从Zygmund空间和F(p,q,s)空间到Bμ空间的广义复合算子


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	谭春雨

	王茂发

从Zygmund空间和F(p,q,s)空间到B^μ空间的广义复合算子

谭春雨, 王茂发

武汉大学数学与统计学院, 武汉 430072

收稿日期: 2014-11-4

基金项目: 国家自然科学基金(11271293)资助

作者简介: 谭春雨, cytan@whu.edu.cn;王茂发, mfwang.math@whu.edu.cn

摘要: 讨论了单位圆盘上从Zygmund空间, F(p,q,s)空间到B^μ空间上的广义复合算子, 给出了其有界性和紧性的一些判别条件. 同时也刻画了对应小空间上的广义复合算子的有界性和紧性.

关键词: 广义复合算子 Zygmund空间 F(p,q,s)空间 B^μ空间有界性紧性

Generalized Composition Operators from Zygmund Spaces and F(p,q,s) Spaces to B^μ Spaces

Tan Chunyu, Wang Maofa

School of Mathematics and Statistics, Wuhan University, Wuhan 430072

Received: 2014-11-04

Abstract: In this paper, we study generalized composition operators from Zygmund spaces and F(p,q,s) spaces to B^μ spaces and obtain some criterions of the boundedness and compactness, respectively. We also consider the generalized composition operators on the corresponding little spaces

Key words: Generalized compostion operator Zygmund space F(p,q,s) space B^μ space boundedness Compactness

1 引言

设${\Bbb D}$为复平面${\Bbb C}$上的单位圆盘. $H({\Bbb D})$表示${\Bbb D}$上解析函数全体组成的函数空间. $\alpha$-Bloch空间 ${\cal B}^{\alpha}$和小$\alpha$-Bloch空间($0<\alpha<\infty$)分别定义为 $$ {\cal B}^{\alpha}=\Big\{f\in H({\Bbb D}): \sup\limits_{z\in{\Bbb D}}(1-|z|^{2})^{\alpha}|f'(z)|<\infty\Big\}, $$ $$ {\cal B}_{0}^{\alpha}=\Big\{f\in H({\Bbb D}):\lim\limits_{|z|\rightarrow1}(1-|z|^{2})^{\alpha}|f'(z)|=0\Big\}. $$ 易知在范数$\| f\| _{{\cal B}^{\alpha}}=|f(0)|+\sup\limits_{z\in{\Bbb D}}(1-|z|^{2})^{\alpha}|f'(z)|$ 下,${\cal B}^{\alpha}$成为Banach空间，${\cal B}_{0}^{\alpha}$是${\cal B}^{\alpha}$ 的闭子空间,$\alpha=1$时,${\cal B}^{1}={\cal B}$就是经典的 Bloch空间(见文献^[18]). 设$\mu$是$[0,1)$上的一个正的连续函数,若存在正数$s$,$t$,$0<s<t$, 和$\delta\in[0,1)$使得$\mu$满足 $$ \frac{\mu}{(1-r)^{s}} \mbox{ 在$[\delta,1)$上单调下降,} \lim\limits_{r\rightarrow1}\frac{\mu}{(1-r)^{s}}=0, $$ $$ \frac{\mu}{(1-r)^{t}} \mbox{ 在$[\delta,1)$上单调上升,} \lim\limits_{r\rightarrow1}\frac{\mu}{(1-r)^{t}}=\infty, $$ 则$\mu$称为正规权函数(见文献^[11]). $\mu$-Bloch空间${B^\mu }$和小$\mu$-Bloch空间$B_0^\mu$分别定义为 $$ {B^\mu }=\Big\{f\in H({\Bbb D}):\sup\limits_{z\in{\Bbb D}}\mu(|z|)|f'(z)|<\infty\Big\}, $$ $$ B_0^\mu =\Big\{f\in H({\Bbb D}):\lim\limits_{z\rightarrow1}\mu(|z|)|f'(z)|=0\Big\}. $$ 在赋以范数$\| f\| _{{B^\mu }}=|f(0)|+\sup\limits_{z\in{\Bbb D}}\mu(|z|)|f'(z)|$下,${B^\mu }$成为Banach空间, $B_0^\mu$是${B^\mu }$的闭子空间,当$\mu(z)=(1-z^{2})^{\alpha}$时,${B^\mu }$就是$\alpha$-Bloch空间 ${\cal B}^{\alpha}$ (见文献^{[2, 5, 12]}).

