设${\mathbb D}=\{z\in {\mathbb C}: |z|<1\}$是复平面上的单位圆盘, $H({\Bbb D})$表示${\Bbb D}$上的所有解析函数的集合, $\varphi$是${\Bbb D}$到${\Bbb D}$的一个解析自映射.

如果存在正数$a$, $b$ ($0< a< b$)和$t_0\in[0, 1)$满足

$\begin{eqnarray*}&&\frac{\phi(r)}{(1-r)^{a}}\ \hbox{在$ [t_0, 1)$上单调递减且}\ \lim\limits_{r\rightarrow1}\frac{\phi(r)}{(1-r)^{a}}=0, \\&&\frac{\phi(r)}{(1-r)^b}\ \hbox{在$ [t_0, 1)$上单调递增且}\ \lim\limits_{r\rightarrow1}\frac{\phi(r)}{(1-r)^b}=\infty, \end{eqnarray*}$

则称$[0, 1)$上的正连续函数$\phi$是一个正规函数.

对于$p, \, q\in (0, \infty)$及正规函数$\phi$,令

$\|f\|_{p, \, q, \, \phi}=\left(\int_{0}^{1}M^{p}_{q}(f, r)\frac{\phi^{p}(r)}{1-r}r{\rm d}r\right)^{1/p}, $

其中

$M_q(f, r)=\left(\frac{1}{2\pi}\int_0^{2\pi}|f(r{\rm e}^{{\rm i}\theta})|^q{\rm d}\theta\right)^{1/q}, \quad 0\leq r<1.$

令$L(p, \, q, \, \phi)$表示${\Bbb D}$上使得$\|f\|_{p, \, q, \, \phi}<\infty$的所有可测函数的集合称为混合模空间. $H(p, \, q, \, \phi)$$=L(p, \, q, \, \phi)\cap H({\Bbb D})$.当$1\leq p<\infty$时, $H(p, \, q, \, \phi)$是以$\|\cdot\|_{p, \, q, \, \phi}$为范数的Banach空间;当$0<p<1$时, $H(p, \, q, \, \phi)$是以$\|\cdot\|_{p, \, q, \, \phi}$为半范数的Fréchet空间,但不是Banach空间.

设$\mu$是一个权函数,即${\Bbb D}$上的正连续函数.令

$\|f\|_{{\cal Z}_{\mu}}=|f(0)|+|f'(0)|+\sup\limits_{z\in{\Bbb D}}\mu(z)|f''(z)|. $

若$f\in H({\Bbb D})$且$\|f\|_{{\cal Z}_{\mu}}<\infty$,则称$f$属于Zygmund-型空间${{\mathcal Z}_\mu}$. $({{\mathcal Z}_\mu}, \|\cdot\|_{{\cal Z}_{\mu}})$也是一个Banach空间.如果$\mu(z)=1-|z|^2$, Zygmund-型空间${\cal Z}_{\mu}$就是经典的Zygmund空间${\mathcal Z}$.

近年来,人们在研究混合模空间与Zygmund-型空间之间某些算子的性质,参见论文[1-13]及它们中的参考文献.对于单位球上全纯函数空间上某些算子的研究参见文献[14-20].

最近, Stević, Sharma及Krishan在文献[21]中引入算子

$T^n_{\psi_{1}, \psi_{2}, \varphi}f(z)=\psi_1(z)f^{(n)}(\varphi(z))+\psi_2(z)f^{(n+1)}(\varphi(z)), f\in H({\Bbb D}), $

其中$\psi_1, \psi_2\in H({\Bbb D}), $$n$是非负整数, $\varphi$是${\Bbb D}$到${\Bbb D}$的一个解析自映射.他们研究了从一般空间到Bloch-型空间$T^n_{\psi_{1}, \psi_{2}, \varphi}$的有界性与紧性.对$\psi_2\equiv0$,可以得到加权微分复合算子,它的研究见论文[22-28].当$n=0$时, $T^n_{\psi_{1}, \psi_{2}, \varphi}=T_{\psi_{1}, \psi_{2}, \varphi}$,后者是由Stević等在文献[29]中引进的.

受论文[16, 21, 24, 26, 30-31]的启发,本文研究算子$T^n_{\psi_{1}, \psi_{2}, \varphi}: H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$的有界性与紧性特征.

先给出一些预备知识.

引理2.1^[32]  设$p, q \in (0, \infty)$, $\phi$是正规的且$f\in H(p, q, \phi)$.则对于$n\in{\mathbb N}_0$,存在一个与$f$无关的常数$C$使得

$|f^{(n)}(z)|\leq C\frac{\|f\|_{p, q, \phi}}{\phi(|z|)(1-|z|^2)^{1/q+n}}, z\in {\Bbb D}.$

下面的紧性准则是有用的工具,它的证明可以参见文献[33]中的命题3.11.

引理2.2  设$\psi_1, \psi_2\in H({\Bbb D})$, $n\in{\mathbb N}_0$, $\varphi$是${\Bbb D}$到${\Bbb D}$的一个解析自映射, $\mu$是一个权函数.则$T^n_{\psi_{1}, \psi_{2}, \varphi}:H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是紧的充要条件$T^n_{\psi_{1}, \psi_{2}, \varphi}: H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是有界的且对于在${\Bbb D}$的任意紧子集上一致收敛于零($k\rightarrow \infty$)的有界序列$\{f_k\}\subset H(p, q, \phi)$,有$\|T^n_{\psi_{1}, \psi_{2}, \varphi}f_k\|_{{\cal Z}_{_{\mu}}}\rightarrow 0\ (k\rightarrow \infty)$.

引理2.3^[34]  对任意的实数$\beta$,设

$J_\beta(z)=\int_0^{2\pi}\frac{{\rm d}\theta}{|1-z{\rm e}^{-{\rm i}\theta}|^{1+\beta}}, \, z\in {\Bbb D}, $

则

$J_\beta(z)\asymp \left\{\begin{array}{lll}1,& \beta<0, \\[2mm] \log\frac{1}{1-|z|^2}, ~~& \beta=0, \hspace{5mm} \ \ (\, |z|\rightarrow 1^-).\\[3mm] \frac{1}{(1-|z|^2)^\beta},& \beta>0, \end{array} \right.$

引理2.4^[35]  对于$\beta>-1$及$\gamma> 1+\beta$,则

$\int_0^1\frac{(1-r)^\beta}{(1-r\rho)^\gamma}{\rm d}r\leq C(1-\rho)^{1+\beta-\gamma}, \ 0<\rho<1.$

本节将给出我们的主要结果与证明.

