设D是单位圆, $H(D)$表示D上的解析函数全体, $H^{\infty}$表示D上的有界解析函数类, 并赋以范数$||f||_{\infty}=\sup\{|f(z)|: \ z\in D\}$; dv为标准体测度, 满足$\displaystyle{\int_{D}dv(z)=1}$.

设$\alpha>0$, α-Bloch型空间$\beta_{\alpha}$和对数权Bloch型空间$\beta_{L}$分别定义如下:

$\begin{eqnarray*} &\;& \beta_{\alpha}=\{f:f\in H(D) \ 且 \ ||f||_{\alpha}=|f(0)|+\ \sup_{z\in D}\ (1-|z|^{2})^{\alpha}\ |f'(z)|<\infty \} ;\\ &\;& \beta_{L}=\{f:f\in H(D) \ 且 \ ||f||_{L}=|f(0)|+\ \sup_{z\in D}\ (1-|z|^{2})\left(\ln\frac{2}{1-|z|}\right)|f'(z)|<\infty \} \ . \end{eqnarray*}$

设$\gamma>-1$和$p>0$, D上加权Bergman空间定义如下:

$A_{\gamma }^{p}=\left\{ f:\ f\in H\left(D\ \right)\ \right.$且${{\left\| f \right\|}_{A_{\gamma }^{p}}}=\left. {{\left(\int_{D}{{{\left| f\left(z \right) \right|}^{p}}d{{v}_{\gamma }}\left(z \right)} \right)}^{\frac{1}{p}}} < \infty \right\}$,

这里$dv_{\gamma}(z)=c_{\gamma}(1-|z|^{2})^{\gamma}\ dv(z)$, 常数$c_{\gamma}$满足$\displaystyle{\int_{D}dv_{\gamma}(z)=1}$.当$\gamma=0$时$A_{\gamma}^{p}$就是Bergman空间$A^{p}$.

设$u\in H(D)$, $\varphi$为D上的解析自映射, X和Y为两个解析函数空间, 则X到Y的加权复合算子$uC_{\varphi}$定义如下:

$(uC_{\varphi}f)(z)=u(z)f[\varphi(z)] \ \ \ (f\in X)\ .$

当$u(z)=1$时就是复合算子$C_{\varphi}$; 当$\varphi(z)=z$时就是乘子算子$M_{u}$.

在上世纪九十年代, Madigan和Matheson在文献[1]和^[2]中研究了D上Lipschitz空间、Bloch空间和小Bloch空间上复合算子$C_{\varphi}$的有界性和紧性问题, 他们证明了$C_{\varphi}$在Bloch空间上总是有界的和$C_{\varphi}$在小Bloch空间上有界当且仅当$\varphi$在小Bloch空间中等结论; 在2000年, 史济怀先生和罗罗博士在文献[3]中将Bloch空间的结论推广到了$C^{n}$中的齐性域上; 接下来在2001年, Ohno和赵如汉在文献[4]中就Bloch空间和小Bloch空间讨论了加权复合算子的有界性和紧性, 给出了比较完整的结果; 在2003年, 张学军在文献[5]中讨论了p-Bloch空间和q-Bloch空间之间加权复合算子为有界算子和紧算子的条件, 获得了较好的结果但尚不完整; 在2007年, 叶善力在文献[6]中探讨了单位圆中对数权Bloch型空间$\beta_{L}$与$\alpha$-Bloch型空间$\beta_{\alpha}$之间加权复合算子的问题, 他给出了如下结果.