Zygmund空间$\ Z$是由满足下列性质的$f\in H({\Bbb D})\cap C(\overline{{\Bbb D}})$组成的函数空间 $$ \sup\limits_{{\rm e}^{{\rm i}\theta}\in\partial{\Bbb D},h>0}\frac{|f({\rm e}^{{\rm i}(\theta+h)})+f({\rm e}^{{\rm i}(\theta-h)})-2f({\rm e}^{{\rm i}\theta})|}{h}<\infty. $$ 根据文献[3,定理5.3],我们知道$f\in\ Z$当且仅当 $\sup\limits_{z\in{\Bbb D}}(1-|z|^{2})|f''(z)|<\infty$. 在赋以范数$\| f\| _{\ Z}=|f(0)|+|f'(0)|+\sup\limits_{z\in{\Bbb D}}(1-|z|^{2})|f''(z)|$下,$\ Z$成为Banach空间. 小Zygmund空间${{\ Z}_0}:=\{f\in\ Z: \lim\limits_{|z|\rightarrow1}(1-|z|^{2})|f''(z)|=0\}$. 对于$f\in\ Z$,根据文献^[13],有

\begin{equation}\label{aa} |f'(z)|\leq C\| f\| _{\ Z}\ln\frac{2}{1-|z|^{2}}. \end{equation}

(1.1)

空间$F(p,q,s)$ 和小${{\ F}_0}$($0<p,s<\infty$,$-2<q<\infty$) (见文献^[17])分别定义为 $$ \ F=\bigg\{f\in H({\Bbb D}):\sup\limits_{a\in{\Bbb D}}\int_{{\Bbb D}}|f'(z)|^{p}(1-|z|^{2})^{q} (1-|\sigma_{a}(z)|^{2})^{s}dm(z)<\infty\bigg\}, $$ $$ {{\ F}_0}=\bigg\{f\in H({\Bbb D}):\lim\limits_{|a|\rightarrow1}\int_{{\Bbb D}}|f'(z)|^{p}(1-|z|^{2})^{q} (1-|\sigma_{a}(z)|^{2})^{s}dm(z)=0\bigg\}, $$ 其中$dm$是${\Bbb D}$上的规范化Lebesgue面积测度,$\sigma_{a}(z)=\frac{a-z}{1-\bar{a}z}$是 ${\Bbb D}$上的M\"{o}bius变换. 在赋以范数$\| f\| _{\ F}=|f(0)|+\{\sup\limits_{a\in{\Bbb D}}\int_{{\Bbb D}} |f'(z)|^{p}(1-|z|^{2})^{q}(1-|\sigma_{a}(z)|^{2})^{s}dm(z)\}^{\frac{1}{p}}$ 下, $\ F$成为Banach空间,${{\ F}_0}$ 是$\ F$的子空间且 $\ F$ 推广了许多经典的函数空间. 诸如$s>1$,$\ F={\cal B}^{\frac{q+2}{p}}$,${{\ F}_0}={\cal B}^{\frac{q+2}{p}}_{0}$; 对于$0<s\leq1$,$\ F\subset{\cal B}^{\frac{q+2}{p}}$, ${{\ F}_0}\subset{\cal B}^{\frac{q+2}{p}}_{0}$; 而$F(2,0,s)={\cal Q}_{s}$, $F_{0}(2,0,s)={\cal Q}_{s,0}$; 此外$F(2,0,1)=BMOA$,$F_{0}(2,0,1)=VMOA$; 如果$q+s\leq1$,则$\ F$是一个常数函数空间.

设$\varphi$是${\Bbb D}$到自身的一个解析映射,则$\varphi$按下式 $$ (C_{\varphi}f)(z)=f(\varphi(z)),f\in H({\Bbb D}),z\in{\Bbb D}, $$ 导出$H({\Bbb D})$上的一个线性算子$C_{\varphi}$,称之为复合算子. 不同函数空间上的复合算子多年来受到了广泛的研究(见文献^{[6, 10]}), 特别地,在文献 ^[7]中,Li S和 Stevic S首次引入了广义复合算子的概念, 并研究了Zygmund空间和$\alpha$-Bloch空间上广义复合算子的一些基本性质. 对于$g\in H({\Bbb D})$和解析自映射$\varphi: {\Bbb D}\to{\Bbb D}$,广义复合算子$ C_{\varphi}^{g}$定义为 $$ (C_{\varphi}^{g}f)(z)=\int_{0}^{z}f'(\varphi(t))g(t){\rm d}t. $$ 显然当$g=\varphi'$时,$C_{\varphi}^{g}$本质上就是复合算子. 广义复合算子由于与一些经典函数空间上的等距表示的紧密联系而日益受到广泛研究 (参见文献^{[4, 7]}).

本文主要研究了从Zygmund空间到${\cal B}^{\mu}$空间,$\ F$空间到${\cal B}^{\mu}$ 空间及其相应小空间上的广义复合算子的有界性和紧性,得到了一些充分必要的判别条件, 拓展了文献^[7]中的主要结果. 下文中用$C$表示常数,约定它是绝对正常数并且在不同的地方可以不同.

