\hbox{Lip}}_{\beta}(X), $则

$ \|f\|_{{\hbox{Lip}}_{\beta}(X)}\quad\approx\quad\sup\limits_{B}\frac{1}{\mu(B)^{1+\beta}}\int_{B}|f(x)-f_{B}|d\mu(x)\\ \quad\quad\quad\quad\quad\quad\approx\quad\sup\limits_{B}\frac{1}{\mu(B)^{\beta}}\bigg(\frac{1}{\mu(B)}\int_{B}|f(x)-f_{B}|^{p}d\mu(x)\bigg)^{1/p}, $

其中$f_{B}=\frac{1}{\mu(B)}\displaystyle\int_{B}f(x)d\mu(x)$.

引理 2.3  ^[15] 设$(X, d, \mu)$是齐型空间, $B_{1}\subset B_{2}$, $f\in {\rm Lip}_{\beta}(X)$, 则

$ |f_{B_{1}}-f_{B_{2}}| \leq C\|f\|_{{\hbox{Lip}}_{\beta}(X)} \mu(B_{2})^{\beta}, $

其中$f_{B}=\frac{1}{\mu(B)}\displaystyle\int_{B}f(x)d\mu(x)$.

引理 2.4  ^[4] 设$(X, d, \mu)$是正规齐型空间, $0<l<1, 1<q<\infty, 1-1/q=l$,

(1) 存在常数$C>0, $使得对每个$f\in L^{1}(X), \gamma>0, $有

$ \mu\{x\in X:|I_{l}(f)(x)|>\gamma\}\leq C(\frac{\|f\|_{L^{1}(X)}}{\gamma})^{q}. $

(2) 若$1<p<1/l, f\in L^{p}(X), b\in L^{p'}(X), $则存在常数$C>0, $使得对$\gamma>0, $有

$ \mu\{x\in X:|[b, I_{l}](f)(x)|>\gamma\}\leq C(\frac{\|b\|_{L^{p'}(X)}\|f\|_{L^{p}(X)}}{\gamma})^{q}. $

引理2.5  设$(X, d, \mu)$是正规齐型空间, $b\in {\rm Lip}_{\beta}(X), 0<\beta, l<1, 1<q<\infty, 1-1/q=l+\beta<1$, 则交换子$[b, I_{l}]$从$L^{1}(X)$到$L^{q, \infty}(X)$有界.

证  注意到$\mu(B(x, d(x, y)))\sim d(x, y), y\in B(x, d(x, y)), $于是

$ |[b, I_{l}]f(x)|\quad\leq\quad\int_{X}|b(x)-b(y)|\mu(B(x, d(x, y)))^{l-1}|f(y)|d\mu(y)\\ \quad\quad\quad\quad\quad\leq\quad \int_{X}\frac{|b(x)-b(y)|}{d(x, y)^{\beta}}\frac{d(x, y)^{\beta}}{\mu(B(x, d(x, y)))^{1-l}}|f(y)|d\mu(y)\\ \quad\quad\quad\quad\quad\leq\quad C\|b\|_{{\hbox{Lip}}_{\beta}(X)}\int_{X}\mu(B(x, r))^{1-l-\beta}|f(y)|d\mu(y)\\ \quad\quad\quad\quad\quad\leq\quad C\|b\|_{{\hbox{Lip}}_{\beta}(X)}I_{l+\beta}(|f|)(x). $

由引理2.4知$I_{l+\beta}$从$L^{1}(X)$到$L^{q, \infty}(X)$有界, 从而有

$ \|[b, I_{l}](f)\|_{L^{q, \infty}(X)}\leq C\|b\|_{{\hbox{Lip}}_{\beta}(X)}\|f\|_{L^{1}(X)}. $

即引理2.5成立.

主要结果如下:

定理2.1  设$(X, d, \mu)$为齐型空间, 令$0\leq \lambda < \infty, 0\leq l<1, 1<q_{1}<1/l, 1/q_{2}=1/q_{1}-l, \lambda -1/q_{2}<\alpha<\lambda+1-1/q_{1}$且$0<p_{1}\leq p_{2}<\infty$, 若次线性算子$T_{l}$满足尺寸条件(ⅰ),(ⅱ)且$T_{l}$从$L^{q_{1}}(X)$到$L^{q_{2}, \infty}(X)$有界, 则$T_{l}$从$M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)$到$WM\dot{K}_{p_{2}, q_{2}}^{\alpha, \lambda}(X)$有界.