定理3.1  设$\psi_1, \psi_2\in H({\Bbb D})$, $n\in{\mathbb N}_0$, $\varphi$是${\Bbb D}$到${\Bbb D}$的一个解析自映射, $\mu$是一个权函数.则$T^n_{\psi_{1}, \psi_{2}, \varphi}:H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是有界的充要条件

(3.1) $\begin{equation} \label{qwe1}\sup\limits_{z\in{\Bbb D}}\frac{\mu(z)|\psi_1''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n}}<\infty, \end{equation}$

(3.2) $\begin{equation}\label{qwe2}\sup\limits_{z\in{\Bbb D}}\frac{\mu(z)|\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+1}}<\infty, \end{equation}$

(3.3) $\begin{equation}\label{qwe3}\sup\limits_{z\in{\Bbb D}}\frac{\mu(z)|\psi_1(z)(\varphi'(z))^2+2\psi_2'(z)\varphi'(z)+\psi_2(z)\varphi''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+2}}<\infty, \end{equation}$

及

(3.4) $\begin{equation}\label{qwe3b}\sup\limits_{z\in{\Bbb D}}\frac{\mu(z)|\psi_2(z)||\varphi'(z)|^2}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+3}}<\infty.\end{equation}$

证  充分性.假设(3.1)-(3.4)式成立.由于

$\begin{eqnarray*}\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f\right)'(z)&=&\psi_1'(z)f^{(n)}(\varphi(z)) \\&& +\left(\psi_1(z)\varphi'(z)+\psi_2'(z)\right)f^{(n+1)}(\varphi(z)) +\psi_2(z)\varphi'(z)f^{(n+2)}(\varphi(z)), \end{eqnarray*}$

因此对$z\in{\Bbb D}$及$f\in H(p, q, \phi)$,根据引理2.1,我们有

(3.5) $\begin{eqnarray}\label{qwe4} &&\mu(z)\left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f\right)^{''}(z)\right|\nonumber\\&\leq&\mu(z)|\psi_1''(z)||f^{(n)}(\varphi(z))|+\mu(z)\left|\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z)\right|\left|f^{(n+1)}(\varphi(z))\right|\nonumber\\&& +\mu(z)|\psi_1(z)(\varphi'(z))^2+2\psi_2'(z)\varphi'(z)+\psi_2(z)\varphi''(z)|\left|f^{(n+2)}(\varphi(z))\right|\nonumber\\&& +\mu(z)\left|\psi_2(z)(\varphi'(z))^2\right|\left|f^{(n+3)}(\varphi(z))\right|\nonumber\\&\leq& C\|f\|_{p, q, \phi}\frac{\mu(z)|\psi_1''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n}}+C\|f\|_{{\cal B}_{\log}}\frac{\mu(z)|\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+1}}\nonumber\\&& +C\|f\|_{p, q, \phi}\frac{\mu(z)\left|\psi_1(z)(\varphi'(z))^2+2\psi_2'(z)\varphi'(z)+\psi_2(z)\varphi''(z)\right|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+2}}\nonumber\\&& +C\|f\|_{p, q, \phi}\frac{\mu(z)|\psi_2(z)||\varphi'(z)|^2}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+3}}\nonumber\\&\leq& C\|f\|_{p, q, \phi}.\end{eqnarray}$

另一方面

(3.6) $\begin{eqnarray}\label{qwe5}\left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f\right)(0)\right|&=&\left|\psi_1(0)f^{(n)}(\varphi(0))+\psi_2(0)f^{(n+1)}(\varphi(0))\right|\nonumber\\&\leq&C\frac{|\psi_1(0)|+|\psi_2(0)|}{\phi(|\varphi(0)|)\left(1-\left|\varphi(0)\right|^{2}\right)^{1/q+n+1}}\|f\|_{p, q, \phi}, \end{eqnarray}$

及

(3.7) $\begin{eqnarray}\label{qwe6}&&\left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f\right)'(0)\right|\nonumber\\&=&\left|\psi_1'(0)f^{(n)}(\varphi(0))+(\psi_1(0)\varphi'(0)+\psi_2'(0))f^{(n+1)}(\varphi(0))+\psi_2(0)\varphi'(0)f^{(n+2)}(\varphi(0))\right|\nonumber\\&\leq&C\frac{|\psi_1'(0)|+|\psi_1(0)\varphi'(0)+\psi_2'(0)|+|\psi_2(0)\varphi'(0)|}{\phi(|\varphi(0)|)\left(1-\left|\varphi(0)\right|^{2}\right)^{1/q+n+2}}\|f\|_{p, q, \phi}.\end{eqnarray}$

利用(3.5), (3.6)及(3.7)式,我们得到算子$T^n_{\psi_{1}, \psi_{2}, \varphi}:H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是有界的.

必要性.设$T^n_{\psi_{1}, \psi_{2}, \varphi}: H(p, q, \phi)\rightarrow\mathcal{Z}_{\mu} $是有界的,则存在常数$C$使得

(3.8) $\begin{equation}\label{qweyou}\|T^n_{\psi_{1}, \psi_{2}, \varphi}f\|_{{\cal Z}_{\mu}}\leq C\|f\|_{p, q, \phi}, \end{equation}$

对任意$f\in H(p, q, \phi)$.在(3.8)式中取$f(z)=\frac{z^{n}}{n!} \in H(p, q, \phi)$,我们有

(3.9) $\begin{equation}\label{qwe7} K_{1}:=\sup\limits_{z\in{\Bbb D}}\mu(z)|\psi_1''(z)|<\infty.\end{equation}$

在(3.8)式中取$f(z)=\frac{z^{n+1}}{(n+1)!} \in H(p, q, \phi)$,可得

(3.10) $\begin{equation}\label{qwe8} \sup\limits_{z\in{\Bbb D}}\mu(z)|\psi_1''(z)\varphi(z)+\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z)|<\infty.\end{equation}$

由于(3.9)式, (3.10)式以及函数$\varphi(z)$是有界的,所以

(3.11) $\begin{equation}\label{qwe9} K_{2}:=\sup\limits_{z\in{\Bbb D}}\mu(z)|\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z)|<\infty.\end{equation}$

再在(3.8)式中取$f(z)=\frac{z^{n+2}}{(n+2)!} \in H(p, q, \phi)$,又可得到

(3.12) $\begin{eqnarray}\label{qwe10} &&\sup\limits_{z\in{\Bbb D}}\mu(z)\Big|\frac{1}{2}\psi_1''(z)(\varphi(z))^2+(\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z))\varphi(z)\nonumber\\&& +(\psi_1(z)(\varphi'(z))^2+2\psi_2'(z)\varphi'(z)+\psi_2(z)\varphi''(z)) \Big|<\infty.\end{eqnarray}$