2 $C_{\varphi}^{g}:\ Z\rightarrow{B^\mu }$ 的有界性与紧性}

由于后面主要定理证明的需要,在本节中我们首先给出下面几个引理.

引理2.1 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 则$C_{\varphi}^{g}:\ Z$ (或${{\ Z}_0})\rightarrow{B^\mu }$ (或$B_0^\mu )$是紧算子当且仅当 $C_{\varphi}^{g}:\ Z$ (或${{\ Z}_0})\rightarrow{B^\mu }$ (或$B_0^\mu )$有界并且对于 $\ Z$ (或${{\ Z}_0})$中的任意有界序列$\{f_{n}\}_{n\in\ N}$,$f_{n}$在${\Bbb D}$的紧子集上一致收敛于$0$时,有$\| C_{\varphi}^{g}f_{n}\| _{{B^\mu }}\rightarrow0$.

注此定理可以用文献^[1]中的常规方法类似证明,故我们略去其证明过程.

引理2.2^[9] $B_0^\mu$中的闭集${\Bbb K}$是紧的当且仅当${\Bbb K}$有界并且 $$ \lim\limits_{|z|\rightarrow1}\sup\limits_{f\in{\Bbb K}}\mu(|z|)|f'(z)|=0. $$

定理2.1 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权,则下面条件等价.

(1) $C_{\varphi}^{g}:\ Z\rightarrow{B^\mu }$有界.

(2) $C_{\varphi}^{g}:{{\ Z}_0}\rightarrow{B^\mu }$有界.

(3)

\begin{equation}\label{a1} \sup\limits_{z\in{\Bbb D}}\mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^{2}}<\infty. \end{equation}

(2.1)

证 $(1)\Rightarrow(2)$ 显然.

$(2)\Rightarrow(3)$~ 设$C_{\varphi}^{g}:Z_{0}\rightarrow{B^\mu }$有界. 则对$f\in Z_{0}$存在常数$C$使得 $$ \| C_{\varphi}^{g}f\| _{{B^\mu }}\leq C\| f\| _{\ Z}. $$ 取$f(z)=z\in{{\ Z}_0}$,则有

\begin{equation}\label{a2} \sup\limits_{z\in{\Bbb D}}\mu(|z|)|g(z)|<\infty. \end{equation}

(2.2)

令

\begin{equation}\label{a3} h(z)=(z-1)\bigg{[}\Big(1+\ln\frac{1}{1-z}\Big)^{2}+1\bigg{]}, \end{equation}

(2.3)

\begin{equation}\label{a4} h_{a}(z)=\frac{h(\bar{a}z)}{\bar{a}}\Big(\ln\frac{1}{1-|a|^{2}}\Big)^{-1}. \end{equation}

(2.4)

其中$a\in{\Bbb D}\setminus\{0\}$. 则$h_{a}\in{{\ Z}_0}$ (见文献^[13]). 计算易知 $$ h'_{a}(z)=\Big(\ln\frac{1}{1-\bar{a}z}\Big)^{2}\Big(\ln\frac{1}{1-|a|^2}\Big)^{-1}. $$ 如果$|\varphi(\lambda)|>\frac{1}{2}$,我们有

\begin{eqnarray}\label{a5} \infty & >&\| C_{\varphi}^{g}h_{\varphi(\lambda)}\| _{{B^\mu }}\geq\mu(|\lambda|)| (C_{\varphi}^{g}h_{\varphi(\lambda)})'(\lambda)|\nonumber\\ & =&\mu(|\lambda|)|h'_{\varphi(\lambda)}(\varphi(\lambda))\| g(\lambda)| =\mu(|\lambda|)|g(\lambda)|\ln\frac{1}{1-|\varphi(\lambda)|^2}. \end{eqnarray}

(2.5)

另外,如果$|\varphi(\lambda)|\leq\frac{1}{2}$,由(2.2)式,

\begin{equation}\label{a6} \sup\limits_{|\varphi(\lambda)|\leq\frac{1}{2}}\mu(|\lambda|)|g(\lambda)|\ln\frac{1}{1-|\varphi(\lambda)|^2}\leq\sup\limits_{\lambda\in{\Bbb D}}\mu(|\lambda|)|g(\lambda)|\ln\frac{4}{3}<\infty. \end{equation}

(2.6)

由(2.2)式,(2.5)式和(2.6)式,即得到(2.1)式.

$(3)\Rightarrow(1)$~ 假设(2.1)式成立. 则对任意$f\in\ Z$,由(1.1)式得

\begin{equation}\label{a7} \mu(|z|)|(C_{\varphi}^{g}f)'(z)|=\mu(|z|)|f'(\varphi(z))\| g(z)|\leq C\| f\| _{\ Z}\mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^2}. \end{equation}

(2.7)

在上式中对$z\in{\Bbb D}$取上确界,结合(2.1)式. 即得$C_{\varphi}^{g}:\ Z\rightarrow{B^\mu }$有界.