证  若$p_{1}\leq p_{2}, $则$WM\dot{K}_{p_{1}, q_{2}}^{\alpha, \lambda}(X)\subseteq WM\dot{K}_{p_{2}, q_{2}}^{\alpha, \lambda}(X).$故对于本定理, 我们只需证明$p_{1}=p_{2}$的情形.令

$ f=f\chi_{B_{k-K}}+f\chi_{B_{k+K-1}\backslash B_{k-K}}+f\chi_{X\backslash B_{k+K-1}}\triangleq F_{1}+F_{2}+F_{3}, $

其中$K=2+[\log_{a}k_{d}], k_{d}$是(1.1) 式中的常数, $[]$是取整函数, 则

$ |T_{l}(f)|\leq C(|T_{l}(F_{1})|+|T_{l}(F_{2})|+|T_{l}(F_{3})|), $

故有

$ \quad\quad\|T_{l}(f)\|^{p_{1}}_{WM\dot{K}_{p_{1}, q_{2}}^{\alpha, \lambda}(X)}\\ \leq\quad\sum\limits_{i=1}^{3}\sup\limits_{\gamma>0}\gamma^{p_{1}}\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}} \sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\mu\{x\in C_{k}:|T_{l}(F_{i})(x)|>\gamma/3\}^{p_{1}/q_{2}}\\ \triangleq\quad I_{1}+I_{2}+I_{3}. $

对于$I_{2}$, 由$T_{l}$从$L^{q_{1}}(X)$到$L^{q_{2}, \infty}(X)$有界, 可知

$ I_{2}\quad\leq\quad C\sup\limits_{\gamma>0}\gamma^{p_{1}}\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}} \sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}[\sum\limits_{j=k-K+1}^{k+K-1}\frac{\|f_{j}\|_{L^{q_{1}}(X)}}{\gamma}]^{p_{1}}\\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\|f_{k}\|_{L^{q_{1}}(X)}^{p_{1}} \leq C\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}. $

对于$I_{i}, i=1, 3$, 由$1/q_{2}=1/q_{1}-l, $可知$q_{2}\geq q_{1}> 1, $于是

$ I_{i}\quad\leq\quad \sup\limits_{\gamma>0}\gamma^{p_{1}}\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}} \sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\left(\int_{C_{k}}\frac{|T_{l}(F_{i})(x)|}{\gamma}d\mu(x)\right)^{p_{1}/q_{2}}\\ \quad\quad\leq\quad \sup\limits_{\gamma>0}\gamma^{p_{1}}\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}} \sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\left(\int_{C_{k}}\frac{|T_{l}(F_{i})(x)|^{q_{2}}}{\gamma^{q_{2}}}d\mu(x)\right)^{p_{1}/q_{2}}\\ \quad\quad\leq\quad \sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\|T_{l}(F_{i})\chi_{k}\|_{L^{q_{2}}(X)}^{p_{1}}. $

对于$I_{1}$, 注意到$j\leq k-K, x\in C_{k}, 1/q_{2}=1/q_{1}-l, $由条件(ⅰ)可得

$ |T_{l}(F_{1})(x)|\quad\leq\quad\sum\limits_{j=-\infty}^{k-K}|T_{l}(f_{j})(x)| \leq C\sum\limits_{j=-\infty}^{k-K}\mu(B(x_{0}, d(x_{0}, x)))^{l-1}\|f_{j}\|_{L^{1}(X)}\\ \quad\quad\quad\quad\quad\quad\leq\quad C\sum\limits_{j=-\infty}^{k-K}\mu(B_{k})^{l-1}\|f_{j}\|_{L^{1}(X)}. $

故

$ \|T_{l}(F_{1})\chi_{k}\|_{L^{q_{2}}(X)}\leq C\sum\limits_{j=-\infty}^{k-K}\mu(B_{k})^{l-1+1/q_{2}}\|f_{j}\|_{L^{1}(X)} =C\sum\limits_{j=-\infty}^{k-K}\mu(B_{k})^{1/q_{1}-1}\|f_{j}\|_{L^{1}(X)}. $

