设$ D = \{z:|z|<1\} $为复平面上的单位圆盘, $ \partial D = \{z:|z| = 1\} $是$ D $的边界, $ H(D) $和$ H(D, D) $分别表示$ D $上的解析函数全体和解析映射全体.

定义1.1  对$ [0, 1) $上的连续函数$ \mu(r)>0 $, 如果存在常数$ 0<s<t $, 使得

(ⅰ) $ \frac{\mu(r)}{(1-r)^s} $在$ [0, 1) $上单调递减且$ \lim\limits_{r\rightarrow 1^-}\frac{\mu(r)}{(1-r)^s} = 0, $

(ⅱ) $ \frac{\mu(r)}{(1-r)^t} $在$ [0, 1) $上单调递增且$ \lim\limits_{r\rightarrow1^-}\frac{\mu(r)}{(1-r)^t} = \infty, $

则称$ \mu $为$ [0, 1) $上的正规函数^[1].

定义1.2   设$ \mu $为$ [0, 1) $上的正规函数, 对$ 0<p<\infty, $记

$ H(p, \mu) = \{f\in H(D):\|f\|_{H(p, \mu)}^p = \int_D|f(z)|^p\frac{\mu^p(|z|)}{1-|z|}\mathrm{d}A(z)<\infty\} $

表示$ D $上的$ \mu- $Bergman空间. 当$ 1\leq p<\infty $时, $ H(p, \mu) $在范数$ \|\cdot\|_{H(p, \mu)} $下是Banach空间.当$ 0<p<1 $时, $ H(p, \mu) $在范数$ \|\cdot\|_{H(p, \mu)} $下为度量空间, 也是Frechet空间. 特别地, 当$ \mu(r) = (1-r)^q, q>-1 $时, $ H(p, \mu) $就是经典的加权Bergman空间$ \mathcal{A}_q^p $ ^[2].

定义1.3   $ D $上的Zygmund空间$ Z $是指满足$ f\in H(D)\bigcap C(\overline{D}) $且

$ \sup\limits_{0<\theta\leq2\pi}\frac{|f(e^{i(\theta+h)})+f(e^{i(\theta-h)})-2f(e^{i\theta})|}{h}<\infty $

的函数全体, 其中$ e^{i\theta}\in \partial D, h>0. $由文献[3]中的定理5.3和闭图像定理得$ f\in Z $当且仅当$ \sup\limits_{z\in D}(1-|z|^2)|f''(z)|<\infty. $

定义范数

$ \|f\|_Z = |f(0)|+|f'(0)|+\sup\limits_{z\in D}(1-|z|^2)|f''(z)|, $

则Zygmund空间在此范数下是Banach空间. 小Zygmund空间是指满足$ f\in Z $且$ \lim\limits_{|z|\rightarrow1^-}(1-|z|^2)|f''(z)| = 0 $的函数全体，记为$ Z_0, $显然$ Z_0 $是$ Z $的闭子空间.

对$ \alpha>0, D $上的Zygmund型空间是指满足$ f\in H(D) $且$ \sup\limits_{z\in D}(1-|z|^2)^\alpha|f''(z)|<\infty $的函数全体, 记为$ Z^\alpha $. 定义范数

$ \|f\|_{Z^\alpha} = |f(0)|+|f'(0)|+\sup\limits_{z\in D}(1-|z|^2)^\alpha|f''(z)|, $

则Zygmund型空间在此范数下是Banach空间. 小Zygmund型空间是指满足$ f\in Z^\alpha $且$ \lim\limits_{|z|\rightarrow1^-}(1-|z|^2)^\alpha|f''(z)| = 0 $的函数全体，记为$ Z_0^\alpha, $显然$ Z_0^\alpha $是$ Z^\alpha $的闭子空间.

定义1.4  设$ \varphi $是$ D $到$ D $上的非常值解析自映射, $ \psi\in H(D) $, $ H(D) $上的加权复合算子$ T_{\varphi, \psi} $定义如下:

$ T_{\varphi, \psi} = \psi(z) f\circ\varphi(z), \forall f\in H(D). $

显然, 加权复合算子$ T_{\varphi, \psi} $是线性算子. 特别地, 当$ \psi = 1 $时即为复合算子$ T_{\varphi} $.

单位圆盘上函数空间中的复合算子和加权复合算子的有界性和紧性问题已有非常广泛的研究, 得到了很好的结论, 如文献[4-8]讨论了单位圆盘上与Zygmund型空间有关的复合算子问题.

文献[2, 9-11]讨论了Bergman空间上的复合算子的有界性和紧性问题. 而对于单位球上Bergman空间上的复合算子, 文献[12-16]有了一些较好的结果. 本文受文献[2、7]工作的启发, 将讨论$ D $上的$ \mu- $Bergman空间到Zygmund型空间上的加权复合算子的有界性和紧性. 本文中$ z = (z_{1}, \cdots, z_{n}), \omega = (\omega_{1}, \cdots, \omega_{n}), \langle z, \omega\rangle = \Sigma_{j = 1}^{n}z_{j}\overline{\omega}_{j} $, 我们将用记号$ C, C_1, C_2, C_3 $来表示与变量$ z, \omega $无关的正数, $ C, C_1, C_2, C_3 $可以与某些范数或有界量有关, 不同的地方可以表示不同的正常数.