定理2.2 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$ 是正规权,则下面条件等价.

(1) $C_{\varphi}^{g}:\ Z\rightarrow{B^\mu }$是紧算子.

(2) $C_{\varphi}^{g}:{{\ Z}_0}\rightarrow{B^\mu }$是紧算子.

(3) $C_{\varphi}^{g}:\ Z\rightarrow{B^\mu }$有界,并且

\begin{equation}\label{b1} \lim\limits_{|\varphi(z)|\rightarrow1}\mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^{2}}=0. \end{equation}

(2.8)

证 $(1)\Rightarrow(2)$ 显然.

$(2)\Rightarrow(3)$ 设$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow{B^\mu }$是紧算子, 则$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow{B^\mu }$有界. 由定理2.1知 $C_{\varphi}^{g}:\ Z\rightarrow{B^\mu }$有界. 设$\{z_{n}\}_{n\in\mathbb{N}}$是${\Bbb D}$中的序列满足 $n\rightarrow\infty$时,$|\varphi(z_n)|\rightarrow1$并且$\varphi(z_n)\neq0(n\in\mathbb{N})$.

令 $$ h_{n}(z)=\frac{h(\overline{\varphi(z_n)}z)}{\overline{\varphi(z_n)}} \Big(\ln\frac{1}{1-|\varphi(z_n)|^{2}}\Big)^{-1},n\in\mathbb{N}. $$ 从定理2.1的证明中可知对每个$n\in\mathbb{N}$有$h_n\in{{\ Z}_0}$. 此外,当$n\rightarrow\infty$时, $h_n$在${\Bbb D}$的紧子集上一致收敛于$0$.

由于 \begin{eqnarray*} \| C_{\varphi}^{g}h_n\| _{{B^\mu }} &=&\sup\limits_{z\in{\Bbb D}}\mu(|z|)|(C_{\varphi}^{g}h_{n})'(z)|\geq\mu(|z_n|)|h'_{n}(\varphi(z_n))\| g(z_n)| \\ &=&\mu(|z_n|)|g(z_n)|\ln\frac{1}{1-|\varphi(z_n)|^2}, \end{eqnarray*}由引理2.1即有$\lim\limits_{n\rightarrow\infty}\mu(|z_n|)|g(z_n)|\ln\frac{1}{1-|\varphi(z_n)|^2}=0$,由此式也有$\lim\limits_{n\rightarrow\infty}\mu(|z_n|)|g(z_n)|=0$,这就得到了(2.8)式.

$(3)\Rightarrow(1)$ 设$C_{\varphi}^{g}:\ Z\rightarrow{B^\mu }$有界并且(2.8)式成立. 由定理2.1知

\begin{equation}\label{b2} L:=\sup\limits_{z\in{\Bbb D}}\mu(|z|)|g(z)|<\infty. \end{equation}

(2.9)

设$\{f_n\}_{n\in\mathbb{N}}$是$\ Z$中的序列满足$\sup\limits_{n\in\mathbb{N}}\| f_n\| _{\ Z}\leq M$,$M>0$, 并且当$n\rightarrow\infty$时,$f_n$在${\Bbb D}$的紧子集上一致收敛于$0$. 由(2.8)式,对任意$\varepsilon>0$,存在常数$\delta\in(0,1)$, 使得当$\delta<|\varphi(z)|<1$有 $$ \mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^{2}}<\varepsilon/M. $$ 令$K=\{w\in{\Bbb D}:|w|\leq\delta\}$,结合(1.1)式有 \begin{eqnarray*} \| C_{\varphi}^{g}f_n\| _{{B^\mu }} & =&\sup\limits_{z\in{\Bbb D}}\mu(|z|)|f_n'(\varphi(z))\| g(z)| \\ & \leq&\sup\limits_{\{z\in{\Bbb D}:|\varphi(z)|\leq\delta\}}\mu(|z|)|f_n'(\varphi(z))\| g(z)|+\sup\limits_{\{z\in{\Bbb D}:\delta<|\varphi(z)|<1\}}\mu(|z|)|f_n'(\varphi(z))\| g(z)|\\ & \leq& L\sup\limits_{\{w\in K\}}|f'_{n}(w)|+C\| f_n\| _{\ Z}\sup\limits_{\{z\in{\Bbb D}:\delta<|\varphi(z)|<1\}}\mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^2}\\ & \leq &L\sup\limits_{\{w\in K\}}|f'_{n}(w)|+C\varepsilon. \end{eqnarray*} 由Cauchy估计,若$\{f_n\}_{n\in\mathbb{N}}$在${\Bbb D}$的紧子集上一致收敛于$0$, 则$\{f'_n\}_{n\in\mathbb{N}}$在${\Bbb D}$的紧子集上也一致收敛于$0$. $K$是紧集, $\lim\limits_{n\rightarrow\infty}\sup\limits_{w\in K}|f'_{n}(w)|=0$. 在上面最后一个不等式中令$n\rightarrow\infty$, 则$\lim\limits_{n\rightarrow\infty}\| C_{\varphi}^{g}f_n\| _{{B^\mu }}\leq C\varepsilon$. $\varepsilon$是任意的,此极限等于$0$. 根据引理2.1即得结论.