又注意到$\alpha<\lambda+1-1/q_{1}, $$\|f_{j}\|_{L^{q}(X)}^{p}\leq \mu(B_{j})^{-\alpha p}\sum\limits_{l=-\infty}^{j}\mu(B_{l})^{\alpha p}\|f_{l}\|_{L^{q}(X)}^{p}, $从而有

$ I_{1}\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{(\alpha+1/q_{1}-1) p_{1}}\left(\sum\limits_{j=-\infty}^{k-K}\|f_{j}\|_{L^{1}(X)}\right)^{p_{1}}\\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{(\alpha+1/q_{1}-1) p_{1}}\left(\sum\limits_{j=-\infty}^{k-K}\|f_{j}\|_{L^{q_{1}}(X)}\mu(B_{j})^{(1-1/q_{1})}\right)^{p_{1}}\\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\sum\limits_{k=-\infty}^{k_{0}}\left(\frac{\mu(B_{k})}{\mu(B_{k_{0}})}\right)^{\lambda p_{1}}\left\{\sum\limits_{j=-\infty}^{k-K}\left(\frac{\mu(B_{j})}{\mu(B_{k})}\right)^{\lambda-1/q_{1}+1-\alpha}\mu(B_{j})^{-\lambda}\right.\\ \quad\quad\quad\quad\left.\times\left\{\sum\limits_{l=-\infty}^{j}\mu(B_{l})^{\alpha p_{1}}\|f_{l}\|^{p_{1}}_{L^{q_{1}}(X)}\right\}^{1/p_{1}}\right\}^{p_{1}}\\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\sum\limits_{k=-\infty}^{k_{0}}A^{(k-k_{0})\lambda p_{1}}\{\sum\limits_{j=-\infty}^{k-K}A^{(j-k)(\lambda-1/q_{1}+1-\alpha)}\}^{p_{1}}\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)} \leq C\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}. $

对于$I_{3}$, 注意到$j\geq k+K, x\in C_{k}, 1/q_{2}=1/q_{1}-l, $由条件(ⅱ)可得

$ |T_{l}(F_{3})(x)|\leq\sum\limits_{j=k+K}^{\infty}|T_{l}(f_{j})(x)| \leq C\sum\limits_{j=k+K}^{\infty}\mu(B_{j})^{l-1}\|f_{j}\|_{L^{1}(X)}. $

故

$ \|T_{l}(F_{3})\chi_{k}\|_{L^{q_{2}}(X)}\leq C\mu(B_{j})^{l-1}\mu(B_{k})^{1/q_{2}}\sum\limits_{j=k+K}^{\infty}\|f_{j}\|_{L^{1}(X)}. $

又注意到$\alpha>\lambda-1/q_{2}, $由对$I_{1}$的讨论, 有

$ I_{3}\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\left(\mu(B_{j})^{l-1}\mu(B_{k})^{1/q_{2}}\sum\limits_{j=k+K}^{\infty}\|f_{j}\|_{L^{1}(X)}\right)^{p_{1}}\\ \quad\quad\leq\quad C\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}. $

综上所述, 定理2.1得证.

定理 2.2  设$(X, d, \mu)$为齐型空间, 令$b\in {\hbox{Lip}}_{\beta}(X)$, $0\leq \lambda < \infty, 0\leq l<1, 0<\beta<1, l+\beta<1, 1<q_{1}<1/(\beta+l), 1/q_{2}=1/q_{1}-l-\beta, \lambda -1/q_{2}<\alpha<\lambda+1-1/q_{1}$且$0<p_{1}\leq p_{2}<\infty$, 若次线性算子$T_{l}$满足尺寸条件(ⅰ),(ⅱ)且交换子$[b, T_{l}]$从$L^{q_{1}}(X)$到$L^{q_{2}, \infty}(X)$有界, 即$\|[b, T_{l}]f\|_{L^{q_{2}, \infty}(X)}\leq C\|b\|_{{\hbox{Lip}}_{\beta}}\|f\|_{L^{q_{1}}(X)}, $则$[b, T_{l}]$从$M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)$到$WM\dot{K}_{p_{2}, q_{2}}^{\alpha, \lambda}(X)$有界.