引理2.1^[11]  设$ 0<p<\infty, \forall f\in H(p, \mu), z\in D $, 有

$|f(z)|\leq \frac{C\|f\|_{H(p, \mu)}}{\mu(|z|)(1-|z|^2)^{\frac{1}{p}}}. $

(2.1)

由文献[17]中的定理2, 令$ m = 1, 2, p = q, $则分别得以下结论：

引理2.2  $ 0<p<\infty, $ $ \mu $为$ [0, 1) $上的正规函数, $ \forall f\in H(D) $, 则有

(ⅰ) $ \{\int_D|f(z)|^p\frac{\mu^p(|z|)}{1-|z|}dA(z)\}^\frac{1}{p}\approx |f(0)|+\{\int_D|f'(z)|^p(1-|z|^2)^p\frac{\mu^p(|z|)}{1-|z|}dA(z)\}^\frac{1}{p}, $

(ⅱ) $ \{\int_D|f(z)|^p\frac{\mu^p(|z|)}{1-|z|}dA(z)\}^\frac{1}{p}\approx |f(0)|+|f'(0)|+\{\int_D|f''(z)|^p(1-|z|^2)^{2p}\frac{\mu^p(|z|)}{1-|z|}dA(z)\}^\frac{1}{p}. $

设$ f\in H(p, \mu), $则由引理2.1和引理2.2可知, $ \forall z\in D $, 有

$|f'(z)|\leq \frac{C\|f\|_{H(p, \mu)}}{\mu(|z|)(1-|z|^2)^{\frac{1}{p}+1}}, |f''(z)|\leq \frac{C\|f\|_{H(p, \mu)}}{\mu(|z|)(1-|z|^2)^{\frac{1}{p}+2}}. $

(2.2)

引理2.3  设$ 0<p<\infty, \alpha>0, f\in H(D) $, 则$ T_{\varphi, \psi} $是$ H(p, \mu) $空间到$ Z^\alpha $型空间上的紧算子的充要条件是对任意满足条件: (1) 在$ H(p, \mu) $中有界; (2) 在$ D $的任一紧子集上一致收敛于0的序列$ \{f_{j}\} $, 都有

$ \|T_{\varphi, \psi}(f_{j})\|_{H(p, \mu)}\rightarrow0 \quad (j\rightarrow\infty). $

证  由引理2.1和Montel定理按紧算子的定义可证.

定理3.1  设$ \varphi\in H(D, D), \psi\in H(D), $则$ T_{\varphi, \psi}:H(p, \mu)\rightarrow Z^\alpha $为有界算子的充要条件是下列条件同时成立:

$M = \sup\limits_{z\in D} \frac{(1-|z|^2)^\alpha|\psi''(z)|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}}}<\infty, $

(3.1)

$N = \sup\limits_{z\in D} \frac{(1-|z|^2)^\alpha|2\psi'(z)\varphi'(z)+\psi(z)\varphi''(z)|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}+1}}<\infty, $

(3.2)

$K = \sup\limits_{z\in D} \frac{(1-|z|^2)^\alpha|\psi(z)(\varphi'(z))^2|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}+2}}<\infty. $

(3.3)

证  先证充分性  设(3.1)-(3.3)式成立, $ \forall z\in D, f\in H(p, \mu), $由三角不等式和(2.1)-(2.2)式有:

$\begin{align} &(1-|z|^2)^\alpha|(T_{\varphi, \psi}f)''(z)| = (1-|z|^2)^\alpha|[\psi'(z)f(\varphi(z))+\psi(z)f'(\varphi(z))\varphi'(z)]'| \\ \leq&(1-|z|^2)^\alpha\{|\psi''(z)f(\varphi(z))|+|(2\psi'(z)\varphi'(z)+\psi(z)\varphi''(z))f'(\varphi(z))|+\psi(z)(\varphi'(z))^2f''(\varphi(z))|\} \\ \leq& C(1-|z|^2)^\alpha\{\frac{|\psi''(z)|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}}}+ \frac{|2\psi'(z)\varphi'(z)+\psi(z)\varphi''(z)|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}+1}} \\ &+\frac{|\psi(z)(\varphi'(z))^2|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}+2}}\}\|f\|_{H(p, \mu)} \\ \leq& C(M+N+K)\|f\|_{H(p, \mu)}. \end{align} $

(3.4)

另一方面,

$|(T_{\varphi, \psi}f)(0)| = |\psi(0)||f(\varphi(0))|\leq\frac{C|\psi(0)|\|f\|_{H(p, \mu)}}{\mu(|\varphi(0)|)(1-|\varphi(0)|^2)^{\frac{1}{p}}}, \\ |(T_{\varphi, \psi}f)'(0)| = |\psi'(0)f(\varphi(0))+\psi(0)f'(\varphi(0))\varphi'(0)| $

(3.5)

$\leq C[\frac{|\psi'(0)|}{\mu(|\varphi(0)|)(1-|\varphi(0)|^2)^{\frac{1}{p}}} +\frac{|\psi(0)\varphi'(0)|}{\mu(|\varphi(0)|)(1-|\varphi(0)|^2)^{\frac{1}{p}+1}}]\|f\|_{H(p, \mu)}. $

(3.6)

所以由(3.4)-(3.6)式知: $ T_{\varphi, \psi}:H(p, \mu)\rightarrow Z^\alpha $为有界算子.