定理2.3 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权, 则$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow B_0^\mu $有界当且仅当$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow{B^\mu }$有界并且

\begin{equation}\label{c1} \lim\limits_{|z|\rightarrow1}\mu(|z|)|g(z)|=0. \end{equation}

(2.10)

证设$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow B_0^\mu $有界,易知$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow{B^\mu }$有界. 取函数$f(z)=z\in{{\ Z}_0}$,则立即可得(2.10)式.

反过来,设$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow{B^\mu }$有界,并且(2.10)式成立. 则对任意多项式$P$ 有

\begin{equation}\label{c2} \mu(|z|)|(C_{\varphi}^{g}P)'(z)|=\mu(|z|)|P'(\varphi(z))\| g(z)|. \end{equation}

(2.11)

而

\begin{equation}\label{c3} \sup\limits_{w\in{\Bbb D}}|P'(w)|<\infty. \end{equation}

(2.12)

由(2.11)和(2.12)式结合(2.10)式知$C_{\varphi}^{g}P\in B_0^\mu$. 又因为多项式在$ {{\ Z}_0} $中稠密(见文献^[8]),则对任意$f\in{{\ Z}_0}$, 存在一个多项式序列$\{P_n\}_{n\in\mathbb{N}}$ 使得当$n\rightarrow\infty$时, $\| f-P_n\| _{\ Z}\rightarrow0$. 则当$n\rightarrow\infty$时有 $$ \| C_{\varphi}^{g}f-C_{\varphi}^{g}P_{n}\| _{{B^\mu }}\leq\| C_{\varphi}^{g}\| _{{{\ Z}_0}\rightarrow{B^\mu }}\| f-P_n\| _{\ Z}\rightarrow0. $$ 而$B_0^\mu$是${B^\mu }$的闭子集,得到$C_{\varphi}^{g}({{\ Z}_0})\subseteq B_0^\mu $, 由此证明了$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow B_0^\mu $有界.

定理2.4 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权,则下面条件等价.

(1) $C_{\varphi}^{g}:\ Z\rightarrow B_0^\mu $是紧算子.

(2) $C_{\varphi}^{g}:{{\ Z}_0}\rightarrow B_0^\mu $是紧算子.

(3) $C_{\varphi}^{g}:\ Z\rightarrow{B^\mu }$有界,并且

\begin{equation}\label{d1} \lim\limits_{|z|\rightarrow1}\mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^{2}}=0. \end{equation}

(2.13)

证 $(1)\Rightarrow(2)$显然.

$(2)\Rightarrow(3)$ 设 $C_{\varphi}^{g}:{{\ Z}_0}\rightarrow B_0^\mu $是紧算子, 则$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow B_0^\mu $有界. 由定理2.3知(2.10)式成立.

当$\| \varphi\| _{\infty}<1$时,根据(2.10)式得 $$ \lim\limits_{|z|\rightarrow1}\mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^{2}}\leq\ln\frac{2}{1-\| \varphi\| _{\infty}^{2}}\lim\limits_{|z|\rightarrow1}\mu(|z|)|g(z)|=0. $$ 即得(2.13)式.

现在设$\| \varphi\| _{\infty}=1$. 又设$\{z_{n}\}_{n\in\mathbb{N}}$是${\Bbb D}$中的序列满足当 $n\rightarrow\infty$时,$|\varphi(z_n)|\rightarrow1$. 由于$C_{\varphi}^{g}:{{\ Z}_0}\rightarrow{B^\mu }$是紧算子. 由定理2.2有 $$ \lim\limits_{|\varphi(z)|\rightarrow1}\mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^{2}}=0, $$ 则对任意$\varepsilon>0$,存在$r\in(0,1)$使得当$r<|\varphi(z)|<1$时有 $\mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^{2}}<\varepsilon$. 从(2.10)式又可得存在$\sigma\in(0,1)$使得当$\sigma<|z|<1$时有 $\mu(|z|)|g(z)|\leq\varepsilon/\ln\frac{2}{1-r^{2}}$.

因此当$\sigma<|z|<1$,$r<|\varphi(z)|<1$时有

\begin{equation}\label{d2} \mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^{2}}<\varepsilon. \end{equation}

(2.14)

$\sigma<|z|<1$,$|\varphi(z)|\leq r$时

\begin{equation}\label{d3} \mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^{2}}\leq\mu(|z|)|g(z)|\ln\frac{2}{1-r^{2}}<\varepsilon. \end{equation}

(2.15)

由(2.14)式与(2.15)式即得(2.13)式.