证  类似定理2.1, 只需证明$p_{1}=p_{2}$的情形.令

$ f=f\chi_{B_{k-K}}+f\chi_{B_{k+K-1}\backslash B_{k-K}}+f\chi_{X\backslash B_{k+K-1}}\triangleq F_{1}+F_{2}+F_{3}, $

其中$K=2+[\log_{a}k_{d}], k_{d}$是(1.1) 式中的常数, $[] $是取整函数, 则

$ |[b, T_{l}](f)|\leq C(|[b, T_{l}](F_{1})|+|[b, T_{l}](F_{2})|+|[b, T_{l}](F_{3})|), $

故有

$ \quad\quad\|[b, T_{l}](f)\|^{p_{1}}_{WM\dot{K}_{p_{1}, q_{2}}^{\alpha, \lambda}(X)}\\ \leq\quad\sum\limits_{i=1}^{3}\sup\limits_{\gamma>0}\gamma^{p_{1}}\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}} \sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\mu\{x\in C_{k}:|[b, T_{l}](F_{i})(x)|>\gamma/3\}^{p_{1}/q_{2}}\\ \triangleq\quad J_{1}+J_{2}+J_{3}. $

对于$J_{2}$, 由$T_{l}$从$L^{q_{1}}(X)$到$L^{q_{2}, \infty}(X)$有界, 可知

$ J_{2}\quad\leq\quad C\|b\|_{Lip_{\beta}}\sup\limits_{\gamma>0}\gamma^{p_{1}}\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}} \sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}[\sum\limits_{j=k-K+1}^{k+K-1}\frac{\|f_{j}\|_{L^{q_{1}}(X)}}{\gamma}]^{p_{1}}\\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\|f_{k}\|_{L^{q_{1}}(X)}^{p_{1}}\\ \quad\quad\leq\quad C\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}. $

对于$J_{i}, i=1, 3$, 由$1/q_{2}=1/q_{1}-l-\beta, $可知$q_{2}\geq q_{1}> 1, $于是由定理2.1的证明过程可知

$ J_{i}\leq \sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\|[b, T_{l}](F_{i})\chi_{k}\|_{L^{q_{2}}(X)}^{p_{1}}. $

对于$J_{1}$, 注意到$j\leq k-K, x\in C_{k}, 1/q_{2}=1/q_{1}-l-\beta, $由条件(ⅰ)可得

$ |[b, T_{l}](f_{j})(x)| \leq C\mu(B(x_{0}, d(x_{0}, x)))^{l-1}\|(b(\cdot)-b)f_{j}\|_{L^{1}(X)}\leq C\mu(B_{k})^{l-1}\|(b(\cdot)-b)f_{j}\|_{L^{1}(X)}, $

从而有

$ \quad\quad\|[b, T_{l}](f_{j})\chi_{k}\|_{L^{q_{2}}(X)}^{q_{2}}\\ \leq\quad C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b(x)-b(y)||f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x)\\ \leq\quad C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b(x)-b_{k}||f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x)\\ \quad\quad+ C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b_{k}-b_{j}||f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x)\\ \quad\quad+ C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b(y)-b_{j}||f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x). $