定理A  设$\alpha>0$, u在单位圆D上解析, $\varphi$是D上的解析自映射, 则$uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的有界算子之充要条件为

$ \begin{array}{*{20}{c}} {{\rm{sup}}}\\ {z \in D} \end{array}{\mkern 1mu} {(1 - |z{|^2})^\alpha }\left( {\ln \ln \frac{2}{{1 - |\varphi (z){|^2}}}} \right)|u'(z)| < \infty $

且

$\begin{array}{*{20}{c}}{{\rm{sup}}}\\{z \in D}\end{array}{(1 - |z{|^2})^\alpha }{\left\{ {(1 - |\varphi (z){|^2})\ln \frac{2}{{1 - |\varphi (z)|}}} \right\}^{ - 1}}|u(z)\varphi '(z)| < \infty .$

定理B  设$\alpha>0$, u在单位圆D上解析, $\varphi$是D上的解析自映射, $uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的有界算子, 则$uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的紧算子之充要条件为

$\mathop {\lim }\limits_{\left| {\varphi \left( z \right)} \right| \to 1} {(1 - |z{|^2})^\alpha }\left( {\ln \ln \frac{2}{{1 - |\varphi (z){|^2}}}} \right)|u'(z)| = 0$

且

$\mathop {\lim }\limits_{\left| {\varphi \left( z \right)} \right| \to 1} {(1 - |z{|^2})^\alpha }{\left\{ {(1 - |\varphi (z){|^2})\ln \frac{2}{{1 - |\varphi (z)|}}} \right\}^{ - 1}}|u(z)\varphi '(z)| = 0.$

实际上, 当$\alpha < 1$时上述两定理讨论的是大空间到小空间的问题, 若$\varphi$为D上的自同构, 则u必须恒为0, 真正意义较大的是$\alpha\geq1$时, 在讨论中发现, 当$\alpha>1$时定理A和定理B中的这两个条件不是独立的.另外, 紧算子的确先是有界算子, 但有界性也是要通过u和$\varphi$满足一定条件来刻画的, 所以定理B中可以不必先给一个有界性的先决条件, 而是通过u和$\varphi$直接进行刻画, 如果$||\varphi||_{\infty} < 1$的话, 定理B中后两个条件是不存在的, 因而紧性条件可以视$||\varphi||_{\infty} < 1$和$||\varphi||_{\infty}=1$而定.本文的主要工作就是给出了$\alpha>1$时, 定理A中较简捷的充要条件和定理B中不同的充要条件.

本文中c、$c_{1}$、$c_{2}$、$c_{3}$等表示与变量z、w等无关的常数, 为方便起见, 不同的位置可以表示不同的数.

引理2.1  设$\alpha>1$, u在D上解析, $\varphi$是D上的解析自映射.

(1) 若$ \displaystyle{\begin{array}{*{20}{c}}{{\rm{sup}}}\\{z \in D}\end{array}(1-|z|^{2})^{\alpha-1}\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right)|u(z)|=M < \infty\, }$则有

$\begin{array}{*{20}{c}}{{\rm{sup}}}\\{z \in D}\end{array}(1-|z|^{2})^{\alpha}\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right)|u'(z)|\leq cM .$

(2) 若$ \displaystyle{\lim_{|z|\rightarrow 1-0}(1-|z|^{2})^{\alpha-1}\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right)|u(z)|=0, }$则有

$\mathop {\lim }\limits_{|z| \to 1 - 0} {(1 - |z{|^2})^\alpha }\left( {\ln \ln \frac{4}{{1 - |\varphi (z){|^2}}}} \right)|u'(z)| = 0.$

证 (1) 因为当$0\leq x < 1$时,

$\displaystyle{h(x)=\ln\ln\frac{4}{1-x}\left(\ln\ln\frac{4}{1-x^{2}}\right)^{-1}\geq 1} $

且连续, 又

$\mathop {\lim }\limits_{x \to 1 - 0} \ln \ln \frac{4}{{1 - x}}{\left( {\ln \ln \frac{4}{{1 - {x^2}}}} \right)^{ - 1}} = 1{\text{ }}.$

这样就有

$\displaystyle{1=\inf_{0\leq x < 1}h(x) < M_{0}=\sup_{0\leq x < 1}h(x) < \infty}$

(2.1)

此外, 对任何复数$|\xi|\geq 1$, 就对数主支($\ln1=0$)有

$\ln|\xi|\leq|\ln\xi|=\{(\ln|\xi|)^{2}+(\arg\xi)^{2}\}^{\frac{1}{2}} \leq \ln|\xi|+\pi, $