必要性  设$ T_{\varphi, \psi}:H(p, \mu)\rightarrow Z^\alpha $为有界算子, 则存在常数$ C $使$ \forall f\in H(p, \mu), $有$ \|T_{\varphi, \psi}(f)\|_{Z^\alpha}\leq C\|f\|_{H(p, \mu)}. $

固定$ \omega\in D $及常数$ a, b $, 取

$f_\omega(z) = \frac{a(1-|\varphi(\omega)|^2)}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+1}} -\frac{(1-|\varphi(\omega)|^2)^2}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+2}} +\frac{b(1-|\varphi(\omega)|^2)^3}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+3}}. $

(3.7)

由文献[18]中的引理2.2和文献[19]中的定理1.12有:

$\begin{align*}&\|f_\omega\|^p_{H(p, \mu)} = \int_D|f_\omega(z)|^p\frac{\mu^p(|z|)}{1-|z|}dA(z)\\ \leq& C\int_D[\frac{|a|^p(1-|\varphi(\omega)|^2)^p}{\mu^p(|\varphi(\omega)|)|1-\overline{\varphi(\omega)}z|^{p+1}} +\frac{(1-|\varphi(\omega)|^2)^{2p}}{\mu^p(|\varphi(\omega)|)|1-\overline{\varphi(\omega)}z|^{1+2p}} +\frac{|b|^p(1-|\varphi(\omega)|^2)^{3p}}{\mu^p(|\varphi(\omega)|)|1-\overline{\varphi(\omega)}z|^{1+3p}}]\frac{\mu^p(|z|)}{1-|z|}dA(z) \\ \leq& C\int_D\{\frac{(1-|z|^2)^{sp}}{(1-|\varphi(\omega)|^2)^{sp}}+\frac{(1-|z|^2)^{tp}}{(1-|\varphi(\omega)|^2)^{tp}}\} [\frac{|a|^p(1-|\varphi(\omega)|^2)^p}{|1-\overline{\varphi(\omega)}z|^{p+1}({1-|z|})} \\ &+\frac{(1-|\varphi(\omega)|^2)^{2p}}{|1-\overline{\varphi(\omega)}z|^{1+2p}({1-|z|})} +\frac{|b|^p(1-|\varphi(\omega)|^2)^{3p}}{|1-\overline{\varphi(\omega)}z|^{1+3p}({1-|z|})}]dA(z) \\ \leq& 2C\{|a|^p[(1-|\varphi(\omega)|^2)^{p-sp}\int_D\frac{(1-|z|^2)^{sp-1}}{|1-\overline{\varphi(\omega)}z|^{p+1}}dA(z) +(1-|\varphi(\omega)|^2)^{p-tp}\int_D\frac{(1-|z|^2)^{tp-1}}{|1-\overline{\varphi(\omega)}z|^{p+1}}dA(z)] \\ &+[(1-|\varphi(\omega)|^2)^{2p-sp}\int_D\frac{(1-|z|^2)^{sp-1}}{|1-\overline{\varphi(\omega)}z|^{2p+1}}dA(z) +(1-|\varphi(\omega)|^2)^{2p-tp}\int_D\frac{(1-|z|^2)^{tp-1}}{|1-\overline{\varphi(\omega)}z|^{2p+1}}dA(z)] \\ &+|b|^p[(1-|\varphi(\omega)|^2)^{3p-sp}\int_D\frac{(1-|z|^2)^{sp-1}}{|1-\overline{\varphi(\omega)}z|^{3p+1}}dA(z) +(1-|\varphi(\omega)|^2)^{3p-tp}\int_D\frac{(1-|z|^2)^{tp-1}}{|1-\overline{\varphi(\omega)}z|^{3p+1}}dA(z)]\} \\ \approx& 4C_1(|a|^p+1+|b|^p) \leq C.\end{align*} $

(3.8)

所以由(3.8)式知$ f_\omega\in H(p, \mu), $且$ \|f_\omega\|_{H(p, \mu)}\leq C $. 直接计算有:

$f'_\omega(z) = \overline{\varphi(\omega)}\{\frac{a(\frac{1}{p}+1)(1-|\varphi(\omega)|^2)}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+2}} -\frac{(\frac{1}{p}+2)(1-|\varphi(\omega)|^2)^2}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+3}} +\frac{b(\frac{1}{p}+3)(1-|\varphi(\omega)|^2)^3}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+4}}\}. $

(3.9)

$f''_\omega(z) = (\overline{\varphi(\omega)})^2\{\frac{a(\frac{1}{p}+1)(\frac{1}{p}+2)(1-|\varphi(\omega)|^2)} {\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+3}} -\frac{(\frac{1}{p}+2)(\frac{1}{p}+3)(1-|\varphi(\omega)|^2)^2}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+4}}\\ +\frac{b(\frac{1}{p}+3)(\frac{1}{p}+4)(1-|\varphi(\omega)|^2)^3}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+5}}\}. $

(3.10)

由(3.7)式和(3.9)-(3.10)式得:

$f_\omega(\varphi(\omega)) = \frac{a-1+b}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}}}, $

(3.11)

$f'_\omega(\varphi(\omega)) = \frac{\overline{\varphi(\omega)}[a(\frac{1}{p}+1)-(\frac{1}{p}+2)+b(\frac{1}{p}+3)]} {\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+1}}, $

(3.12)

$f''_\omega(\varphi(\omega)) = \frac{(\overline{\varphi(\omega)})^2[a(\frac{1}{p}+1)(\frac{1}{p}+2)-(\frac{1}{p}+2)(\frac{1}{p}+3)+b(\frac{1}{p}+3)(\frac{1}{p}+4)]} {\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+2}}. $

(3.13)

设$ v = \frac{1}{p}+1, $则由$ av-(v+1)+b(v+2) = 0, av(v+1)-(v+1)(v+2)+b(v+2)(v+3) = 0 $得: $ a = \frac{v+1}{2v}, b = \frac{v+1}{2v+4} $, 此时$ a-1+b = \frac{1}{v(v+2)} = \frac{p^2}{(1+p)(1+3p)}. $由(3.11)-(3.13)式有

$f_\omega(\varphi(\omega)) = \frac{\frac{p^2}{(1+p)(1+3p)}}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}}}, f'_\omega(\varphi(\omega)) = f''_\omega(\varphi(\omega)) = 0. $

(3.14)

由$ T_{\varphi, \psi} $是$ H(p, \mu) $到$ Z^\alpha $的有界算子及(3.14)式知

$ C\geq\|(T_{\varphi, \psi}f_\omega)(z)\|_{Z^\alpha}\geq (1-|\omega|^2)^\alpha |(T_{\varphi, \psi}f_\omega)''(\omega)| = (1-|\omega|^2)^\alpha \frac{|\psi''(\omega)|\frac{p^2}{(1+p)(1+3p)}}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}}}. $

即

$\frac{(1-|\omega|^2)^\alpha|\psi''(\omega)|}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}}}\leq C. $

(3.15)

由$ \omega $的任意性及(3.15)式知(3.1)式成立.