故由引理2.2, 引理2.3及Hölder不等式, 可知

$ \quad\quad\|[b, T_{l}](f_{j})\chi_{k}\|_{L^{q_{2}}(X)}^{q_{2}}\\ \leq\quad C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}|b(x)-b_{k}|^{q_{2}}d\mu(x)[\int_{C_{j}}|f_{j}(y)|^{q_{1}}d\mu(y)]^{q_{2}/q_{1}}\mu(B_{j})^{q_{2}/q_{1}'}\\ \quad\quad+C\mu(B_{k})^{(l-1+\beta)q_{2}+1}\|b\|_{Lip_{\beta}}^{q_{2}}[\int_{C_{j}}|f_{j}(y)|^{q_{1}}d\mu(y)]^{q_{2}/q_{1}}\mu(B_{j})^{q_{2}/q_{1}'}\\ \quad\quad+C\mu(B_{k})^{(l-1)q_{2}+1}[\int_{C_{j}}|f_{j}(y)|^{q_{1}}d\mu(y)]^{q_{2}/q_{1}}[\int_{C_{j}}|b(y)-b_{j}|^{q_{1}'}d\mu(y)]^{q_{2}/q_{1}'}\\ \leq\quad C\mu(B_{k})^{(l-1+\beta)q_{2}+1}\mu(B_{j})^{q_{2}/q_{1}'}\|b\|_{Lip_{\beta}}^{q_{2}}\|f_{j}\|_{L^{q_{1}}(X)}^{q_{2}}\\ \quad\quad+C\mu(B_{k})^{(l-1+\beta)q_{2}+1}\mu(B_{j})^{q_{2}/q_{1}'}\|b\|_{Lip_{\beta}}^{q_{2}}\|f_{j}\|_{L^{q_{1}}(X)}^{q_{2}}\\ \quad\quad+C\mu(B_{k})^{(l-1)q_{2}+1}\mu(B_{j})^{q_{2}/q_{1}'+\beta q_{2}}\|b\|_{Lip_{\beta}}^{q_{2}}\|f_{j}\|_{L^{q_{1}}(X)}^{q_{2}}\\ \leq\quad C\left(\frac{\mu(B_{j})}{\mu(B_{k})}\right)^{q_{2}(1-1/q_{1})}\left(2+\frac{\mu(B_{j})^{\beta q_{2}}}{\mu(B_{k})^{\beta q_{2}}}\right)\|b\|_{Lip_{\beta}}^{q_{2}}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}\\ \leq\quad C\left(\frac{\mu(B_{j})}{\mu(B_{k})}\right)^{q_{2}(1-1/q_{1})}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}. $

又注意到$\alpha<\lambda+1-1/q_{1}, $类似于对$I_{1}$的讨论, 有

$ J_{1}\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\left\{\sum\limits_{j=-\infty}^{k-K}\left(\frac{\mu(B_{j})}{\mu(B_{k})}\right)^{1-1/q_{1}}\|f_{j}\|_{L^{q_{1}}(X)}\right\}^{p_{1}}\\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\sum\limits_{k=-\infty}^{k_{0}}\left(\frac{\mu(B_{k})}{\mu(B_{k_{0}})}\right)^{\lambda p_{1}}\left\{\sum\limits_{j=-\infty}^{k-K}\left(\frac{\mu(B_{j})}{\mu(B_{k})}\right)^{\lambda-1/q_{1}+1-\alpha}\mu(B_{j})^{-\lambda}\right.\\ \quad\quad\quad\quad\left.\times\left\{\sum\limits_{l=-\infty}^{j}\mu(B_{l})^{\alpha p_{1}}\|f_{l}\|^{p_{1}}_{L^{q_{1}}(X)}\right\}^{1/p_{1}}\right\}^{p_{1}}\\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\sum\limits_{k=-\infty}^{k_{0}}A^{(k-k_{0})\lambda p_{1}}\{\sum\limits_{j=-\infty}^{k-K}A^{(j-k)(\lambda-1/q_{1}+1-\alpha)}\}^{p_{1}}\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}\\ \quad\quad\leq\quad C\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}. $

对于$J_{3}$, 注意到$j\geq k+K, x\in C_{k}, 1/q_{2}=1/q_{1}-l-\beta, $由条件(ⅱ)可得

$ |[b, T_{l}](f_{j})(x)|\leq C\mu(B_{j})^{l-1}\|(b(\cdot)-b)f_{j}\|_{L^{1}(X)}. $

于是类似于$J_{1}$的证明, 有

$ \quad\quad\|[b, T_{l}](f_{j})\chi_{k}\|_{L^{q_{2}}(X)}^{q_{2}}\\ \leq\quad C\mu(B_{j})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b(x)-b(y)||f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x)\\ \leq\quad C\frac{\mu(B_{j})}{\mu(B_{k})}\left(2+\frac{\mu(B_{k})}{\mu(B_{k})}\right)\|b\|_{Lip_{\beta}}^{q_{2}}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}\\ \leq\quad C\frac{\mu(B_{k})}{\mu(B_{j})}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}. $

又注意到$\alpha>\lambda-1/q_{2}, $类似于$J_{1}$的讨论, 有

$ J_{3}\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\left\{\sum\limits_{j=k+K}^{\infty}\frac{\mu(B_{k})}{\mu(B_{j})}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}\right\}^{p_{1}}\\ \quad\quad\leq\quad C\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}. $

综上所述, 定理2.2得证.