利用(3.9), (3.11)和(3.12)式及$\varphi(z)$的有界性,最后得到

(3.13) $\begin{equation}\label{qwe11} K_{3}:=\sup\limits_{z\in{\Bbb D}}\mu(z)\left|\psi_1(z)(\varphi'(z))^2+2\psi_2'(z)\varphi'(z)+\psi_2(z)\varphi''(z)\right|<\infty.\end{equation}$

类似可得

(3.14) $\begin{equation} \label{qwe11b} K_{4}:=\sup\limits_{z\in{\Bbb D}}\mu(z)|\psi_2(z)||\varphi'(z)|^2<\infty.\end{equation}$

对固定的$w\in{\Bbb D}$以及常数$A, B, C$,令

(3.15) $\begin{equation}\label{qwe12} f_{w}(z)=\frac{(1-|w|^2)^{b+1}}{\phi(|w|)}\left(\frac{1}{(1-\overline{w}z)^{\alpha}}+A\frac{(1-|w|^2)}{(1-\overline{w}z)^{\alpha+1}}+ B\frac{(1-|w|^2)^2}{(1-\overline{w}z)^{\alpha+2}} +C\frac{(1-|w|^2)^3}{(1-\overline{w}z)^{\alpha+3}}\right), \end{equation}$

其中$b$由函数$\phi$确定且$\alpha=1/q+b+1$.简单计算可得

(3.16) $\begin{eqnarray}\label{qwe13}f^{(n)}_{w}(z)&=&\frac{(1-|w|^2)^{b+4}(\overline{w})^n}{\phi(|w|)(1-\overline{w}z)^{\alpha+n}}\left(\frac{\alpha\cdots(\alpha+n-1)}{(1-|w|^2)^3}+\frac{A(\alpha+1)\cdots(\alpha+n)}{(1-|w|^2)^2(1-\overline{w}z)}\right.\nonumber\\&&+\left.\frac{B(\alpha+2)\cdots(\alpha+n+1)}{(1-|w|^2)(1-\overline{w}z)^2} +\frac{C(\alpha+3)\cdots(\alpha+n+2)}{(1-\overline{w}z)^3}\right);\end{eqnarray}$

(3.17) $\begin{eqnarray}\label{qwe14}f^{(n+1)}_{w}(z)&=&\frac{(1-|w|^2)^{b+4}(\overline{w})^{n+1}}{\phi(|w|)(1-\overline{w}z)^{\alpha+n+1}}\left(\frac{\alpha\cdots(\alpha+n)}{(1-|w|^2)^3}+\frac{A(\alpha+1)\cdots(\alpha+n+1)}{(1-|w|^2)^2(1-\overline{w}z)}\right.\nonumber\\&&+\left.\frac{B(\alpha+2)\cdots(\alpha+n+2)}{(1-|w|^2)(1-\overline{w}z)^2} +\frac{C(\alpha+3)\cdots(\alpha+n+3)}{(1-\overline{w}z)^3}\right);\end{eqnarray}$

(3.18) $\begin{eqnarray}\label{qwe15}f^{(n+2)}_{w}(z)&=&\frac{(1-|w|^2)^{b+4}(\overline{w})^{n+2}}{\phi(|w|)(1-\overline{w}z)^{\alpha+n+2}}\left(\frac{\alpha\cdots(\alpha+n+1)}{(1-|w|^2)^3}+\frac{A(\alpha+1)\cdots(\alpha+n+2)}{(1-|w|^2)^2(1-\overline{w}z)}\right.\nonumber\\&&+\left.\frac{B(\alpha+2)\cdots(\alpha+n+3)}{(1-|w|^2)(1-\overline{w}z)^2} +\frac{C(\alpha+3)\cdots(\alpha+n+4)}{(1-\overline{w}z)^3}\right);\end{eqnarray}$

(3.19) $\begin{eqnarray} \label{qwe15b}f^{(n+3)}_{w}(z)&=&\frac{(1-|w|^2)^{b+4}(\overline{w})^{n+3}}{\phi(|w|)(1-\overline{w}z)^{\alpha+n+3}}\left(\frac{\alpha\cdots(\alpha+n+2)}{(1-|w|^2)^3}+\frac{A(\alpha+1)\cdots(\alpha+n+3)}{(1-|w|^2)^2(1-\overline{w}z)}\right.\nonumber\\&&+\left.\frac{B(\alpha+2)\cdots(\alpha+n+4)}{(1-|w|^2)(1-\overline{w}z)^2} +\frac{C(\alpha+3)\cdots(\alpha+n+5)}{(1-\overline{w}z)^3}\right);\end{eqnarray}$

利用引理2.3,可得

(3.20) $\begin{eqnarray}\label{qwee123} &&M_q(f_w, r)\leq C\frac{(1-|w|^2)^{b+1}}{\phi(|w|)(1-r|w|)^{b+1}}. \end{eqnarray}$

根据$\phi$的正规性及引理2.4,可得(参见文献[26, 32])

(3.21) $\begin{eqnarray}\label{eq10} \sup\limits_{w\in \rm {\Bbb D}}\|f_w\|_{p, q, \phi}\leq C.\end{eqnarray}$

为证明我们的结果,首先考察方程组

(3.22) $\begin{eqnarray}\label{sys1} && \alpha\cdots(\alpha+n) +A(\alpha+1)\cdots(\alpha+n+1)\nonumber \\ &&+B(\alpha+2)\cdots(\alpha+n+2)+C(\alpha+3)\cdots(\alpha+n+3)=0, \nonumber \\ && \alpha\cdots(\alpha+n+1) +A(\alpha+1)\cdots(\alpha+n+2)\nonumber \\ &&+B(\alpha+2)\cdots(\alpha+n+3)+C(\alpha+3)\cdots(\alpha+n+4)=0, \\ && \alpha\cdots(\alpha+n+2) +A(\alpha+1)\cdots(\alpha+n+3)\nonumber \\ &&+B(\alpha+2)\cdots(\alpha+n+4)+C(\alpha+3)\cdots(\alpha+n+5)=0.\nonumber\end{eqnarray}$

方程组(3.22)变形为

(3.23) $\begin{eqnarray} \label{sys2} && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n+1)\nonumber \\ &&+B(\alpha+2)(\alpha+n+1)(\alpha+n+2)+C(\alpha+n+1)(\alpha+n+2)(\alpha+n+3)=0, \nonumber \\ && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n+2)\nonumber \\ &&+B(\alpha+2)(\alpha+n+2)(\alpha+n+3)+C(\alpha+n+2)(\alpha+n+3)(\alpha+n+4)=0, \\ && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n+3)\nonumber \\ &&+B(\alpha+2)(\alpha+n+3)(\alpha+n+4)+C(\alpha+n+3)(\alpha+n+4)(\alpha+n+5)=0.\nonumber\end{eqnarray}$