下面证明(3.2)式成立. 固定$ \omega\in D $及常数$ c, d $, 取

$g_\omega(z) = -\frac{1-|\varphi(\omega)|^2}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+1}} +\frac{c(1-|\varphi(\omega)|^2)^2}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+2}} +\frac{d(1-|\varphi(\omega)|^2)^3}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+3}}. $

(3.16)

类似$ f_\omega(z) $的证明知$ g_\omega\in H(p, \mu), $且$ \|g_\omega\|_{H(p, \mu)}\leq C. $直接计算得

$g'_\omega(z) = \frac{\overline{\varphi(\omega)}}{\mu(|\varphi(\omega)|)}[-\frac{(\frac{1}{p}+1)(1-|\varphi(\omega)|^2)} {(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+2}} +\frac{c(\frac{1}{p}+2)(1-|\varphi(\omega)|^2)^2}{(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+3}} +\frac{d(\frac{1}{p}+3)(1-|\varphi(\omega)|^2)^3}{(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+4}}], $

(3.17)

$ \begin{align} g''_\omega(z) = &\frac{(\overline{\varphi(\omega)})^2}{\mu(|\varphi(\omega)|)}[-\frac{(\frac{1}{p}+1) (\frac{1}{p}+2)(1-|\varphi(\omega)|^2)} {(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+3}}\\ &+\frac{c(\frac{1}{p}+2)(\frac{1}{p}+3)(1-|\varphi(\omega)|^2)^2}{(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+4}} +\frac{d(\frac{1}{p}+3)(\frac{1}{p}+4)(1-|\varphi(\omega)|^2)^3}{(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+5}}]. \end{align} $

(3.18)

且由(3.16)-(3.18)式有

$g_\omega(\varphi(\omega)) = \frac{c+d-1}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}}}, $

(3.19)

$g'_\omega(\varphi(\omega)) = \frac{\overline{\varphi(\omega)}[-(\frac{1}{p}+1)+c(\frac{1}{p}+2)+d(\frac{1}{p}+3)]} {\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+1}}, $

(3.20)

$g''_\omega(\varphi(\omega)) = \frac{(\overline{\varphi(\omega)})^2[-(\frac{1}{p}+1)(\frac{1}{p}+2)+c(\frac{1}{p}+2)(\frac{1}{p}+3) +d(\frac{1}{p}+3)(\frac{1}{p}+4)]} {\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+2}}. $

(3.21)

设$ v = \frac{1}{p}+1, $则由$ c+d-1 = 0, -v(v+1)+c(v+1)(v+2)+d(v+2)(v+3) = 0 $得$ c = \frac{2v+3}{v+2}, d = -\frac{v+1}{v+2} $, 此时$ -v+c(v+1)+d(v+2) = \frac{1}{v+2} = \frac{p}{1+3p}. $由(3.19)-(3.21)式有

$g'_\omega(\varphi(\omega)) = \frac{\frac{p}{(1+3p)}\overline{\varphi(\omega)}}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}}}, \quad g_\omega(\varphi(\omega)) = g''_\omega(\varphi(\omega)) = 0. $

(3.22)

由$ T_{\varphi, \psi} $是$ H(p, \mu) $到$ Z^\alpha $的有界算子及(3.22)式知

$ \begin{align} C\geq&\|(T_{\varphi, \psi}g_\omega)(z)\|_{Z^\alpha}\geq (1-|\omega|^2)^\alpha |(T_{\varphi, \psi}g_\omega)''(\omega)| \\ = &(1-|\omega|^2)^\alpha|2\psi'(\omega)\varphi'(\omega) +\psi(\omega)\varphi''(\omega)||g'_\omega\varphi(\omega)| \\ = & \frac{(1-|\omega|^2)^\alpha|2\psi'(\omega)\varphi'(\omega) +\psi(\omega)\varphi''(\omega)||\varphi(\omega)|\frac{p}{(1+3p)}}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+1}}. \end{align} $

(3.23)

当$ |\varphi(\omega)|>\frac{1}{2} $时, 由(3.23)式有

$\frac{(1-|\omega|^2)^\alpha|2\psi'(\omega)\varphi'(\omega) +\psi(\omega)\varphi''(\omega)|}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+1}}\leq C. $

(3.24)

当$ |\varphi(\omega)|\leq\frac{1}{2} $时, 取$ g_\omega(z) = \frac{z}{\mu(|\varphi(\omega)|)}, $由文献[18]中的引理2.2, 直接计算得

$ \begin{align*} \|g_\omega\|_{H(p, \mu)}^p = &\int_D\frac{|z|^p\mu^p(|z|)}{\mu^p(|\varphi(\omega)|)(1-|z|)}dA(z) \\ \leq& 2C\int_D[\frac{(1-|z|^2)^{sp-1}}{(1-|\varphi(\omega)|^2)^{sp}}+\frac{(1-|z|^2)^{tp-1}}{(1-|\varphi(\omega)|^2)^{tp}}]dA(z) \\ \leq& \frac{2C\cdot2\pi}{(1-|\varphi(\omega)|^2)^{sp}}\int_0^1(1-r^2)^{sp-1}rdr+\frac{2C\cdot2\pi}{(1-|\varphi(\omega)|^2)^{tp}}\int_0^1(1-r^2)^{tp-1}rdr \\ = &\frac{2\pi C}{sp(1-|\varphi(\omega)|^2)^{sp}}+\frac{2\pi C}{tp(1-|\varphi(\omega)|^2)^{tp}}\leq C, \end{align*} $