定理 2.3  设$(X, d, \mu)$为齐型空间, 令$b\in {\hbox{BMO}}(X), 0\leq \lambda < \infty, 0\leq l<1, 1<q_{1}<1/l, 1/q_{2}=1/q_{1}-l, \lambda -1/q_{2}<\alpha<\lambda+1-1/q_{1}$且$0<p_{1}\leq p_{2}<\infty$, 若次线性算子$T_{l}$满足尺寸条件(ⅰ),(ⅱ)且交换子$[b, T_{l}]$从$L^{q_{1}}(X)$到$L^{q_{2}, \infty}(X)$有界, 即$\|[b, T_{l}]f\|_{L^{q_{2}, \infty}(X)}\leq C\|b\|_{*}\|f\|_{L^{q_{1}}(X)}, $则$[b, T_{l}]$从$M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)$到$WM\dot{K}_{p_{2}, q_{2}}^{\alpha, \lambda}(X)$有界.

证  类似定理2.1, 只需证明$p_{1}=p_{2}$的情形.令

$ f=f\chi_{B_{k-K}}+f\chi_{B_{k+K-1}\backslash B_{k-K}}+f\chi_{X\backslash B_{k+K-1}}\triangleq F_{1}+F_{2}+F_{3}, $

其中$K=2+[\log_{a}k_{d}], k_{d}$是(1.1) 式中的常数, $[]$是取整函数.则

$ |[b, T_{l}](f)|\leq C(|[b, T_{l}](F_{1})|+|[b, T_{l}](F_{2})|+|[b, T_{l}](F_{3})|), $

故有

$ \quad\quad\|[b, T_{l}](f)\|^{p_{1}}_{WM\dot{K}_{p_{1}, q_{2}}^{\alpha, \lambda}(X)}\\ \leq\quad\sum\limits_{i=1}^{3}\sup\limits_{\gamma>0}\gamma^{p_{1}}\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}} \sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\mu\{x\in C_{k}:|[b, T_{l}](F_{i})(x)|>\gamma/3\}^{p_{1}/q_{2}}\\ \triangleq\quad M_{1}+M_{2}+M_{3}. $

对于$M_{2}$, 由$T_{l}$从$L^{q_{1}}(X)$到$L^{q_{2}, \infty}(X)$有界, 可知

$ M_{2}\quad\leq\quad C\|b\|_{\ast}\sup\limits_{\gamma>0}\gamma^{p_{1}}\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}} \sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}[\sum\limits_{j=k-K+1}^{k+K-1}\frac{\|f_{j}\|_{L^{q_{1}}(X)}}{\gamma}]^{p_{1}}\\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\|f_{k}\|_{L^{q_{1}}(X)}^{p_{1}}\\ \quad\quad\leq\quad C\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}. $

对于$M_{i}, i=1, 3$, 由$1/q_{2}=1/q_{1}-l, $可知$q_{2}/q_{1}\geq 1, $于是由定理2.1的证明过程可知

$ M_{i}\leq \sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\|[b, T_{l}](F_{i})\chi_{k}\|_{L^{q_{2}}(X)}^{p_{1}}. $

对于$M_{1}$, 注意到$j\leq k-K, x\in C_{k}, 1/q_{2}=1/q_{1}-l, $由条件(ⅰ)可得

$ |[b, T_{l}](f_{j})(x)| \leq C\mu(B(x_{0}, d(x_{0}, x)))^{l-1}\|(b(\cdot)-b)f_{j}\|_{L^{1}(X)}\leq C\mu(B_{k})^{l-1}\|(b(\cdot)-b)f_{j}\|_{L^{1}(X)}, $

从而有

$ \quad\quad\|[b, T_{l}](f_{j})\chi_{k}\|_{L^{q_{2}}(X)}^{q_{2}}\\ \leq\quad C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b(x)-b(y)||f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x)\\ \leq\quad C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b(x)-b_{k}||f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x)\\ \quad\quad+C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b_{k}-b_{j}||f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x)\\ \quad\quad+C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b(y)-b_{j}||f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x), $

