设$ \mathbb{D} $为复平面$ \mathbb{C} $中的单位圆盘, $ H(\mathbb{D}) $为$ \mathbb{D} $上所有解析函数构成的集合, $ S(\mathbb{D}) $表示$ \mathbb{D} $上解析自映射构成的集合. 定义在$ \mathbb{D} $上的调和函数$ f $是复值函数且满足

$ \Delta f:=4\frac{\partial^2f}{\partial z \partial\overline{z}}\equiv0, $

记$ \mathcal{H}ar(\mathbb{D}) $为$ \mathbb{D} $上所有调和函数组成的集合. 由调和函数和解析函数的关系知$ f\in\mathcal{H}ar(\mathbb{D}). $当且仅当$ f $有唯一分解$ f=g+\overline{h} $, 其中$ g, h\in H(\mathbb{D}) $且$ h(0)=0. $记$ \varphi_{a}(z):=(a-z)/(1-\bar{a} z) $为$ \mathbb{D} $上交换0和$ a\in \mathbb{D} $的对合自同构. 对于给定的$ z, w \in \mathbb{D} $, $ z $与$ w $之间的伪双曲度量为

$ \begin{eqnarray*} \rho(z, w):=\left|\varphi_{w}(z)\right|=\left|\frac{z-w}{1-\bar{w}z}\right|. \end{eqnarray*} $

为简便起见, 对于$ \varphi, \psi \in S(\mathbb{D}) $, 记$ \rho(z):=\rho(\varphi(z), \psi(z)) $. 对于每个$ \varphi \in S(\mathbb{D}) $都可以定义一个复合算子$ C_{\varphi} $为$ C_{\varphi} f =f\circ \varphi $, $ f\in \mathcal{H}ar(\mathbb{D}). $此时这样的算子保持了调和性. 对于任意两个赋范线性空间$ X $和$ Y $, 线性算子$ T:X\longrightarrow Y $是有界的, 当且仅当存在$ C>0 $使得对于任意$ f\in X $成立$ \|Tf\|_Y\leq C\|f\|_X. $如果$ T $把$ X $中的有界集映成$ Y $中的列紧集, 则$ T $是紧算子. 长期以来, 一些全纯函数空间上复合算子$ C_\varphi $与复合算子差分$ C_\varphi-C_\psi $的算子性质得到了广泛的研究, 可参考^[1–15]. 但是对于调和函数构成的空间上相关的研究仍然有限. 因此, 本文主要研究从(小)调和Zygmund型空间到(小)调和Bloch型空间的复合算子$ C_\varphi $与复合算子差分$ C_\varphi-C_\psi $的有界性与紧性.

对于$ 0<\alpha<\infty $, 调和Zygmund型空间$ \mathcal{Z}_{\mathcal{H}}^{\alpha} $包含所有$ f \in \mathcal{H}ar(\mathbb{D}) $满足范数

$ \begin{eqnarray*} \|f\|_{\mathcal{Z}_{\mathcal{H}}^{\alpha}}=|f(0)|+\left|\frac{\partial f}{\partial z}(0)\right|+\left|\frac{\partial f}{\partial\overline{z}}(0)\right|+\sup\limits_{z\in\mathbb{D}}(1-|z|^2)^\alpha \left(\left|\frac{\partial^2f}{\partial z^2}(z)\right|+\left|\frac{\partial^2f}{\partial\overline{z}^2}(z)\right|\right)<\infty \end{eqnarray*} $

成立. 进一步地, 小调和Zygmund型空间$ \mathcal{Z}_{\mathcal{H}, 0}^{\alpha} $定义为

$ \begin{eqnarray*} \mathcal{Z}_{\mathcal{H}, 0}^{\alpha}=\left\{f \in \mathcal{Z}_{\mathcal{H}}^{\alpha}: \lim _{|z| \rightarrow 1}(1-|z|^2)^\alpha \left(\left|\frac{\partial^2f}{\partial z^2}(z)\right|+\left|\frac{\partial^2f}{\partial\overline{z}^2}(z)\right|\right)=0\right\}. \end{eqnarray*} $

对于$ 0<\beta<\infty $, 调和Bloch型空间$ \mathcal{B}_{\mathcal{H}}^{\beta} $包含所有$ f \in \mathcal{H}ar(\mathbb{D}) $使得范数