所以$ g_\omega\in H(p, \mu), $且$ \|g_\omega\|_{H(p, \mu)}\leq C. $

因为$ T_{\varphi, \psi} $是$ H(p, \mu) $到$ Z^\alpha $的有界算子, 且$ g_\omega(\varphi(\omega)) = \frac{\varphi(\omega)}{\mu(|\varphi(\omega)|)}, g'_\omega(\varphi(\omega)) = \frac{1}{\mu(|\varphi(\omega)|)}, g''_\omega(\varphi(\omega)) = 0 $, 所以

$ C\geq\|(T_{\varphi, \psi}g_\omega)\|_{Z^\alpha}\geq(1-|\omega|^2)^\alpha |\psi''(\omega)g_\omega(\varphi(\omega))+(2\psi'(\omega)\varphi'(\omega)+\psi(\omega)\varphi''(\omega))g'_\omega(\varphi(\omega))|. $

从而由三角不等式有

$ \frac{(1-|\omega|^2)^\alpha |(2\psi'(\omega)\varphi'(\omega)+\psi(\omega)\varphi''(\omega))|} {\mu(|\varphi(\omega)|)}\leq C+\frac{(1-|\omega|^2)^\alpha |\psi''(\omega)\varphi(\omega)|}{\mu(|\varphi(\omega)|)}. $

上式两边同乘以$ \frac{1}{(1-|\varphi(\omega)|^2)^{\frac{1}{p}+1}}, $结合(3.1)式及$ \frac{1}{1-|\varphi(\omega)|^2}\leq\frac{4}{3} $有

$ \begin{align} &\frac{(1-|\omega|^2)^\alpha |(2\psi'(\omega)\varphi'(\omega)+\psi(\omega)\varphi''(\omega))|} {\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+1}} \\ \leq& \frac{C}{(1-|\varphi(\omega)|^2)^{\frac{1}{p}+1}}+\frac{(1-|\omega|^2)^\alpha |\psi''(\omega)}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}}}\frac{\varphi(\omega)|}{(1-|\varphi(\omega)|^2)} \leq C(\frac{4}{3})^{\frac{1}{p}+1}+\frac{2}{3}M\leq C. \end{align} $

(3.25)

结合(3.24)-(3.25)式知(3.2)式成立.

最后证明(3.3)式成立. 当$ |\varphi(\omega)|>\frac{1}{2} $时, 取

$ h_\omega(z) = \frac{e(1-|\varphi(\omega)|^2)}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+1}} +\frac{f(1-|\varphi(\omega)|^2)^2}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+2}}- \frac{(1-|\varphi(\omega)|^2)^3}{\mu(|\varphi(\omega)|)(1-\overline{\varphi(\omega)}z)^{\frac{1}{p}+3}}. $

易知$ h_\omega\in H(p, \mu), $且$ \|h_\omega\|_{H(p, \mu)}\leq C. $直接计算得

$ h_\omega(\varphi(\omega)) = \frac{e+f-1}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}}}, \quad h'_\omega(\varphi(\omega)) = \frac{\overline{\varphi(\omega)}[e(\frac{1}{p}+1)+f(\frac{1}{p}+2)-(\frac{1}{p}+3)]} {\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+1}}, $

$ h''_\omega(\varphi(\omega)) = \frac{(\overline{\varphi(\omega)})^2[e(\frac{1}{p}+1)(\frac{1}{p}+2)+f(\frac{1}{p}+2)(\frac{1}{p}+3) -(\frac{1}{p}+3)(\frac{1}{p}+4)]}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+2}}. $

设$ v = \frac{1}{p}+1, $则由$ e+f-1 = 0, ev+f(v+1)-(v+2) = 0 $得$ e = -1, f = 2 $, 此时$ e(\frac{1}{p}+1)(\frac{1}{p}+2)+f(\frac{1}{p}+2)(\frac{1}{p}+3) -(\frac{1}{p}+3)(\frac{1}{p}+4) = -2, h_\omega(\varphi(\omega)) = h'_\omega(\varphi(\omega)) = 0, h''_\omega(\varphi(\omega)) = \frac{-2(\overline{\varphi(\omega)})^2}{\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+2}}. $因为$ T_{\varphi, \psi} $是$ H(p, \mu) $到$ Z^\alpha $的有界算子, 所以

$ C\geq\|(T_{\varphi, \psi}h_\omega)\|_{Z^\alpha}\geq\frac{(1-|\omega|^2)^\alpha|\psi(\omega)(\varphi'(\omega))^2||\varphi(\omega)|^2} {\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+2} }, $

即

$\frac{(1-|\omega|^2)^\alpha|\psi(\omega)(\varphi'(\omega))^2|} {\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+2}}\leq C. $

(3.26)

当$ |\varphi(\omega)|\leq\frac{1}{2} $时, 取$ h_\omega(z) = \frac{z^2}{\mu(|\varphi(\omega)|)}, $易知$ h_\omega\in H(p, \mu), $且$ \|h_\omega\|_{H(p, \mu)}\leq C. $