由于方程组(3.23)的系数行列式

$\begin{eqnarray*} &&\left| \begin{array}{ccc} (\alpha+1)(\alpha+2)(\alpha+n+1)&(\alpha+2)(\alpha+n+1)(\alpha+n+2)&(\alpha+n+1)(\alpha+n+2)(\alpha+n+3) \\ (\alpha+1)(\alpha+2)(\alpha+n+2)&(\alpha+2)(\alpha+n+2)(\alpha+n+3)&(\alpha+n+2)(\alpha+n+3)(\alpha+n+4) \\ (\alpha+1)(\alpha+2)(\alpha+n+3)&(\alpha+2)(\alpha+n+3)(\alpha+n+4)&(\alpha+n+3)(\alpha+n+4)(\alpha+n+5) \\ \end{array} \right|\\&&=(\alpha+1)(\alpha+2)^2(\alpha+n+1)(\alpha+n+2)(\alpha+n+3)\left| \begin{array}{ccc} 1&(\alpha+n+2)&(\alpha+n+2)(\alpha+n+3) \\ 1&(\alpha+n+3)&(\alpha+n+3)(\alpha+n+4) \\ 1&(\alpha+n+4)&(\alpha+n+4)(\alpha+n+5) \\ \end{array} \right|\\&&=2(\alpha+1)(\alpha+2)^2(\alpha+n+1)(\alpha+n+2)(\alpha+n+3)\neq 0, \end{eqnarray*}$

根据克兰姆法则,方程组(3.23)有非零解.由(3.16)-(3.19)式,存在常数$A, B, C$使得$f_{w}^{(n+1)}(w)=f_{w}^{(n+2)}(w)=f_{w}^{(n+3)}(w)=0$及

$f^{(n)}_{w}(w)=C_1(A, B, C, \alpha, n)\frac{\left(\overline{w}\right)^{n}}{\phi(|w|)(1-|w|^2)^{1/q+n}}, $

其中$C_1(A, B, C, \alpha, n)=\alpha\cdots(\alpha+n-1) +A(\alpha+1)\cdots(\alpha+n) +B(\alpha+2)\cdots(\alpha+n+1)+C(\alpha+3)\cdots(\alpha+n+2)\neq 0$.因此对$\omega\in {\Bbb D}$及$\varphi(\omega)\neq 0$,取检验函数$f_{\varphi(\omega)}$,我们得到

(3.24) $\begin{eqnarray}\label{qwe16}C&\geq&\|T^n_{\psi_{1}, \psi_{2}, \varphi}f_{\varphi(\omega)}\|_{{\cal Z}_{\mu}}\geq \mu(\omega)\left|\psi_1''(\omega)f^{(n)}_{\varphi(\omega)}(\varphi(\omega))\right|\nonumber\\&=&|C_1(A, B, C, \alpha, n)|\frac{\mu(\omega)|\psi_1''(\omega)|\left|\overline{\varphi(\omega)}\right|^{n}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n}}.\end{eqnarray}$

由(3.24)式,对$r\in(t_0, 1)$,有

(3.25) $\begin{eqnarray}\label{qwe18}\sup\limits_{r<|\varphi(\omega)|<1}\frac{\mu(\omega)|\psi_1''(\omega)|}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n}}&\leq& \frac{1}{r^n}\sup\limits_{r<|\varphi(\omega)|<1}\frac{\mu(\omega)|\psi_1''(\omega)|\left|\overline{\varphi(\omega)}\right|^{n}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n}}\nonumber\\&\leq& \frac{1}{r^n}\sup\limits_{\omega\in{\Bbb D}}\frac{\mu(\omega)|\psi_1''(\omega)|\left|\overline{\varphi(\omega)}\right|^{n}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n}}\nonumber\\&\leq &C<\infty.\end{eqnarray}$

又从(3.9)式可得

(3.26) $\begin{eqnarray}\label{qwe17}\sup\limits_{|\varphi(\omega)|\leq r}\frac{\mu(\omega)|\psi_1''(\omega)|}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n}}\leq C\sup\limits_{|\varphi(\omega)|\leq r}\mu(\omega)|\psi_1''(\omega)|\leq CK_{1}<\infty.\end{eqnarray}$

因此把(3.26)与(3.25)式结合起来可得(3.1)式成立.

其次考察方程组

(3.27) $ \begin{eqnarray}\label{sys3} && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n)\nonumber \\ &&+B(\alpha+2)(\alpha+n)(\alpha+n+1)+C(\alpha+n)(\alpha+n+1)(\alpha+n+2)=0, \nonumber \\ && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n+2)\nonumber \\ &&+B(\alpha+2)(\alpha+n+2)(\alpha+n+3)+C(\alpha+n+2)(\alpha+n+3)(\alpha+n+4)=0, \\ && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n+3)\nonumber \\ &&+B(\alpha+2)(\alpha+n+3)(\alpha+n+4)+C(\alpha+n+3)(\alpha+n+4)(\alpha+n+5)=0.\nonumber \end{eqnarray}$

由于

$\begin{eqnarray*}&&\left| \begin{array}{ccc} (\alpha+1)(\alpha+2)(\alpha+n)&(\alpha+2)(\alpha+n)(\alpha+n+1)&(\alpha+n)(\alpha+n+1)(\alpha+n+2) \\ (\alpha+1)(\alpha+2)(\alpha+n+2)&(\alpha+2)(\alpha+n+2)(\alpha+n+3)&(\alpha+n+2)(\alpha+n+3)(\alpha+n+4) \\ (\alpha+1)(\alpha+2)(\alpha+n+3)&(\alpha+2)(\alpha+n+3)(\alpha+n+4)&(\alpha+n+3)(\alpha+n+4)(\alpha+n+5) \\ \end{array} \right|\\&&=(\alpha+1)(\alpha+2)^2(\alpha+n)(\alpha+n+2)(\alpha+n+3)\left| \begin{array}{ccc} 1&(\alpha+n+1)&(\alpha+n+1)(\alpha+n+2) \\ 1&(\alpha+n+3)&(\alpha+n+3)(\alpha+n+4) \\ 1&(\alpha+n+4)&(\alpha+n+4)(\alpha+n+5) \\ \end{array} \right|\\&&=6(\alpha+1)(\alpha+2)^2(\alpha+n)(\alpha+n+2)(\alpha+n+3) \neq 0, \end{eqnarray*}$

根据克兰姆法则,方程组(3.27)有非零解.由(3.16)-(3.19)式,知在(3.15)式中存在常数$A, B, C$使得$g_{w}^{(n)}(w)=g_{w}^{(n+2)}(w)=g_{w}^{(n+3)}(w)=0$及