$ \begin{eqnarray*} \|f\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}=|f(0)|+\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left(\left|\frac{\partial f}{\partial z}(z)\right|+\left|\frac{\partial f}{\partial\bar{z}}(z)\right|\right)<\infty \end{eqnarray*} $

成立. 进而小调和Bloch型空间$ \mathcal{B}_{\mathcal{H}, 0}^{\beta} $定义为

$ \begin{eqnarray*} \mathcal{B}_{\mathcal{H}, 0}^{\beta}=\left\{f \in \mathcal{B}_{\mathcal{H}}^{\beta}: \lim _{|z| \rightarrow 1}\left(1-|z|^{2}\right)^{\beta} \left(\left|\frac{\partial f}{\partial z}(z)\right|+\left|\frac{\partial f}{\partial\bar{z}}(z)\right|\right)=0\right\}. \end{eqnarray*} $

特别地, 当$ f\in H(\mathbb{D}) $时, $ \frac{\partial f}{\partial z}=f^\prime $且$ \frac{\partial f}{\partial\overline{z}}=\frac{\partial^2f}{\partial\overline{z}^2}=0. $于是, 对于所有的$ 0<\alpha<\infty $, $ \mathcal{Z}_{\mathcal{H}}^{\alpha} $中所有解析函数组成的空间是经典的Zygmund型空间$ \mathcal{Z}^{\alpha} $, 其范数为

$ \begin{eqnarray*} \|f\|_{\mathcal{Z}^{\alpha}}=|f(0)|+|f^\prime(0)|+\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\alpha}\left|{f}''(z)\right|. \end{eqnarray*} $

对于所有的$ 0<\beta<\infty $, $ \mathcal{B}_{\mathcal{H}}^{\beta} $中所有解析函数组成的空间是经典的Bloch型空间$ \mathcal{B}^{\beta} $, 其范数为$ \|f\|_{\mathcal{B}^{\beta}}=|f(0)|+\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left|f^\prime(z)\right|. $由[16, 定理19]知, 对于$ 0<\alpha<\infty, f\in \mathcal{H}ar(\mathbb{D}) $,

$ \begin{eqnarray} \|f\|_{\mathcal{B}_{\mathcal{H}}^{\alpha}}\approx \|f\|_{\mathcal{Z}_{\mathcal{H}}^{\alpha+1}}. \end{eqnarray} $

(1.1)

关于上述空间的更多资料, 请参阅^{[13, 14, 17–22]}.

本文的结构如下：在第二节中, 我们引用了一些相关引理为后续证明做铺垫；在第三节中刻画了$ \alpha>1, \beta>0 $时复合算子$ C_{\varphi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $的有界性与紧性；在第四节中建立了复合算子差分$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $相关性质的等价刻画.

在本文中, 符号$ A \lesssim B $与$ A \gtrsim B $意味着存在正整数$ C $（其确切值可能不同）, 使得$ A \leqslant C B $与$ A \geqslant C B $分别成立. $ A\approx B $当且仅当$ A \lesssim B $且$ A \gtrsim B $.

在本节中, 我们提供了几个引理用于证明第二节和第三节中的定理.

引理2.1   [8, 引理2.1] 设$ 1<\alpha<\infty $. 对于任意$ f \in \mathcal{Z^{\alpha}} $, 有

$ \begin{eqnarray*} \left|\left(1-|z|^{2}\right)^{\alpha-1} f^{\prime}(z)-\left(1-|w|^{2}\right)^{\alpha-1} f^{\prime}(w)\right|\lesssim\|f\|_{\mathcal{Z_{\alpha}}} \rho(z, w), \quad z, w \in \mathbb{D}. \end{eqnarray*} $

对[21, 定理1.14]稍作修改可得到如下引理.