因为$ T_{\varphi, \psi} $是$ H(p, \mu) $到$ Z^\alpha $的有界算子, 所以

$ \begin{align} C\geq&\|(T_{\varphi, \psi}h_\omega)\|_{Z^\alpha}\\ \geq&(1-|\omega|^2)^\alpha |\psi''(\omega)h_\omega(\varphi(\omega)) +(2\psi'(\omega)\varphi'(\omega)+\psi(\omega)\varphi''(\omega))h'_\omega(\varphi(\omega)) +\psi(\omega)(\varphi'(\omega))^2h''_\omega(\varphi(\omega))|. \end{align} $

(3.27)

又由$ h_\omega(\varphi(\omega)) = \frac{(\varphi(\omega))^2}{\mu(|\varphi(\omega)|)}, g'_\omega(\varphi(\omega)) = \frac{2\varphi(\omega)}{\mu(|\varphi(\omega)|)}, g''_\omega(\varphi(\omega)) = \frac{2}{\mu(|\varphi(\omega)|)} $及三角不等式和(3.27)式有

$ \begin{align*} &\frac{2(1-|\omega|^2)^\alpha|\psi(\omega)(\varphi'(\omega))^2|}{\mu(|\varphi(\omega)|)}\\ \leq& C+\frac{(1-|\omega|^2)^\alpha |\psi''(\omega)||\varphi(\omega)|^2}{\mu(|\varphi(\omega)|)} +\frac{2(1-|\omega|^2)^\alpha |(2\psi'(\omega)\varphi'(\omega)+\psi(\omega)\varphi''(\omega))||\varphi(\omega)|} {\mu(|\varphi(\omega)|)}. \end{align*} $

上式两边同乘以$ \frac{1}{(1-|\varphi(\omega)|^2)^{\frac{1}{p}+2}}, $结合(3.1)-(3.2)式及$ \frac{1}{1-|\varphi(\omega)|^2}\leq\frac{4}{3} $有

$\frac{(1-|\omega|^2)^\alpha |\psi(\omega)||\varphi'(\omega)|^2} {\mu(|\varphi(\omega)|)(1-|\varphi(\omega)|^2)^{\frac{1}{p}+2}}\leq C. $

(3.28)

结合(3.26)式和(3.28)式知(3.3)式成立. 定理3.1证毕.

定理3.2  设$ \varphi\in H(D, D), \psi\in H(D), $则$ T_{\varphi, \psi}:H(p, \mu)\rightarrow Z^\alpha $为紧算子的充要条件是$ T_{\varphi, \psi} $为有界算子且下列条件同时成立:

$\lim\limits_{|\varphi(z)|\rightarrow1^-} \frac{(1-|z|^2)^\alpha|\psi''(z)|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}}} = 0, $

(3.29)

$\lim\limits_{|\varphi(z)|\rightarrow1^-} \frac{(1-|z|^2)^\alpha|2\psi'(z)\varphi'(z)+\psi(z)\varphi''(z)|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}+1}} = 0, $

(3.30)

$\lim\limits_{|\varphi(z)|\rightarrow1^-} \frac{(1-|z|^2)^\alpha|\psi(z)(\varphi'(z))^2|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}+2}} = 0. $

(3.31)

证  先证充分性  假设(3.29)-(3.31)式同时成立, 则$ \forall \varepsilon >0 $, 存在$ 0<\delta<1 $, 当$ \delta<|\varphi(z)|<1 $时, 有

$\frac{(1-|z|^2)^\alpha}{\mu(|\varphi(z)|)}[\frac{|\psi''(z)|}{(1-|\varphi(z)|^2)^{\frac{1}{p}}}+ \frac{|2\psi'(z)\varphi'(z)+\psi(z)\varphi''(z)|}{(1-|\varphi(z)|^2)^{\frac{1}{p}+1}} +\frac{|\psi(z)(\varphi'(z))^2|}{(1-|\varphi(z)|^2)^{\frac{1}{p}+2}}]<\varepsilon. $

(3.32)

由$ T_{\varphi, \psi} $为有界算子及(3.1)式有

$ \begin{align} &(1-|z|^2)^\alpha|\psi''(z)|\leq \frac{(1-|z|^2)^\alpha|\psi''(z)|}{(1-|\varphi(z)|)^s}\frac{\mu(|\varphi(z)|)}{\mu(|\varphi(z)|)} \\ \leq& \frac{(1-|z|^2)^\alpha|\psi''(z)|}{\mu(|\varphi(z)|)}\frac{\mu(\delta)}{(1-\delta)^s} \leq \frac{(1-|z|^2)^\alpha|\psi''(z)|}{\mu(|\varphi(z)|)(1-|\varphi(z)|^2)^{\frac{1}{p}}}\frac{\mu(\delta)}{(1-\delta)^s} \leq C. \end{align} $

(3.33)

同理由(3.2)式和(3.3)式有

$(1-|z|^2)^\alpha|2\psi'(z)\varphi'(z)+\psi(z)\varphi''(z)|\leq C, (1-|z|^2)^\alpha|\psi(z)(\varphi'(z))^2|\leq C. $

(3.34)

设$ \{f_{j}\} $是在$ D $的任一紧子集上一致趋于0且满足$ \|f_{j}\|_{H(p, \mu)}\leq1 $的任一解析函数序列, 则$ \{f'_j\} $和$ \{f''_j\} $在$ E = \{\omega:|\omega|\leq \delta\} $上一致收敛于0. 由(3.33)式和(3.34)式知, 存在正整数$ N $, 当$ j>N $时, 有