又由$1/q_{2}=1/q_{1}-l, $引理2.1及Hölder不等式, 可得

$ \quad\quad\|[b, T_{l}](f_{j})\chi_{k}\|_{L^{q_{2}}(X)}^{q_{2}}\\ \leq\quad C\mu(B_{k})^{(l-1)q_{2}}\int_{C_{k}}|b(x)-b_{k}|^{q_{2}}d\mu(x)[\int_{C_{j}}|f_{j}(y)|^{q_{1}}d\mu(y)]^{q_{2}/q_{1}}\mu(B_{j})^{q_{2}/q_{1}'}\\ \quad\quad+C\mu(B_{k})^{(l-1)q_{2}+1}(k-j)^{q_{2}}\|b\|_{\ast}^{q_{2}}[\int_{C_{j}}|f_{j}(y)|^{q_{1}}d\mu(y)]^{q_{2}/q_{1}}\mu(B_{j})^{q_{2}/q_{1}'}\\ \quad\quad+C\mu(B_{k})^{(l-1)q_{2}}[\int_{C_{j}}|f_{j}(y)|^{q_{1}}d\mu(y)]^{q_{2}/q_{1}}[\int_{C_{j}}|b(y)-b_{j}|^{q_{1}'}d\mu(y)]^{q_{2}/q_{1}'}\\ \leq\quad C\mu(B_{k})^{(l-1)q_{2}+1}\mu(B_{j})^{q_{2}/q_{1}'}\|b\|_{\ast}^{q_{2}}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}\\ \quad\quad+C\mu(B_{k})^{(l-1)q_{2}+1}\mu(B_{j})^{q_{2}/q_{1}'}(k-j)^{q_{2}}\|b\|_{\ast}^{q_{2}}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}\\ \quad\quad+C\mu(B_{k})^{(l-1)q_{2}+1}\mu(B_{j})^{q_{2}/q_{1}'}\|b\|_{\ast}^{q_{2}}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}\\ \leq\quad C\left(\frac{\mu(B_{j})}{\mu(B_{k})}\right)^{q_{2}(1-1/q_{1})}[2+(k-j)^{q_{2}}]\|b\|_{\ast}^{q_{2}}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}\\ \leq \quad C\left(\frac{\mu(B_{j})}{\mu(B_{k})}\right)^{q_{2}(1-1/q_{1})}(k-j)^{q_{2}}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}. $

注意到$\alpha<\lambda+1-1/q_{1}, $$\|f_{j}\|_{L^{q}(X)}^{p}\leq \mu(B_{j})^{-\alpha p}\sum\limits_{l=-\infty}^{j}\mu(B_{l})^{\alpha p}\|f_{l}\|_{L^{q}(X)}^{p}$及(1.2) 式, 有

$ M_{1}\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\left\{\sum\limits_{j=-\infty}^{k-K}(k-j)\left(\frac{\mu(B_{j})}{\mu(B_{k})}\right)^{1-1/q_{1}}\|f_{j}\|_{L^{q_{1}}(X)}\right\}^{p_{1}}\\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\sum\limits_{k=-\infty}^{k_{0}}\left(\frac{\mu(B_{k})}{\mu(B_{k_{0}})}\right)^{\lambda p_{1}}\left\{\sum\limits_{j=-\infty}^{k-K}(k-j)\left(\frac{\mu(B_{j})}{\mu(B_{k})}\right)^{\lambda-1/q_{1}+1-\alpha}\mu(B_{j})^{-\lambda}\right.\\ \quad\quad\quad\quad\left.\times\left\{\sum\limits_{l=-\infty}^{j}\mu(B_{l})^{\alpha p_{1}}\|f_{l}\|^{p_{1}}_{L^{q_{1}}(X)}\right\}^{1/p_{1}}\right\}^{p_{1}}\\ \\ \quad\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\sum\limits_{k=-\infty}^{k_{0}}A^{(k-k_{0})\lambda p_{1}}\{\sum\limits_{j=-\infty}^{k-K}(k-j)A^{(k-j)(\alpha-\lambda+1/q-1)}\}^{p_{1}}\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}\\ \quad\quad\leq\quad C\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}. $