(2.2)

以及对任意$z\in D$和$k\geq 2$有

$\left| {\ln \frac{k}{{1 - z}}} \right| = \left| {\ln k + \sum\limits_{n = 1}^\infty {\frac{{{z^n}}}{n}} } \right| \leqslant \ln k + \sum\limits_{n = 1}^\infty {\frac{{|z{|^n}}}{n}} = \ln \frac{k}{{1 - |z|}}.$

(2.3)

当$\displaystyle\begin{array}{*{20}{c}}{{\rm{sup}}}\\{z \in D}\end{array}(1-|z|^{2})^{\alpha-1}\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right)|u(z)|=M < \infty$时, 对任意$w\in D$, 令

$\displaystyle{F_{w}(z)=\left(\ln\ln\frac{4}{1-\overline{\varphi(w)}\ \varphi(z)}\right)u(z)}, $

由上式和(2.1)-(2.3) 式知

$\left| {{F_w}\left( z \right)} \right| \le \left( {\ln \ln \frac{4}{{1 - \left| {\varphi \left( z \right)} \right|}} + \pi } \right)\left| {u\left( z \right)} \right| \le \left( {1 + \frac{\pi }{{\ln \ln 4}}} \right)\frac{{{M_0}M}}{{{{(1 - {{\left| z \right|}^2})}^{\alpha - 1}}}}\;.$

这样对一切$\gamma>\alpha-2$有$F_{w}\in A_{\gamma}^{1}$, 由文献[7]中的定理2.2 ($n=1$的情形)知

$F_{w}(z)=\int_{D}\frac{F_{w}(\xi)}{(1-\overline{\xi}z)^{2+\gamma}}\ dv_{\gamma}(\xi) \ \ \ (z\in D)\ .$

根据文献[8]中命题1.4.10 ($n=1$的情形)可得

$\ \ \ \ |F_{w}'(z)|=\left|\int_{D}\frac{(2+\gamma)\ \overline{\xi}\ F_{w}(\xi)}{(1-\overline{\xi}z)^{3+\gamma}}\ dv_{\gamma}(\xi) \right|\\ \leq \int_{D}\frac{c_{1}M(1-|\xi|^{2})^{\gamma-\alpha+1}}{|1-\overline{\xi}z|^{3+\gamma}}\ dv(\xi)\leq \frac{c_{2}M}{(1-|z|^{2})^{\alpha}}\ .$

(2.4)

在(2.4) 式中取$z=w$经过计算并结合Pick引理可得

$\begin{eqnarray*} &\;& (1-|w|^{2})^{\alpha}\left(\ln\ln\frac{4}{1-|\varphi(w)|^{2}}\right)|u'(w)|\\ &\leq& c_{2}M+\frac{(1-|w|^{2})\ |\varphi'(w)|}{1-|\varphi(w)|^{2}}\ (1-|w|^{2})^{\alpha-1}\left(\ln\frac{4}{1-|\varphi(w)|^{2}}\right)^{-1}|u(w)||\varphi(w)|\\ &\leq& c_{2}M+\left(\ln\frac{4}{1-|\varphi(w)|^{2}}\right)^{-1}\left(\ln\ln\frac{4}{1-|\varphi(w)|^{2}}\right)^{-1}M\leq cM\ . \end{eqnarray*}$

由w的任意性可知

$\mathop {\sup }\limits_{w \in D} {(1 - |w{|^2})^\alpha }\left( {\ln \ln \frac{4}{{1 - |\varphi (w){|^2}}}} \right)|u'(w)| \leqslant cM\;.$

(2.5)

(2) 由$ \displaystyle{\lim_{|z|\rightarrow 1-0}(1-|z|^{2})^{\alpha-1}\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right)|u(z)|=0} $知, 对任意$\varepsilon>0$, 存在$0 < r_{0} < 1$, 当$r_{0} < |z| < 1$时, 有