$(3)\Rightarrow(1)$ 设$f\in\ Z$,由(1.1)式知 $$ \mu(|z|)|(C_{\varphi}^{g}f)'(z)|=\mu(|z|)|f'(\varphi(z))\| g(z)|\leq C\| f\| _{\ Z}\mu(|z|)|g(z)|\ln\frac{2}{1-|\varphi(z)|^2}. $$ 对上面不等式关于$f\in\ Z$,$\| f\| _{\ Z}\leq1$取上确界. 并令$|z|\rightarrow1$,由(2.13)式得 $$ \lim\limits_{|z|\rightarrow1}\sup\limits_{\| f\| _{\ Z}\leq1}\mu(|z|)|(C_{\varphi}^{g}f)'(z)|=0 $$ 由引理2.2得到$C_{\varphi}^{g}:\ Z\rightarrow B_0^\mu $是紧算子.

3 $C_\varphi ^g:F(p,q,s) \to {B^\mu }$的有界性与紧性

在这一节中,我们对从$\ F$(p,q,s) (或${{\ F}_0}$(p,q,s))到${\cal B}^{\mu}$ (或$B_0^\mu)$上的广义复合算子的有界性、紧性进行刻画,首先介绍下面引理.

引理3.1 ^[17] 设$0<p,s<\infty$,$-2<q<\infty$,$q+s>-1$并且$f\in\ F$(p,q,s), 则存在常数$C>0$使得$\| f\| _{{\beta ^\alpha }}\leq C\| f\| _{F}$,此外,如果$f\in{{\ F}_0}$(p,q,s), 则$f\in\beta _0^\alpha$. 这里$\alpha=\frac{q+2}{p}$.

类似引理2.1,我们易证下面的结论.

引理3.2 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$,$-2<q<\infty$,$q+s>-1$. 则$C_{\varphi}^{g}:\ F$ (p,q,s)(或$\;{F_0}(p,q,s) \to {B^\mu }$)(或$B_0^\mu)$ 是紧算子当且仅当$C_\varphi ^g:F(p,q,s)$(或${F_0}(p,q,s) \to {B^\mu }$ (或$B_0^\mu)$ 有界并且对于${\ F}(p,q,s)$ (或${{\ F}_0}(p,q,s))$中的任意有界序列$\{f_{n}\}_{n\in\mathbb{N}}$, $f_{n}$在${\Bbb D}$ 的紧子集上一致收敛于$0$时,有$\| C_{\varphi}^{g}f_{n}\| _{{B^\mu }}\rightarrow0$.

定理3.1 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$, $-2<q<\infty$,$q+s>-1$. 则$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$有界当且仅当

\begin{equation}\label{e1} \sup\limits_{z\in{\Bbb D}}\frac{\mu(|z|)|g(z)|}{(1-|\varphi(z)|^{2})^{\alpha}}<\infty. \end{equation}

(3.1)

证设$C_\varphi ^g:F(p,q,s) \to {B^\mu }$有界. 令$\alpha=\frac{q+2}{p}$,固定$w\in{\Bbb D}$, 考虑函数

\begin{equation}\label{e2} f_{w}(z)=\frac{(z-\varphi(w))(1-|\varphi(w)|^{2})}{(1-\overline{\varphi(w)}z)^{\alpha+1}}. \end{equation}

(3.2)

根据文献^[16]有${\left\| {{f_w}} \right\|_{F(p,q,s)}} \leqslant C$,这里的常数$C$与$w$的取值无关. 由计算知$f_{w}(\varphi(w))=0$,$f'(\varphi(w))=\frac{1}{(1-|\varphi(w)|^2)^{\alpha}}$. 则 $$ \frac{\mu(|w|)|g(w)|}{(1-|\varphi(w)|^{2})^{\alpha}}=\mu(|w|)|f'(\varphi(w))\| g(w)|\leq\| C_{\varphi}^{g}f_{w}\| _{{B^\mu }}<\infty. $$ 即(3.1)式成立.