$ \begin{align} &|(T_{\varphi, \psi}f_j)(0)|+|(T_{\varphi, \psi}f_j)'(0)|+\sup\limits_{|\varphi(z)|\leq \delta}(1-|z|^2)^\alpha|(T_{\varphi, \psi}f_j)''(z)| \\ \leq & |\psi(0)||f_j(\varphi(0))|+ |\psi'(0)||f_j(\varphi(0))|+ |\psi(0)||f'_j(\varphi(0))||\varphi'(0)| \\ &+\sup\limits_{|\varphi(z)|\leq \delta}(1-|z|^2)^\alpha[|\psi''(z)f_j(\varphi(z))|+|2\psi'(z)\varphi'(z)+\psi(z)\varphi''(z)| \\ &\times|f'_j(\varphi(z))| +|\psi(z)(\varphi'(z))^2||f''_j(\varphi(z))|] \\ \leq & |\psi(0)||f_j(\varphi(0))|+ |\psi'(0)||f_j(\varphi(0))|+ |\psi(0)||f'_j(\varphi(0))||\varphi'(0)| \\ &+C[\sup\limits_{|\varphi(z)|\leq \delta}|f_j(\varphi(z))|+\sup\limits_{|\varphi(z)|\leq \delta}|f'_j(\varphi(z))| +\sup\limits_{|\varphi(z)|\leq \delta}|f''_j(\varphi(z))|] \\ \leq & C\varepsilon. \end{align} $

(3.35)

由(3.32)式和(3.35)式可得, 当$ j>N $时, 有

$ \begin{align*} \|T_{\varphi, \psi}f_j\|_{Z^\alpha} = &|(T_{\varphi, \psi}f_j)(0)|+|(T_{\varphi, \psi}f_j)'(0)| +\sup\limits_{z\in D}(1-|z|^2)^\alpha|(T_{\varphi, \psi}f_j)''(z)| \\ \leq& [|\psi(0)|+|\psi'(0)|]|f_j(\varphi(0))|+|\psi(0)||\varphi'(0)||f'_j(\varphi(0))| \\ &+\sup\limits_{\delta<|\varphi(z)|<1} (1-|z|^2)^\alpha|(T_{\varphi, \psi}f_j)''(z)|+\sup\limits_{|\varphi(z)|\leq \delta}(1-|z|^2)^\alpha|(T_{\varphi, \psi}f_j)''(z)| \\ \leq& C\varepsilon+C\|f_j\|_{H(p, \mu)}\varepsilon\leq 2C\varepsilon. \end{align*} $

由$ \varepsilon $的任意性知$ \lim\limits_{j\rightarrow \infty}\|T_{\varphi, \psi}f_{j}\|_{Z^\alpha} = 0 $, 由引理2.3知$ T_{\varphi, \psi} $为$ H(p, \mu) $到$ Z^\alpha $的紧算子.

必要性设$ T_{\varphi, \psi} $为$ H(p, \mu) $到$ Z^\alpha $的紧算子, 则$ T_{\varphi, \psi} $为$ H(p, \mu) $到$ Z^\alpha $的有界算子.

先证(3.29)成立. 对于$ D $中满足$ \lim\limits_{j\rightarrow\infty}|\varphi(z_j)| = 1 $的任意序列$ \{z_j\}, (j = 1, 2, \cdots) $, 取

$ f_j(z) = \frac{a(1-|\varphi(z_j)|^2)}{\mu(|\varphi(z_j)|)(1-\overline{\varphi(z_j)}z)^{\frac{1}{p}+1}} -\frac{(1-|\varphi(z_j)|^2)^2}{\mu(|\varphi(z_j)|)(1-\overline{\varphi(z_j)}z)^{\frac{1}{p}+2}} +\frac{b(1-|\varphi(z_j)|^2)^3}{\mu(|\varphi(z_j)|)(1-\overline{\varphi(z_j)}z)^{\frac{1}{p}+3}}. $

类似(3.7)式中的讨论知$ \|f_j\|_{H(p, \mu)}\leq C, $ $ f_j\in H(p, \mu) $, 且当$ a = \frac{1+2p}{2+2p}, b = \frac{1+2p}{2+6p} $时, $ f_j(\varphi(z_j)) = \frac{\frac{p^2}{(1+p)(1+3p)}}{\mu(|\varphi(z_j)|)(1-|\varphi(z_j)|^2)^{\frac{1}{p}}}, f'_j(\varphi(z_j)) = f''_j(\varphi(z_j)) = 0. $

下面证明$ \{f_{j}\} $在$ D $的任一紧子集上一致收敛于0.对任意$ 0<r<1 $, 有

$ \sup\limits_{|z|\leq r}|f_{j}(z)|\leq \sup\limits_{|z|\leq r} \frac{C(1-|\varphi(z_j)|^2)}{\mu(|\varphi(z_j)|)(1-|z|)^{\frac{1}{p}+1}} \leq \frac{C_1(1-|\varphi(z_j)|^2)}{(1-r)^{\frac{1}{p}+1}}\rightarrow0, \quad (j\rightarrow\infty). $

所以$ \{f_{j}\} $在$ D $的任一紧子集上一致收敛于0.由引理2.3有$ \lim\limits_{j\rightarrow \infty}\|(T_{\varphi, \psi}f_{j})\|_{Z^\alpha} = 0 $, 即$ \forall\varepsilon>0, $存在正整数$ N $, 当$ j>N $时, 有

$ C\varepsilon\geq \|(T_{\varphi, \psi}f_j)\|_{Z^\alpha}\geq (1-|z_j|^2)^\alpha|\psi''(z_j)f_j(\varphi(z_j))| = \frac{\frac{p^2}{(1+p)(1+3p)}(1-|z_j|^2)^\alpha|\psi''(z_j)|} {\mu(|\varphi(z_j)|)(1-|\varphi(z_j)|^2)^{\frac{1}{p}}}. $

由此及$ \lim\limits_{j\rightarrow\infty}|\varphi(z_j)| = 1 $得

$ \lim\limits_{|\varphi(z_j)|\rightarrow 1^-} \frac{(1-|z_j|^2)^\alpha|\psi''(z_j)|} {\mu(|\varphi(z_j)|)(1-|\varphi(z_j)|^2)^{\frac{1}{p}}} = 0. $

所以(3.29)式成立.