$(1-|z|^{2})^{\alpha-1}\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right)|u(z)| < \varepsilon\ .$

(2.6)

当$|\xi|\leq r_{0}$时, 由(2.2)-(2.3) 式应用到(1) 中的$F_{w}$可得

$|{F_w}(\xi )| \le \left( {\ln \ln \frac{4}{{1 - {t_0}}} + \pi } \right)\mathop {\max }\limits_{\left| \xi \right| \leqslant {r_{\text{o}}}}|u(\xi )| = {M_1},{\rm{ 这里}}{t_0} = {\max _{|\xi | \le {r_0}}}|\varphi (\xi )| < 1\;.$

(2.7)

根据(2.6)-(2.7) 式和文献[8]中命题1.4.10($n=1$的情形)有

$|F_{w}'(z)|\leq \int_{|\xi|\leq r_{0}}\frac{(2+\gamma)|F_{w}(\xi)|}{|1-z\overline{\xi}|^{3+\gamma}}dv_{\gamma}(\xi) +\int_{r_{0} < |\xi| < 1}\frac{c_{1}\varepsilon(1-|\xi|^{2})^{\gamma-\alpha+1}}{|1-\overline{\xi}z|^{3+\gamma}}\ dv(\xi)$

$\leq \frac{(2+\gamma)M_{1}}{(1-r_{0})^{3+\gamma}}+\int_{D}\frac{c_{1}\varepsilon(1-|\xi|^{2})^{\gamma-\alpha+1}}{|1-\overline{\xi}z|^{3+\gamma}}\ dv(\xi)\leq \frac{(2+\gamma)M_{1}}{(1-r_{0})^{3+\gamma}}+\frac{c_{2}\varepsilon}{(1-|z|^{2})^{\alpha}}\ .$

当$r_{0} < |w| < 1$时, 在上式中取$z=w$经整理就有

$(1-|w|^{2})^{\alpha}\left(\ln\ln\frac{4}{1-|\varphi(w)|^{2}}\right)|u'(w)|\leq c\varepsilon+\frac{(2+\gamma)M_{1}}{(1-r_{0})^{3+\gamma}}(1-|w|^{2})^{\alpha}.$

从而

$\mathop {\overline {\lim } }\limits_{|w| \to 1} \;{(1 - |w{|^2})^\alpha }\left( {\ln \ln \frac{4}{{1 - |\varphi (w){|^2}}}} \right)|u'(w)| \leqslant c\varepsilon \;.$

由$\varepsilon$的任意性可知

$\mathop {\lim }\limits_{|w| \to 1 - 0} {(1 - |w{|^2})^\alpha }\left( {\ln \ln \frac{4}{{1 - |\varphi (w){|^2}}}} \right)|u'(w)| = 0.$

(2.8)

定理2.2  设$\alpha>1$, u在单位圆D上解析, $\varphi$是D上的解析自映射, 则$uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的有界算子之充要条件为

$\;\mathop {\sup }\limits_{z \in D} {(1 - |z{|^2})^{\alpha - 1}}\left( {\ln \ln \frac{4}{{1 - |\varphi (z){|^2}}}} \right)|u(z)|{\text{ < }}\infty \;.$

(2.9)

证  若(2.9) 式成立, 设

$\mathop {\sup }\limits_{z \in D} {(1 - |z{|^2})^{\alpha - 1}}\left( {\ln \ln \frac{4}{{1 - |\varphi (z){|^2}}}} \right)|u(z)| = M\;.$