反之,设(3.1)式成立. 对任意$f \in F(p,q,s)$,根据引理3.1有 \begin{eqnarray*} \mathop {\sup }\limits_{z \in \mathbb{D}} {(1 - |z{|^2})^\alpha }|f'(z)| \leqslant C{\left\| f \right\|_{F{\text{(p,q,s)}}}} \end{eqnarray*} 则 \begin{eqnarray*} {\left\| {C_\varphi ^gf(z)} \right\|_{{B^\mu }}} = \mathop {\sup }\limits_{z \in \mathbb{D}} \mu (|z|)|f'(\varphi (z))\left\| {g(z)|} \right. \hfill \\ = \mathop {\sup }\limits_{z \in \mathbb{D}} {(1 - |\varphi (z){|^2})^\alpha }|f'(\varphi (z))|\frac{{\mu (|z|)|g(z)|}}{{{{(1 - |\varphi (z){|^2})}^\alpha }}} \hfill \\ \leqslant C{\left\| f \right\|_{F{\text{(p,q,s)}}}}\mathop {\sup }\limits_{z \in \mathbb{D}} \frac{{\mu (|z|)|g(z)|}}{{{{(1 - |\varphi (z){|^2})}^\alpha }}} \hfill \\ \end{eqnarray*} 由(3.1)式知$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$有界.

定理3.2 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$, $-2<q<\infty$,$q+s>-1$. 则$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$是紧算子当且仅当

\begin{equation}\label{f1} \lim\limits_{|\varphi(z)|\rightarrow1}\frac{\mu(|z|)|g(z)|}{(1-|\varphi(z)|^{2})^{\alpha}}=0. \end{equation}

(3.3)

证设$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$是紧算子. 令$\alpha=\frac{q+2}{p}$, 并且$\{z_n\}_{z\in\mathbb{N}}$是${\Bbb D}$中的序列满足当$n\rightarrow\infty$时,$|\varphi(z_{n})|\rightarrow1$. 考虑函数 $$ f_{n}(z)=\frac{(z-\varphi(z_{n}))(1-|\varphi(z_{n})|^{2})}{(1-\overline{\varphi(z_{n})}z)^{\alpha+1}}. $$ 同样用文献^[16]中的方法得到${f_n} \in F{\text{(p,q,s)}}$,$(f_{n})_{n\in\mathbb{N}}$是$F(p,q,s)$中的一个有界序列,并且在${\Bbb D}$的紧子集上一致收敛于$0$. 注意到$f_{n}(\varphi(z_n))=0$,$f'(\varphi(z_n))=\frac{1}{(1-|\varphi(z_n)|^2)^{\alpha}}$. 即有 $$ \frac{\mu(|z_{n}|)|g(z_{n})|}{(1-|\varphi(z_{n})|^{2})^{\alpha}}=\mu(|z_{n}|)|f'(\varphi(z_{n}))\| g(z_{n})|\leq\| C_{\varphi}^{g}f_{n}\| _{{B^\mu }}. $$ 根据引理3.2,(3.3)式成立.

反过来,设(3.3)式成立. 令$\{f_{n}\}_{n\in\mathbb{N}}$是$\ F(p,q,s)$中有界序列并在${\Bbb D}$的紧子集上一致收敛于$0$. 令$M = \mathop {\sup }\limits_n {\left\| {{f_n}} \right\|_{F{\text{(p,q,s)}}}}<\infty $,根据引理3.2, 往证$\lim\limits_{n\rightarrow\infty}\| C_{\varphi}^{g}f_{n}\| _{{B^\mu }}=0$. 由(3.3)式,对于任意$\varepsilon>0$,存在$r\in(0,1)$,当$|\varphi(z)|>r$时, 有$\frac{\mu(|z|)|g(z)|}{(1-|\varphi(z)|^{2})^{\alpha}}<\varepsilon$. 由Cauchy 估计, 如果$\{f_{n}\}_{n\in\mathbb{N}}$在${\Bbb D}$的紧子集上一致收敛于$0$,则$\{f'_{n}\}_{n\in\mathbb{N}}$ 在${\Bbb D}$的紧子集上也一致收敛于$0$,即存在$n_{0}>0$使得当$n>n_{0}$ 时, $\sup\limits_{\{|\varphi(w)|\leq r\}}|f'_{n}(\varphi(w))|<\varepsilon$. 由以上叙述及引理3.1得到 \begin{eqnarray*} & &\sup\limits_{w\in{\Bbb D}}\mu(|w|)|f'_{n}(\varphi(w))\| g(w)| \\ & \leq&\sup\limits_{\{|\varphi(w)|\leq r\}}\mu(|w|)|f'_{n}(\varphi(w))\| g(w)| +\sup\limits_{\{|\varphi(w)|>r\}}\mu(|w|)|f'_{n}(\varphi(w))\| g(w)|\\ & =& \sup\limits_{\{|\varphi(w)|\leq r\}}\mu(|w|)|f'_{n}(\varphi(w))\| g(w)|\\ & &+ \sup\limits_{\{|\varphi(w)|>r\}}(1-|\varphi(w)|^2)^{\alpha}|f'_{n}(\varphi(w))|\frac{\mu(|w|)|g(w)|}{(1-|\varphi(w)|^2)^{\alpha}}\\ &\leq &C\sup\limits_{\{|\varphi(w)|\leq r\}}|f_{n}'(\varphi(w))|+\| f_{n}\| _{\ F}\sup\limits_{\{|\varphi(w)|>r\}}\frac{\mu(|w|)|g(w)|}{(1-|\varphi(w)|^{2})^{\alpha}}\\ & \leq& (C+M)\varepsilon, \end{eqnarray*} 即当$n\rightarrow\infty$时,$\| C_{\varphi}^{g}f_{n}\| _{{B^\mu }}=\sup\limits_{w\in{\Bbb D}}\mu(|w|)|f'(\varphi(w))\| g(w)|\rightarrow0$,证明完成.