$g^{(n+1)}_{w}(w)=C_2(A, B, C, \alpha, n)\frac{\left(\overline{w}\right)^{n+1}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+1}}, $

其中$C_2(A, B, C, \alpha, n)=\alpha\cdots(\alpha+n) +A(\alpha+1)\cdots(\alpha+n+1) +B(\alpha+2)\cdots(\alpha+n+2)+C(\alpha+3)\cdots(\alpha+n+3)\neq 0$, $g_{w}$表示相应的函数.因此对$\omega\in {\Bbb D}$且$\varphi(\omega)\neq 0$,考察检验函数$g_{\varphi(\omega)}$,可得

(3.28) $\begin{eqnarray}\label{qwe23}C&\geq&\|T^n_{\psi_{1}, \psi_{2}, \varphi}g_{\varphi(\omega)}\|_{{\cal Z}_{\mu}}\nonumber\\&\geq&\mu(\omega)\left|\psi_1(\omega)\varphi''(\omega)+2\psi_1'(\omega)\varphi'(\omega)+\psi_2''(\omega)\right|\left|g^{(n+1)}_{\varphi(\omega)}(\varphi(\omega))\right|\nonumber\\&=&|C_2(A, B, C, \alpha, n)|\frac{\mu(\omega)\left|\psi_1(\omega)\varphi''(\omega)+2\psi_1'(\omega)\varphi'(\omega)+\psi_2''(\omega)\right|\left|\overline{\varphi(\omega)}\right|^{n+1}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+1}}.\end{eqnarray}$

利用(3.28)式有

(3.29) $\begin{eqnarray}\label{qwe24}&&\sup\limits_{r<|\varphi(\omega)|<1}\frac{\mu(\omega)|\psi_1(\omega)\varphi''(\omega)+2\psi_1'(\omega)\varphi'(\omega)+\psi_2''(\omega)|}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+1}}\nonumber\\&\leq &\frac{1}{r^{n+1}}\sup\limits_{r<|\varphi(\omega)|<1} \frac{\mu(\omega)|\psi_1(\omega)\varphi''(\omega)+2\psi_1'(\omega)\varphi'(\omega)+\psi_2''(\omega)|\left|\overline{\varphi(\omega)}\right|^{n+1}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+1}}\nonumber\\&\leq &\frac{1}{r^{n+1}}\sup\limits_{\omega\in{\Bbb D}} \frac{\mu(\omega)|\psi_1(\omega)\varphi''(\omega)+2\psi_1'(\omega)\varphi'(\omega)+\psi_2''(\omega)|\left|\overline{\varphi(\omega)}\right|^{n+1}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+1}}\nonumber\\ &\leq& C <\infty.\end{eqnarray}$

根据(3.11)式,我们可得

(3.30) $\begin{eqnarray}\label{qwe25}&&\sup\limits_{|\varphi(\omega)|\leq r}\frac{\mu(\omega)|\psi_1(\omega)\varphi''(\omega)+2\psi_1'(\omega)\varphi'(\omega)+\psi_2''(\omega)|}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+1}}\nonumber\\&\leq &C\sup\limits_{|\varphi(\omega)|\leq r}\mu(\omega)|\psi_1(\omega)\varphi''(\omega)+2\psi_1'(\omega)\varphi'(\omega)+\psi_2''(\omega)|\nonumber\\&\leq& CK_{2}<\infty.\end{eqnarray}$

由(3.29)及(3.30)式可得(3.2)式成立.

再次为了证明(3.3)式,可考察方程组

(3.31) $\begin{eqnarray}\label{sys4}&& \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n)\nonumber \\ &&+B(\alpha+2)(\alpha+n)(\alpha+n+1)+C(\alpha+n)(\alpha+n+1)(\alpha+n+2)=0, \nonumber \\ && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n+1)\nonumber \\ &&+B(\alpha+2)(\alpha+n+1)(\alpha+n+2)+C(\alpha+n+1)(\alpha+n+2)(\alpha+n+3)=0, \\ && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n+3)\nonumber \\ &&+B(\alpha+2)(\alpha+n+3)(\alpha+n+4)+C(\alpha+n+3)(\alpha+n+4)(\alpha+n+5)=0.\nonumber \end{eqnarray}$

由于

$\begin{eqnarray*} &&\left| \begin{array}{ccc} (\alpha+1)(\alpha+2)(\alpha+n)&(\alpha+2)(\alpha+n)(\alpha+n+1)&(\alpha+n)(\alpha+n+1)(\alpha+n+2) \\ (\alpha+1)(\alpha+2)(\alpha+n+1)&(\alpha+2)(\alpha+n+1)(\alpha+n+2)&(\alpha+n+1)(\alpha+n+2)(\alpha+n+3) \\ (\alpha+1)(\alpha+2)(\alpha+n+3)&(\alpha+2)(\alpha+n+3)(\alpha+n+4)&(\alpha+n+3)(\alpha+n+4)(\alpha+n+5) \\ \end{array} \right|\\&&=(\alpha+1)(\alpha+2)^2(\alpha+n)(\alpha+n+1)(\alpha+n+3)\left| \begin{array}{ccc} 1&(\alpha+n+1)&(\alpha+n+1)(\alpha+n+2) \\ 1&(\alpha+n+2)&(\alpha+n+2)(\alpha+n+3) \\ 1&(\alpha+n+4)&(\alpha+n+4)(\alpha+n+5) \\ \end{array} \right|\\&&=8(\alpha+1)(\alpha+2)^2(\alpha+n)(\alpha+n+1)(\alpha+n+3)\neq 0, \end{eqnarray*}$

由(3.16)-(3.19)式可知,存在(3.15)式中的常数$A, B, C$使得$h_{w}^{(n)}(w)=h_{w}^{(n+1)}(w)=h_{w}^{(n+3)}(w)=0$及

$h^{(n+2)}_{w}(w)=C_3(A, B, C, \alpha, n)\frac{\left(\overline{w}\right)^{n+2}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+2}}, $

这里$C_3(A, B, C, \alpha, n)=\alpha\cdots(\alpha+n+1) +A(\alpha+1)\cdots(\alpha+n+2) +B(\alpha+2)\cdots(\alpha+n+3)+C(\alpha+3)\cdots(\alpha+n+4)\neq 0$, $h_{w}$表示相应的函数.因此对$\omega\in {\Bbb D}$且$\varphi(\omega)\neq 0$，考察检验函数$h_{\varphi(\omega)}$,可得