对任意$f\in \beta_{L}$, 由文献[6]中的引理2.1、Pick引理以及(2.9) 式和引理2.1中的(2.5) 式可得

$\begin{eqnarray*} &\;& (1-|z|^{2})^{\alpha}|(uC_{\varphi }f)'(z)|\leq (1-|z|^{2})^{\alpha}|f'[\varphi(z)]\varphi'(z)u(z)+f[\varphi(z)]u'(z)|\\ &\leq& \frac{(1-|z|^{2})^{\alpha}}{1-|\varphi(z)|^{2}}\left(\ln\frac{2}{1-|\varphi(z)|}\right)^{-1}|\varphi'(z)u(z)|.||f||_{L}\\ &+&(1-|z|^{2})^{\alpha}\left(2+\ln\ln\frac{2}{1-|\varphi(z)|}\right)|u'(z)|.||f||_{L}\\ &\leq& \left(\ln\frac{2}{1-|\varphi(z)|}\right)^{-1}\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right)^{-1}M||f||_{L}\\ &+&c\left(2+\ln\ln\frac{2}{1-|\varphi(z)|}\right)\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right)^{-1}M||f||_{L}\\ &\leq &\frac{M}{\ln2\ln\ln4} ||f||_{L}+c\left(\frac{2}{\ln\ln4}+1\right)M||f||_{L}\leq c_{1}||f||_{L}\ . \end{eqnarray*}$

这样可得

$\begin{eqnarray*} ||uC_{\varphi}f||_{\alpha}&\leq& |u(0)f[\varphi(0)]|+\frac{M}{\ln2\ln\ln4} ||f||_{L}+c\left(\frac{2}{\ln\ln4}+1\right)M||f||_{L}\\ &\leq& |u(0)|\left(2+\ln\ln\frac{2}{1-|\varphi(0)|}\right)||f||_{L}+c_{1}||f||_{L}\leq c_{2}||f||_{L}\ . \end{eqnarray*}$

因此$uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的有界算子.反之, 若$uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的有界算子, 则$||uC_{\varphi}f||_{\alpha}\leq ||uC_{\varphi}||.||f||_{L}$对$f\in \beta_{L}$成立.

对任意$w\in D$, 取

$\displaystyle{f_{w}(z)=\ln\ln\frac{4}{1-\overline{\varphi(w)}\ z}}.$

记$t=|\varphi(w)|$和$\theta=\arg\overline{\varphi(w)}$及$z_{1}=e^{i\theta}z$, 由(2.2) 式及文献[6]中的引理2.3-2.4得

$\begin{eqnarray*} &\;& (1-|z|^{2})\ln\frac{2}{1-|z|}|f_{w}'(z)|=\frac{(1-|z|^{2})\ |\overline{\varphi(w)}|}{|1-\overline{\varphi(w)}z|}\ln\frac{2}{1-|z|} \left|\ln\frac{4}{1-\overline{\varphi(w)}z}\right|^{-1}\\ &\leq& \frac{2(1-|z|)}{|1-\overline{\varphi(w)}z|}\ln\frac{2}{1-|z|} \left(\ln\frac{4}{|1-\overline{\varphi(w)}z|}\right)^{-1}\\ &=& 2(1-|z_{1}|)\ln\frac{2}{1-|z_{1}|}\left\{(1-|tz_{1}|)\ln\frac{2}{1-|tz_{1}|}\right\}^{-1}\\ &\times & \left\{(1-|tz_{1}|)\ln\frac{2}{1-|tz_{1}|}\right\}\left\{ |1-tz_{1}|\ln\frac{4}{|1-tz_{1}|}\right\}^{-1}\leq 8. \end{eqnarray*}$

这样$||f_{w}||_{L}\leq 8+\ln\ln4.$因此$||uC_{\varphi}f_{w}||_{\alpha}\leq(8+\ln\ln4)||uC_{\varphi}||$.再由文献[5]中引理2.3可得, 对一切$z\in D$有

$|[uC_{\varphi}f_{w}](z)|\leq \frac{c||uC_{\varphi}||}{(1-|z|^{2})^{\alpha-1}}.$

(2.10)

在(2.10) 式中取$z=w$就有

$(1-|w|^{2})^{\alpha-1}\left(\ln\ln\frac{4}{1-|\varphi(w)|^{2}}\right)|u(w)|\leq c||uC_{\varphi}||\ .$

根据w的任意性可知(2.9) 式成立.