定理3.3 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$, $-2<q<\infty$,$q+s>-1$. 则$C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界当且仅当 $C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界并且

\begin{equation}\label{g1} \lim\limits_{|z|\rightarrow1}\mu(|z|)|g(z)|=0. \end{equation}

(3.4)

证设$C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界,则$C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界. 取$f(z) = z \in {F_0}{\text{(p,q,s)}}$,即得到(3.4)式.

反之,设$C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界并且(3.4)式成立. 则对任意多项式$P$有

\begin{equation}\label{g2} \mu(|z|)|(C_{\varphi}^{g}P)'(z)|=\mu(|z|)|P'(\varphi(z))\| g(z)|. \end{equation}

(3.5)

由(3.4)式,(3.5)式及定理2.3中的(2.12)式表明$C_{\varphi}^{g}P\in B_0^\mu $. 而多项式在${F_0}{\text{(p,q,s)}}$中稠密,所以对任意$f \in {F_0}{\text{(p,q,s)}}$存在多项式序列$\{P_{n}\}_{n\in\mathbb{N}}$ 使得当$n\rightarrow\infty$ 时,$\| f-P_{n}\| _{\ F}\rightarrow0$,因此 $$ \| C_{\varphi}^{g}f-C_{\varphi}^{g}P_n\| _{{B^\mu }}\leq\| C_{\varphi}^{g}\| _{{{\ F}_0}\rightarrow{B^\mu }}\| f-P_n\| _{\ F}\rightarrow0. $$ 又$B_0^\mu$是${B^\mu }$的闭子集,$C_{\varphi}^{g}({{\ F}_0})\subseteq B_0^\mu $, 所以有$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu } $有界.

下面我们给出在满足一定条件下小空间与对应空间上的广义复合算子的紧性是等价的.

定理3.4 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$, $-2<q<\infty$,$q+s>-1$. 则下面条件等价.

(1) $C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$是紧算子.

(2) $C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu } $是紧算子.

(3)

\begin{equation}\label{h1} \sup\limits_{|z|\rightarrow1}\frac{\mu(|z|)|g(z)|}{(1-|\varphi(z)|^{2})^{\alpha}}=0. \end{equation}

(3.6)

证 $(1)\Rightarrow(2)$ 显然.

$(2)\Rightarrow(3)$ 令$\alpha=\frac{q+2}{p}$. 设$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu } $是紧算子, 由引理2.2,对$M>0$有 $$ \lim\limits_{|z|\rightarrow1}\sup\limits_{\| f\| _{\ F\leq M,f\in{{\ F}_0}}} \mu(|z|)|(C_{\varphi}^{g}f)'(z)|=0. $$ 注意到在定理3.1的证明中给出的函数$f_{w}(z)=\frac{(z-\varphi(w))(1-|\varphi(w)|^{2})}{(1-\overline{\varphi(w)}z)^{\alpha+1}}$属于${{\ F}_0}$, 并且关于$w$范数一致有界,则 $$ \sup\limits_{|w|\rightarrow1}\frac{\mu(|w|)|g(w)|}{(1-|\varphi(w)|^{2})^{\alpha}}=0. $$ 即得(3.6)式成立.

$(3)\Rightarrow(1)$ 设$f\in\ F$满足$\| f\| _{\ F}\leq1$. 由引理3.1, $f\in{\cal B}^{\alpha}$,$\| f\| _{{\cal B}^{\alpha}}\leq C\| f\| _{\ F}$. \begin{eqnarray*} \mu(|z|)|(C_{\varphi}^{g}f)'(z)|&=&\mu(|z|)|f'(\varphi(z))\| g(z)|\\ &=&(1-|\varphi(z)|^2)^{\alpha}|f'(\varphi(z))|\frac{\mu(|z|)|g(z)|}{(1-|\varphi(z)|^2)^{\alpha}} \\ & \leq &C\| f\| _{\ F}\frac{\mu(|z|)|g(z)|}{(1-|\varphi(z)|^{2})^{\alpha}}. \end{eqnarray*} 由(3.6)式得 $ \lim\limits_{|z|\rightarrow1}\sup\limits_{\| f\| _{\ F\leq1}} \mu(|z|)|(C_{\varphi}^{g}f)'(z)|=0. $ 由引理2.2知$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$ 是紧算子.

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