引理2.2    设$ \alpha>1, \beta>0, $且$ T: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $是一个有界线性算子, 则$ T $是紧算子当且仅当函数序列$ \left\{f_{n}\right\} $在$ \mathcal{Z}_{\mathcal{H}}^{\alpha} $上是有界的, 且当$ f_{n} $在$ \mathbb{D} $的紧子集上一致收敛到$ 0 $时, 有$ \left \| Tf_{n} \right \| _{\mathcal{B_{\mathcal{H}}^{\beta }}}\rightarrow 0, n \rightarrow \infty . $

引理2.3   [23, 引理1.1] 设$ \alpha>1 $, 则对于每个$ f \in \mathcal{Z^{\alpha}} $, 有$ \left|f^{\prime}(z)\right| \lesssim \frac{\|f\|_{\mathcal{Z^{\alpha}}}}{\left(1-|z|^{2}\right)^{\alpha-1}}. $

在本节中, 主要研究了复合算子$ C_{\varphi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $的相关性质, 其结论将会应用于下一节复合算子差分$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $的相关性质中. 对于$ \alpha>1 $, $ \beta>0 $与$ \varphi \in S(\mathbb{D}) $, 记

$ \begin{eqnarray*} \varphi_{\#}^{\alpha, \beta}(z):=\frac{\left(1-|z|^{2}\right)^{\beta}\varphi^{\prime}(z)}{\left(1-|\varphi(z)|^{2}\right)^{\alpha-1}}. \end{eqnarray*} $

定理3.1    设$ \alpha>1 $, $ \beta>0 $且$ \varphi \in S\left( \mathbb{D} \right) $, 则下列条件等价:

$ (1) $$ C_{\varphi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $是有界的.

$ (2) $$ \sup\limits_{j\in \mathbb{N}}\|C_{\varphi}P_{j}\|_\mathcal{B_{\mathcal{H}}^{\beta}}<\infty, P_{j}=(z^{j}+\overline{z}^{j} )j^{\alpha}, j\in \mathbb{N}. $

$ (3) $$ \sup\limits_{z \in \mathbb{D}}|\varphi_{\#}^{\alpha, \beta}(z)|<\infty. $

证   首先证明$ (1)\Rightarrow (2) $. 假设$ C_{\varphi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $是有界的, 从而对任意$ f \in \mathcal{Z}_{\mathcal{H}}^{\alpha} $, 有$ ||C_{\varphi}f||_{B_{\mathcal{H}}^{\beta}}\lesssim||f||_{\mathcal{Z}_{\mathcal{H}}^{\alpha}} $. 由计算得$ P_j \in \mathcal{Z}_{\mathrm{\mathcal{H}}}^{\alpha} $, 于是对任意$ j\in \mathbb{N} $成立$ ||C_{\varphi}P_{j}||_{B_{\mathcal{H}}^{\beta}}\lesssim||P_{j}||_{\mathcal{Z}_{\mathcal{H}}^{\alpha}}<\infty. $故(2)成立.

现假设$ (3) $成立.设$ f \in \mathcal{Z}_{\mathcal{H}}^{\alpha} $, 根据(1.1), 有

$ \begin{align} \left \| C_{\varphi}f \right \| _{\mathcal{B}_{\mathcal{H}}^{\beta}} &=|f(\varphi(0))|+\sup\limits_{z\in \mathbb{D} }(1-|z|^{2})^{\beta}\left(\left|\frac{\partial(C_\varphi f)}{\partial z}(z)\right|+\left|\frac{\partial(C_\varphi f)}{\partial\overline{z}}(z)\right|\right) \\ &=|f(\varphi(0))|+\sup\limits_{z\in \mathbb{D} }(1-|z|^{2})^{\beta} \left | {\varphi}' (z) \right | \left ( \left | \frac{\partial f}{\partial z}(\varphi(z)) \right | + \left | \frac{\partial f}{\partial\overline{z}}(\varphi(z)) \right | \right )\\ &\le |f(\varphi(0))|+\sup\limits_{z\in \mathbb{D} }(1-|z|^{2})^{\beta} \left | {\varphi}' (z) \right | \frac{\left \| f \right \|_{\mathcal{B}_{\mathcal{H}}^{\alpha -1}} }{( 1-\left | \varphi (z) \right | ^{2} ) ^{\alpha -1} } \\ &\approx |f(\varphi(0))|+\left \| f \right \|_{\mathcal{Z}_{\mathcal{H}}^{\alpha}}\cdot \sup _{z \in \mathbb{D}}\frac{\left(1-|z|^{2}\right)^{\beta}\left | \varphi^{\prime}(z) \right | }{\left(1-|\varphi(z)|^{2}\right)^{\alpha-1}} \lesssim \left \| f \right \|_{\mathcal{Z}_{\mathcal{H}}^{\alpha}}. \end{align} $

(3.1)

于是$ C_{\varphi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $有界.因此$ (3) \Rightarrow (1) $成立.