定理2.3  设$\alpha>1$, u在单位圆D上解析, $\varphi$是D上的解析自映射, 则$uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的紧算子之充要条件为:当$||\varphi||_{\infty} < 1$时$u\in \beta_{\alpha}$; 当$||\varphi||_{\infty}= 1$时$u\in \beta_{\alpha}$且

$\mathop {\lim }\limits_{\left| {\varphi \left( z \right)} \right| \to 1} {(1 - |z{|^2})^{\alpha - 1}}\left( {\ln \ln \frac{4}{{1 - |\varphi (z){|^2}}}} \right)|u(z)| = 0$

(2.11)

和

$\mathop {\lim }\limits_{\left| {\varphi \left( z \right)} \right| \to 1} {(1 - |z{|^2})^\alpha }\left( {\ln \ln \frac{4}{{1 - |\varphi (z){|^2}}}} \right)|u'(z)| = 0$

(2.12)

同时成立.

证  若$uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的紧算子, 取$f=1\in \beta_{L}$, 立即可得$u\in \beta_{\alpha}$.

当$||\varphi||_{\infty}=1$时, 设$\{z_{n}\}\subset D$是任意一个使得$\displaystyle{|\varphi(z_{n})|\rightarrow 1} \ \ (n\rightarrow\infty)$的序列, 令

$f_{n}(z)=\left(\ln\ln\frac{4}{1-|\varphi(z_{n})|^{2}}\right)^{-1}\left(\ln\ln\frac{4}{1-\overline{\varphi(z_{n})}\ z}\right)^{2}\, $

则$\{f_{n}\}$在D的任一紧子集上一致收敛于0, 且利用(2.1)-(2.3) 式以及文献[6]中引理2.3-2.4经计算可得$||f_{n}||_{L}\leq 16M_{0}+\ln\ln 4$, 这样

$\displaystyle{\lim_{n\rightarrow\infty} ||uC_{\varphi}f_{n}||_{\alpha}=0}\ .$

由文献[5]中引理2.3可得, 对一切$z\in D$有

$\begin{eqnarray*}&&\displaystyle{|[uC_{\varphi}f_{n}](z)|\leq \frac{c||uC_{\varphi}f_{n}||_{\alpha}}{(1-|z|^{2})^{\alpha-1}}} \ \ \Rightarrow \ \ \mbox{当$n\rightarrow\infty$ 时}, \\ &&(1-|z_{n}|^{2})^{\alpha-1}\left(\ln\ln\frac{4}{1-|\varphi(z_{n})|^{2}}\right)|u(z_{n})|\leq c||uC_{\varphi}f_{n}||_{\alpha}\rightarrow 0. \end{eqnarray*}$

这表明(2.11) 式成立.

另一方面, 经计算且令$z=z_{n}$并利用Pick引理及(2.11) 式可得, 当$n\rightarrow\infty$时,

$\begin{eqnarray*} &&(1-|z_{n}|^{2})^{\alpha}\left(\ln\ln\frac{4}{1-|\varphi(z_{n})|^{2}}\right)|u'(z_{n})|\\ &\leq& ||uC_{\varphi}f_{n}||_{\alpha} +\frac{2}{\ln4 \ln\ln4}(1-|z_{n}|^{2})^{\alpha-1}\left(\ln\ln\frac{4}{1-|\varphi(z_{n})|^{2}}\right)|u(z_{n})|\rightarrow 0. \end{eqnarray*}$

这表明(2.12) 式成立.

反过来, 若$u\in \beta_{\alpha}$, 根据文献[5]中的引理2.3知$(1-|z|^{2})^{\alpha-1}|u(z)|\leq c||u||_{\alpha}$, 又由于$\varphi\in H^{\infty}\subset \beta_{1}$, 故$(1-|z|^{2})|\varphi'(z)|\leq ||\varphi||_{1}$对一切$z\in D$成立.