下面证明(3.30)式成立. 取

$ g_j(z) = -\frac{1-|\varphi(z_j)|^2}{\mu(|\varphi(z_j)|)(1-\overline{\varphi(z_j)}z)^{\frac{1}{p}+1}} +\frac{\frac{2+5p}{1+3p}(1-|\varphi(z_j)|^2)^2}{\mu(|\varphi(z_j)|)(1-\overline{\varphi(z_j)}z)^{\frac{1}{p}+2}} +\frac{\frac{-(1+2p)}{1+3p}(1-|\varphi(z_j)|^2)^3}{\mu(|\varphi(z_j)|)(1-\overline{\varphi(z_j)}z)^{\frac{1}{p}+3}}. $

类似(3.16)式中的讨论知$ \|g_j\|_{H(p, \mu)}\leq C, $ $ g_j\in H(p, \mu) $, 且$ g_j(\varphi(z_j)) = g''_j(\varphi(z_j)) = 0, g'_j(\varphi(z_j)) = \frac{\frac{p}{1+3p}\overline{\varphi(z_j)}}{\mu(|\varphi(z_j)|)(1-|\varphi(z_j)|^2)^{\frac{1}{p}+1}}. $

显然, $ \{g_{j}\} $在$ D $的任一紧子集上一致收敛于0.由引理2.3有$ \lim\limits_{j\rightarrow \infty}\|(T_{\varphi, \psi}g_{j})\|_{Z^\alpha} = 0 $, 即$ \forall\varepsilon>0, $存在正整数$ N $, 当$ j>N $时, 有

$ \begin{eqnarray*} C\varepsilon\geq \|(T_{\varphi, \psi}g_j)\|_{Z^\alpha}&\geq& (1-|z_j|^2)^\alpha|2\psi'(z_j)\varphi'(z_j)+\psi(z_j)\varphi''(z_j)|g'_j(\varphi(z_j))| \\ &\geq& \frac{C_1|\overline{\varphi(z_j)}|(1-|z_j|^2)^\alpha|2\psi'(z_j)\varphi'(z_j)+\psi(z_j)\varphi''(z_j)|} {\mu(|\varphi(z_j)|)(1-|\varphi(z_j)|^2)^{\frac{1}{p}+1}}. \end{eqnarray*} $

由此及$ \lim\limits_{j\rightarrow\infty}|\varphi(z_j)| = 1, \frac{1}{2}<|\varphi(z_j)|\rightarrow1 $得

$ \lim\limits_{|\varphi(z_j)|\rightarrow 1^-} \frac{(1-|z_j|^2)^\alpha|2\psi'(z_j)\varphi'(z_j)+\psi(z_j)\varphi''(z_j)|} {\mu(|\varphi(z_j)|)(1-|\varphi(z_j)|^2)^{\frac{1}{p}+1}} = 0. $

所以(3.30)式成立.

最后证(3.31)式成立. 取

$ h_j(z) = -\frac{1-|\varphi(z_j)|^2}{\mu(|\varphi(z_j)|)(1-\overline{\varphi(z_j)}z)^{\frac{1}{p}+1}} +\frac{2(1-|\varphi(z_j)|^2)^2}{\mu(|\varphi(z_j)|)(1-\overline{\varphi(z_j)}z)^{\frac{1}{p}+2}} +\frac{-(1-|\varphi(z_j)|^2)^3}{\mu(|\varphi(z_j)|)(1-\overline{\varphi(z_j)}z)^{\frac{1}{p}+3}}. $

类似对定理3.1中的$ h_\omega $的讨论知$ \|h_j\|_{H(p, \mu)}\leq C, $ $ h_j\in H(p, \mu) $, 且$ h_j(\varphi(z_j)) = h'_j(\varphi(z_j)) = 0, h''_j(\varphi(z_j)) = \frac{-2(\overline{\varphi(z_j)})^2}{\mu(|\varphi(z_j)|)(1-|\varphi(z_j)|^2)^{\frac{1}{p}+2}}. $

易知, $ \{h_{j}\} $在$ D $的任一紧子集上一致收敛于0.由引理2.3有$ \lim\limits_{j\rightarrow \infty}\|(T_{\varphi, \psi}h_{j})\|_{Z^\alpha} = 0 $, 即$ \forall\varepsilon>0, $存在正整数$ N $, 当$ j>N $时, 有

$ \begin{eqnarray*} C\varepsilon&\geq& \|(T_{\varphi, \psi}h_j)\|_{Z^\alpha}\geq (1-|z_j|^2)^\alpha|\psi(z_j)(\varphi'(z_j))^2||h''_j(\varphi(z_j))| \\ & = & \frac{2|\overline{\varphi(z_j)}|^2(1-|z_j|^2)^\alpha|\psi(z_j)(\varphi'(z_j))^2|} {\mu(|\varphi(z_j)|)(1-|\varphi(z_j)|^2)^{\frac{1}{p}+2}}. \end{eqnarray*} $

由此及$ \lim\limits_{j\rightarrow\infty}|\varphi(z_j)| = 1, \frac{1}{2}<|\varphi(z_j)|\rightarrow1 $得

$ \lim\limits_{|\varphi(z_j)|\rightarrow 1^-} \frac{(1-|z_j|^2)^\alpha|\psi(z_j)(\varphi'(z_j))^2|} {\mu(|\varphi(z_j)|)(1-|\varphi(z_j)|^2)^{\frac{1}{p}+2}} = 0. $

所以(3.31)式成立. 定理3.2证毕.