对于$M_{3}$, 注意到$j\geq k+K, x\in C_{k}, 1/q_{2}=1/q_{1}-l, $由条件(ⅱ)可得

$ |[b, T_{l}](f_{j})(x)|\leq C\mu(B_{j})^{l-1}\|(b(\cdot)-b)f_{j}\|_{L^{1}(X)}. $

于是, 类似于$M_{1}$的证明, 有

$ \quad\quad\|[b, T_{l}](f_{j})\chi_{k}\|_{L^{q_{2}}(X)}^{q_{2}}\\ \leq\quad C\mu(B_{j})^{(l-1)q_{2}}\int_{C_{k}}[\int_{C_{j}}|b(x)-b(y)|\cdot|f_{j}(y)|d\mu(y)]^{q_{2}}d\mu(x)\\ \leq\quad C\frac{\mu(B_{k})}{\mu(B_{j})}(j-k)^{q_{2}}\|f_{j}\|^{q_{2}}_{L^{q_{1}}(X)}. $

注意到$\alpha>\lambda-1/q_{2}, $由对$M_{1}$的讨论过程, 有

$ M_{3}\quad\leq\quad C\sup\limits_{k_{0}\in\mathbb{Z}}\mu(B_{k_{0}})^{-\lambda p_{1}}\sum\limits_{k=-\infty}^{k_{0}}\mu(B_{k})^{\alpha p_{1}}\left\{\sum\limits_{j=-\infty}^{k-K}(j-k)\left(\frac{\mu(B_{k})}{\mu(B_{j})}\right)^{1/q_{2}}\|f_{j}\|_{L^{q_{1}}(X)}\right\}^{p_{1}}\\ \quad\quad\leq\quad C\|f\|^{p_{1}}_{M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)}. $

综上所述, 定理2.3得证.

因为分数次积分算子$I_{l}$满足尺寸条件(ⅰ),(ⅱ), 故由本文定理及引理2.4, 引理2.5, 可得下述推论.

推论 2.1  设$(X, d, \mu)$为正规齐型空间, $0\leq l<1, I_{l}$是分数次积分算子.

(1) 若$0\leq \lambda < \infty, 0\leq l<1, 1/q=1-l, \lambda -1/q<\alpha<\lambda$且$0<p_{1}\leq p_{2}<\infty$, 则$I_{l}$从$M\dot{K}_{p_{1}, 1}^{\alpha, \lambda}(X)$到$WM\dot{K}_{p_{2}, q}^{\alpha, \lambda}(X)$有界.

(2) 若$b\in {\rm Lip}_{\beta}(X)$, $0\leq \lambda < \infty, 0<\beta<1, l+\beta<1, 1/q=l-\beta, \lambda -1/q<\alpha<\lambda$且$0<p_{1}\leq p_{2}<\infty$, 则$[b, I_{l}]$从$M\dot{K}_{p_{1}, 1}^{\alpha, \lambda}(X)$到$WM\dot{K}_{p_{2}, q}^{\alpha, \lambda}(X)$有界.

(3) 若$b\in L^{p'}(X), 0\leq \lambda < \infty, 1<q_{1}<1/l, 1/q_{2}=1-l, \lambda -1/q_{2}<\alpha<\lambda+1-1/q_{1}$且$0<p_{1}\leq p_{2}<\infty$, 则$[b, I_{l}]$从$M\dot{K}_{p_{1}, q_{1}}^{\alpha, \lambda}(X)$到$WM\dot{K}_{p_{2}, q_{2}}^{\alpha, \lambda}(X)$有界.

注  设$(X, d, \mu)$为齐型空间, $0\leq l<1, $若用下式

$ |T_{l}f(x)|\leq C\int_{X}\frac{|f(y)|}{\mu(B(x_{0}, d(x, y)))^{1-l}}d\mu(y), x\not\in {\hbox{supp}}f. $

代替本文中次线性算子$T_{l}$所满足的尺寸条件(ⅰ)和(ⅱ), 则定理2.1-2.3依然成立.具体证明可参阅文献[16].