设$\{f_{n}\}$是$\beta_{L}$中任一有界序列且在D的任一紧子集上一致收敛于0, 则由Cauchy积分公式立即可得$\{f_{n}'\}$也在D的任一紧子集上一致收敛于0.当$||\varphi||_{\infty} < 1$时, 若$n\rightarrow\infty$, 则

$\begin{eqnarray*} && ||uC_{\varphi}f_{n}||_{\alpha}=|u(0)f_{n}[\varphi(0)]|+\begin{array}{*{20}{c}}{{\rm{sup}}}\\{z \in D}\end{array}(1-|z|^{2})^{\alpha}|u'(z)f_{n}[\varphi(z)]+u(z)f_{n}'[\varphi(z)]\varphi'(z)|\\ & \leq& |u(0)f_{n}[\varphi(0)]|+||u||_{\alpha}\sup_{|w|\leq ||\varphi||_{\infty}}|f_{n}(w)|+c||\varphi||_{1}||u||_{\alpha}\sup_{|w|\leq ||\varphi||_{\infty}}|f_{n}'(w)|\rightarrow 0\ . \end{eqnarray*}$

这意味着$uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的紧算子.

当$||\varphi||_{\infty}=1$时, 若还有(2.11)-(2.12) 式成立, 故对任意$\varepsilon>0$, 存在$0 < \delta < 1$, 当$\delta < |\varphi(z)| < 1$时,

$\begin{eqnarray} &&(1-|z|^{2})^{\alpha-1}\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right) |u(z)| < \varepsilon, \end{eqnarray}$

(2.13)

$\begin{eqnarray} &&(1-|z|^{2})^{\alpha}\left(\ln\ln\frac{4}{1-|\varphi(z)|^{2}}\right) |u'(z)| < \varepsilon \end{eqnarray}$

(2.14)

同时成立.

记$\displaystyle{ K=\sup||f_{n}||_{L} }\, $则由文献[6]中的引理2.1、Pick引理以及(2.13)-(2.14) 式可得

$\begin{eqnarray*} &\;& ||uC_{\varphi}f_{n}||_{\alpha}=|u(0)f_{n}[\varphi(0)]|+\sup_{z\in D}(1-|z|^{2})^{\alpha}|u'(z)f_{n}[\varphi(z)]+u(z)f_{n}'[\varphi(z)]\varphi'(z)|\\ &\leq& |u(0)f_{n}[\varphi(0)]|+||u||_{\alpha}\sup_{|w|\leq\delta}|f_{n}(w)|+c_{1}||\varphi||_{1}||u||_{\alpha}\sup_{|w|\leq\delta}|f_{n}'(w)|\\ &&+\sup_{\delta < |\varphi(z)| < 1}(1-|z|^{2})^{\alpha}\left(2+\ln\ln\frac{2}{1-|\varphi(z)|}\right)|u'(z)|.||f_{n}||_{L}\\ &&+\sup_{\delta < |\varphi(z)| < 1}(1-|z|^{2})^{\alpha-1} |u(z)|\left(\ln\frac{2}{1-|\varphi(z)|}\right)^{-1}\frac{(1-|z|^{2})|\varphi'(z)|}{1-|\varphi(z)|^{2}} ||f_{n}||_{L}\\ &\leq& |u(0)f_{n}[\varphi(0)]|+||u||_{\alpha}\sup_{|w|\leq\delta}|f_{n}(w)|+c_{1}||\varphi||_{1}||u||_{\alpha}\sup_{|w|\leq\delta}|f_{n}'(w)|+c_{2}K\varepsilon, \end{eqnarray*}$

这样就有

$\displaystyle{\overline{\lim_{n\rightarrow\infty}}\ ||uC_{\varphi}f_{n}||_{\alpha}\leq c_{2}K\varepsilon}.$

由$\varepsilon$的任意性知

$\displaystyle{\lim_{n\rightarrow\infty}||uC_{\varphi}f_{n}||_{\alpha}=0}.$

这意味着$uC_{\varphi}$是$\beta_{L}$到$\beta_{\alpha}$的紧算子.