(3.32) $\begin{eqnarray}\label{qwe30}C&\geq&\|T^n_{\psi_{1}, \psi_{2}, \varphi}h_{\varphi(\omega)}\|_{{\cal Z}_{\mu}}\nonumber\\&\geq&\mu(\omega)|\psi_1(\omega)(\varphi'(\omega))^2+2\psi_2'(\omega)\varphi'(\omega)+\psi_2(\omega)\varphi''(\omega)|\left|h^{(n+2)}_{\varphi(\omega)}(\varphi(\omega))\right|\nonumber\\&=&|C_4(A, B, C, \alpha, n)|\frac{\mu(\omega)|\psi_1(\omega)(\varphi'(\omega))^2+2\psi_2'(\omega)\varphi'(\omega)+\psi_2(\omega)\varphi''(\omega)|\left|\overline{\varphi(\omega)}\right|^{n+2}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+2}}.\nonumber\\\end{eqnarray}$

利用(3.32)式,对$r\in(t_0, 1)$,有

(3.33) $\begin{eqnarray}\label{qwe31}&&\sup\limits_{r<|\varphi(\omega)|<1}\frac{\mu(\omega)\left|\psi_1(\omega)(\varphi'(\omega))^2+2\psi_2'(\omega)\varphi'(\omega)+\psi_2(\omega)\varphi''(\omega)\right|}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+2}}\nonumber\\&\leq&\frac{1}{r^{n+2}}\sup\limits_{r<|\varphi(\omega)|<1}\frac{\mu(\omega)|\psi_1(\omega)(\varphi'(\omega))^2+2\psi_2'(\omega)\varphi'(\omega)+\psi_2(\omega)\varphi''(\omega)|\left|\overline{\varphi(\omega)}\right|^{n+2}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+2}}\nonumber\\&\leq& \frac{1}{r^{n+2}}\sup\limits_{\omega\in{\Bbb D}}\frac{\mu(\omega)|\psi_1(\omega)(\varphi'(\omega))^2+2\psi_2'(\omega)\varphi'(\omega)+\psi_2(\omega)\varphi''(\omega)|\left|\overline{\varphi(\omega)}\right|^{n+2}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+2}}\nonumber\\&\leq& C<\infty.\end{eqnarray}$

使用(3.13)式,我们有

(3.34) $\begin{eqnarray}\label{qwe32}&&\sup\limits_{|\varphi(\omega)|\leq r}\frac{\mu(\omega)|\psi_1(\omega)(\varphi'(\omega))^2+2\psi_2'(\omega)\varphi'(\omega)+\psi_2(\omega)\varphi''(\omega)|}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+2}}\nonumber\\&\leq&\sup\limits_{|\varphi(\omega)|\leq r}\mu(\omega)|\psi_1(\omega)(\varphi'(\omega))^2+2\psi_2'(\omega)\varphi'(\omega)+\psi_2(\omega)\varphi''(\omega)|\nonumber\\&\leq& CK_{3}<\infty.\end{eqnarray}$

依据(3.33)和(3.34)式, (3.3)式成立.

最后我们证明(3.4)式成立.为此考察下面的方程组

(3.35) $ \begin{eqnarray} && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n)\nonumber \\ &&+B(\alpha+2)(\alpha+n)(\alpha+n+1)+C(\alpha+n)(\alpha+n+1)(\alpha+n+2)=0, \nonumber \\ && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n+1)\nonumber \\ &&+B(\alpha+2)(\alpha+n+1)(\alpha+n+2)+C(\alpha+n+1)(\alpha+n+2)(\alpha+n+3)=0, \\ && \alpha(\alpha+1)(\alpha+2) +A(\alpha+1)(\alpha+2)(\alpha+n+2)\nonumber \\ &&+B(\alpha+2)(\alpha+n+2)(\alpha+n+3)+C(\alpha+n+2)(\alpha+n+3)(\alpha+n+4)=0.\nonumber \end{eqnarray}$

由于

$\begin{eqnarray*}&&\left| \begin{array}{ccc} (\alpha+1)(\alpha+2)(\alpha+n)&(\alpha+2)(\alpha+n)(\alpha+n+1)&(\alpha+n)(\alpha+n+1)(\alpha+n+2) \\ (\alpha+1)(\alpha+2)(\alpha+n+1)&(\alpha+2)(\alpha+n+1)(\alpha+n+2)&(\alpha+n+1)(\alpha+n+2)(\alpha+n+3) \\ (\alpha+1)(\alpha+2)(\alpha+n+2)&(\alpha+2)(\alpha+n+2)(\alpha+n+3)&(\alpha+n+2)(\alpha+n+3)(\alpha+n+4) \\ \end{array} \right|\\&&=(\alpha+1)(\alpha+2)^2(\alpha+n)(\alpha+n+1)(\alpha+n+2)\left| \begin{array}{ccc} 1&(\alpha+n+1)&(\alpha+n+1)(\alpha+n+2) \\ 1&(\alpha+n+2)&(\alpha+n+2)(\alpha+n+3) \\ 1&(\alpha+n+3)&(\alpha+n+3)(\alpha+n+4) \\ \end{array} \right|\\&&=2(\alpha+1)(\alpha+2)^2(\alpha+n)(\alpha+n+1)(\alpha+n+2)\neq 0, \end{eqnarray*}$

从(3.16)-(3.19)式,知在(3.15)式中一定存在$A, B, C$使得$k_{w}^{(n)}(w)=k_{w}^{(n+1)}(w)=k_{w}^{(n+2)}(w)=0$及

$k^{(n+3)}_{w}(w)=C_4(A, B, C, \alpha, n)\frac{\left(\overline{w}\right)^{n+3}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+3}}, $

其中$C_4(A, B, C, \alpha, n)=\alpha\cdots(\alpha+n+2) +A(\alpha+1)\cdots(\alpha+n+3) +B(\alpha+2)\cdots(\alpha+n+4)+C(\alpha+3)\cdots(\alpha+n+5)\neq 0$, $k_{w}$表示相应的函数.因此对$\omega\in {\Bbb D}$且$\varphi(\omega)\neq 0$,对检验函数$k_{\varphi(\omega)}$,有

(3.36) $\begin{eqnarray}\label{qwe30b}C&\geq&\|T^n_{\psi_{1}, \psi_{2}, \varphi}k_{\varphi(\omega)}\|_{{\cal Z}_{\mu}}\nonumber\\&\geq&\mu(\omega)|\psi_2(\omega)||\varphi'(\omega)|^2\left|k^{(n+3)}_{\varphi(\omega)}(\varphi(\omega))\right|\nonumber\\&=&|C_4(A, B, C, \alpha, n)|\frac{\mu(\omega)|\psi_2(\omega)||\varphi'(\omega)|^2\left|\overline{\varphi(\omega)}\right|^{n+3}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+3}}.\end{eqnarray}$

由(3.36)式,得到

(3.37) $\begin{eqnarray}\label{qwe31b}\sup\limits_{r<|\varphi(\omega)|<1}\frac{\mu(\omega)|\psi_2(\omega)||\varphi'(\omega)|^2}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+3}}&\leq&\frac{1}{r^{n+3}}\sup\limits_{r<|\varphi(\omega)|<1}\frac{\mu(\omega)|\psi_2(\omega)||\varphi'(\omega)|^2\left|\overline{\varphi(\omega)}\right|^{n+3}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+3}}\nonumber\\&\leq &\frac{1}{r^{n+3}}\sup\limits_{\omega\in{\Bbb D}}\frac{\mu(\omega)|\psi_2(\omega)||\varphi'(\omega)|^2\left|\overline{\varphi(\omega)}\right|^{n+3}}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+3}}\nonumber\\&\leq &C<\infty.\end{eqnarray}$

另一方面由(3.14)式知

(3.38) $\begin{eqnarray}\label{qwe32b}\sup\limits_{|\varphi(\omega)|\leq r}\frac{\mu(\omega)|\psi_2(\omega)||\varphi'(\omega)|^2}{\phi(|\varphi(\omega)|)\left(1-\left|\varphi(\omega)\right|^{2}\right)^{1/q+n+3}}&\leq&\sup\limits_{|\varphi(\omega)|\leq r} C\mu(\omega)|\psi_2(\omega)||\varphi'(\omega)|^2\\&\leq& CK_{4}<\infty.\end{eqnarray}$

结合(3.37)和(3.38)式即可推出(3.4)式成立,注意在我们的证明中,我们已经使用了以下事实:对固定的$\omega\in{\Bbb D}$且$\varphi(\omega)\neq 0$,函数$g_{\varphi(\omega)}, h_{\varphi(\omega)}, k_{\varphi(\omega)}\in H(p, q, \phi)$.证毕.

定理3.2  设$\psi_1, \psi_2\in H({\Bbb D})$, $n\in{\mathbb N}_0$, $\varphi$是${\Bbb D}$到${\Bbb D}$的一个解析自映射, $\mu$是一个权函数,则$T^n_{\psi_{1}, \psi_{2}, \varphi}:H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是紧的充要条件$T^n_{\psi_{1}, \psi_{2}, \varphi}:H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是有界的,

(3.39) $\begin{equation}\label{qwe33}\lim\limits_{|\varphi(z)|\rightarrow1}\frac{\mu(z)|\psi_1''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n}}=0, \end{equation}$

(3.40) $\begin{equation}\label{qwe34}\lim\limits_{|\varphi(z)|\rightarrow1}\frac{\mu(z)|\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+1}}=0, \end{equation}$

(3.41) $\begin{equation}\label{qwe35}\lim\limits_{|\varphi(z)|\rightarrow1}\frac{\mu(z)|\psi_1(z)(\varphi'(z))^2+2\psi_2'(z)\varphi'(z)+\psi_2(z)\varphi''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+2}}=0, \end{equation}$

及

(3.42) $\begin{equation} \label{qwe35b}\lim\limits_{|\varphi(z)|\rightarrow1}\frac{\mu(z)|\psi_2(z)||\varphi'(z)|^2}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+3}}=0.\end{equation}$

证  充分性.设$T^n_{\psi_{1}, \psi_{2}, \varphi}: H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是有界的且条件(3.39)-(3.42)式成立.对于在${\Bbb D}$的任意紧子集上一致收敛于零($k\rightarrow \infty$)的有界序列$\{f_k\}\subset H(p, q, \phi)$,根据引理2.2,只需证明

$\|T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\|_ {{\cal Z}_{\mu}}\rightarrow 0\ (k\rightarrow\infty).$

不失一般性不妨设$\|f_{k}\|_{p, q, \phi}\leq1$.由(3.39)-(3.42)式,对任意的$\varepsilon>0$,一定存在$\rho \in(0, 1)$,当$ \rho<|\varphi(z)|<1$时,有

(3.43) $\begin{equation}\label{qwe36} \frac{\mu(z)|\psi_1''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n}}<\varepsilon, \end{equation}$

(3.44) $\begin{equation}\label{qwe37}\frac{\mu(z)|\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+1}}<\varepsilon, \end{equation}$

(3.45) $\begin{equation}\label{qwe38}\frac{\mu(z)|\psi_1(z)(\varphi'(z))^2+2\psi_2'(z)\varphi'(z)+\psi_2(z)\varphi''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+2}}<\varepsilon, \end{equation}$

及

(3.46) $\begin{equation} \label{qwe38b}\frac{\mu(z)|\psi_2(z)||\varphi'(z)|^2}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+3}}<\varepsilon.\end{equation}$

由于算子$T^n_{\psi_{1}, \psi_{2}, \varphi}: H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是有界的,根据定理3.1,我们看到(3.9), (3.11), (3.13)及(3.14)式成立.又因为$f_{k}$在${\Bbb D}$的任意紧子集上一致收敛于零($k\rightarrow \infty$), Cauchy估计保证$f^{(n)}_{k}$, $f^{(n+1)}_{k}$, $f^{(n+2)}_{k}$及$f^{(n+3)}_{k}$在${\Bbb D}$的任意紧子集上也一致收敛于零,于是存在常数$K_{0}\in{\Bbb N}$使得当$k>K_{0}$时,对$\rho\in(0, 1)$,有

(3.47) $\begin{eqnarray}\label{qwe39}&&|(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k})(0)|+|(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k})'(0)|+\sup\limits_{|\varphi(z)|\leq\rho}\mu(z)\left|(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k})''(z)\right|\nonumber\\&\leq&|\psi_1(0)|\left|f^{(n)}_{k}(\varphi(0))\right|+\left|\psi_2(0)f^{(n+1)}_{k}(\varphi(0))\right|+\left|\psi'(0)||f^{(n)}_{k}(\varphi(0))\right|\nonumber\\&& +|\psi(0)\varphi'(0)|\left|f^{(n+1)}_{k}(\varphi(0))\right|+\left|\psi_2(0)\varphi'(0)f^{(n+2)}_{k}(\varphi(0))\right|\nonumber\\&& +\sup\limits_{|\varphi(z)|\leq\rho}\mu(z)\left|\psi_1''(z)||f^{(n)}_{k}(\varphi(z))\right|\nonumber\\&& +\sup\limits_{|\varphi(z)|\leq\rho}\mu(z)|\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z)|\left|f^{(n+1)}_{k}(\varphi(z))\right|\nonumber\\&& +\sup\limits_{|\varphi(z)|\leq\rho}\mu(z)|\psi_1(z)(\varphi'(z))^2+2\psi_2'(z)\varphi'(z)+\psi_2(z)\varphi''(z)|\left|f^{(n+2)}_{k}(\varphi(z))\right|\nonumber\\&& +\sup\limits_{|\varphi(z)|\leq\rho}\mu(z)|\psi_2(z)||\varphi'(z)|^2\left|f^{(n+3)}_{k}(\varphi(z))\right|\nonumber\\&\leq& C\varepsilon +K_{1}\sup\limits_{|\varphi(z)|\leq\rho}\left|f^{(n)}_{k}(\varphi(z))\right|+K_{2}\sup\limits_{|\varphi(z)|\leq\rho}\left|f^{(n+1)}_{k}(\varphi(z))\right|\nonumber\\&& +K_{3}\sup\limits_{|\varphi(z)|\leq\rho}|f^{(n+2)}_{k}(\varphi(z))|+K_{4}\sup\limits_{|\varphi(z)|\leq\rho}|f^{(n+3)}_{k}(\varphi(z))|\nonumber\\&<&C\varepsilon.\end{eqnarray}$

利用(3.43)-(3.47)式以及引理2.1,可得当$k>K_{0}$时

(3.48) $\begin{eqnarray}\label{qwe40}\|T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\|_{\mathcal{Z}_{\mu}}&=&\left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\right)(0)\right|+\left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\right)'(0)\right|+\sup\limits_{z\in{\Bbb D}}\mu(z)\left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\right)^{''}(z)\right|\nonumber\\&\leq& \left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\right)(0)\right|+\left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\right)'(0)\right|\nonumber\\&&+\sup\limits_{|\varphi(z)|\leq\rho}\mu(z)\left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\right)^{''}(z)\right|+\sup\limits_{\rho<|\varphi(z)|<1}\mu(z)\left|\left(T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\right)^{''}(z)\right|\nonumber\\&<&C\varepsilon+C\sup\limits_{\rho<|\varphi(z)|<1}\frac{\mu(z)|\psi_1''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n}}\|f\|_{p, q, \phi}\nonumber\\&& +C\sup\limits_{\rho<|\varphi(z)|<1}\frac{\mu(z)|\psi_1(z)\varphi''(z)+2\psi_1'(z)\varphi'(z)+\psi_2''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+1}}\|f\|_{p, q, \phi}\nonumber\\&& +C\sup\limits_{\rho<|\varphi(z)|<1}\frac{\mu(z)|\psi_1(z)(\varphi'(z))^2+2\psi_2'(z)\varphi'(z)+\psi_2(z)\varphi''(z)|}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+2}}\|f\|_{p, q, \phi} \nonumber\\&& +C\sup\limits_{\rho<|\varphi(z)|<1}\frac{\mu(z)|\psi_2(z)||\varphi'(z)|^2}{\phi(|\varphi(z)|)\left(1-\left|\varphi(z)\right|^{2}\right)^{1/q+n+3}}\|f\|_{p, q, \phi} \nonumber\\&<&4C\varepsilon.\end{eqnarray}$

即

$\lim\limits_{k\rightarrow\infty}\|T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\|_ {{\cal Z}_{\mu}}=0.$

因此算子$T^n_{\psi_{1}, \psi_{2}, \varphi}: H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是紧的.

必要性.如果$T^n_{\psi_{1}, \psi_{2}, \varphi}: H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是紧的.显然$T^n_{\psi_{1}, \psi_{2}, \varphi}: H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是有界的.设$\{z_{k}\}$是${\Bbb D}$中的一个点列使得$|\varphi(z_{k})|\rightarrow1$$(k\rightarrow\infty)$.使用检验函数

$\begin{eqnarray*}f_{k}(z):&=&f_{\varphi(z_{k})}(z)\nonumber\\&=&\frac{(1-|\varphi(z_{k})|^2)^{b+1}}{\phi(|\varphi(z_{k})|)}\left(\frac{1}{(1-\overline{\varphi(z_{k})}z)^{\alpha}}+A\frac{(1-|\varphi(z_{k})|^2)}{(1-\overline{\varphi(z_{k})}z)^{\alpha+1}}\right)\nonumber\\&&+\frac{(1-|\varphi(z_{k})|^2)^{b+1}}{\phi(|\varphi(z_{k})|)}\left(B\frac{(1-|\varphi(z_{k})|^2)^2}{(1-\overline{\varphi(z_{k})}z)^{\alpha+2}}+C\frac{(1-|\varphi(z_{k})|^2)^3}{(1-\overline{\varphi(z_{k})}z)^{\alpha+3}}\right).\nonumber\end{eqnarray*}$

注意到$\phi$是正规函数,对$|z|=r<1$,有

$|f_{k}(z)|\leq C(1-|\varphi(z_{k})|)\rightarrow 0 \ (k\rightarrow\infty).\nonumber$

可见${f_{k}}$是$H(p, q, \phi)$中一个有界的函数列且在${\Bbb D}$的任意紧子集上一致收敛于零.根据引理2.2,可得

$\lim\limits_{k\rightarrow\infty}\|T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\|_{{\cal Z}_{\mu}}=0.$

由于

$f_{k}^{(n+1)}(\varphi(z_{k}))=f_{k}^{(n+2)}(\varphi(z_{k}))=f_{k}^{(n+3)}(\varphi(z_{k}))=0, $

$f^{(n)}_{k}(\varphi(z_{k}))=\frac{C_1(A, B, C, \alpha, n)\left(\overline{\varphi(z_{k})}\right)^{n}}{\phi(|\varphi(z_k)|)\left(1-\left|\varphi(z_k)\right|^{2}\right)^{1/q+n}}.$

利用(3.24)式以及算子$T^n_{\psi_{1}, \psi_{2}, \varphi}: H(p, q, \phi)\rightarrow{\cal Z}_{\mu}$是紧的,我们有

(3.49) $\begin{equation}\label{qwe42}|C_1(A, B, C, \alpha, n)|\frac{\mu(z_{k})|\psi_1''(z_{k})|\left|\overline{\varphi(z_{k})}\right|^{n}}{\phi(|\varphi(z_k)|)\left(1-\left|\varphi(z_k)\right|^{2}\right)^{1/q+n}}\leq\|T^n_{\psi_{1}, \psi_{2}, \varphi}f_{k}\|_{{\cal Z}_{\mu}}\rightarrow0~ (k\rightarrow\infty).\end{equation}$

利用(3.49)式和$|\varphi(z_{k})|\rightarrow1\ (k\rightarrow\infty)$,得到

$\lim\limits_{k\rightarrow\infty}\frac{\mu(z_{k})|\psi_1''(z_{k})|}{\phi(|\varphi(z_k)|)\left(1-\left|\varphi(z_k)\right|^{2}\right)^{1/q+n}}=0, $

因此(3.39)式成立.类似可证(3.40), (3.41)以及(3.42)式成立,此处省去有关细节.