下面证明$ (2) \Rightarrow (3) $.假设$ L:=\sup_{j\in \mathbb{N}}\left\|C_{\varphi}P_{j}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}<\infty. $因为$ C_\varphi P_1=\varphi+\overline{\varphi}, $所以对任意$ z\in\mathbb{D}, $ $ \left|\frac{\partial[C_\varphi P_1(z)]}{\partial z}\right|=\left|\frac{\partial[C_\varphi P_1(z)]}{\partial\overline{z}}\right|=|\varphi^{\prime}(z)|. $于是

$ \begin{eqnarray} \sup\limits_{z\in\mathbb{D}}(1-|z|^2)^\beta|\varphi'(z)|\leq\frac{1}{2}\|C_\varphi P_1\|_{\mathcal{B}_{\mathcal{H}}^\beta}\leq\frac{L}{2}. \end{eqnarray} $

(3.2)

令$ 0<s<1, $当$ |\varphi(z)|\leq s $时,

$ \sup\limits_{z\in\mathbb{D}}\frac{(1-|z|^2)^\beta|\varphi'(z)|}{\left ( 1-\left | \varphi(z) \right | ^{2} \right )^{\alpha -1} } \le \frac{L}{2(1-s^{2} )^{\alpha -1} } < \infty . $

当$ |\varphi(z)|>s $时, 对于固定的$ b\in\mathbb{D}, $定义

$ \begin{eqnarray} F_{b}^\alpha(z):=\frac{(1-|b|^2)^3}{(1-\overline{b}z)^{\alpha+1}}+\frac{(1-|b|^2)^3}{(1-b\overline{z})^{\alpha+1}}. \end{eqnarray} $

(3.3)

当$ \left | b \right | \rightarrow 1 $时, $ F_{b}^\alpha(z) $在$ \mathbb{D} $的紧子集上一致收敛到0. 由Stirling公式, $ F_{b}^\alpha(z) $的级数展开式为

$ \begin{align} F_{b}^{\alpha}(z)=(1-|b|^{2})^{3}\sum\limits_{j=0}^{\infty}\frac{\Gamma(j+\alpha+1)}{j!\Gamma(\alpha+1)}\Big\{(\overline{b}z)^{j}+(b\overline{z})^{j}\Big\} \approx (1-|b|^{2})^{3}\sum\limits_{j=0}^{\infty}j^{\alpha}\Big\{(\overline{b}z)^{j}+(b\overline{z})^{j}\Big\}. \end{align} $

(3.4)

于是, 对于$ z\in\mathbb{D}, $ $ \|C_\varphi F_{\varphi(z)}^\alpha\|_{\mathcal{B}_{\mathcal{H}}^\beta}\lesssim(1-|\varphi(z)|^2)^3\sum\limits_{j=0}^\infty |\varphi(z)|^j\|C_\varphi P_j\|_{\mathcal{B}_{\mathcal{H}}^\beta}\le L\sum\limits_{j=0}^\infty|\varphi(z)|^j\lesssim L. $通过直接计算, 对任意$ z\in \mathbb{D}, $我们得到

$ \begin{equation} \frac{\partial\left[C_{\varphi}F_{\varphi(z)}^{\alpha}\right]}{\partial z}\left(z\right)=\frac{\left(\alpha+1\right)\overline{\varphi\left(z\right)}\varphi^{\prime}\left(z\right)}{\left(1-\left|\varphi\left(z\right)\right|^{2}\right)^{\alpha-1}}, \;\; \frac{\partial\left[C_{\varphi}F_{\varphi(z)}^{\alpha}\right]}{\partial\overline{z}}\left(z\right)=\frac{\left(\alpha+1\right)\varphi\left(z\right)\overline{\varphi^{\prime}\left(z\right)}}{\left(1-\left|\varphi\left(z\right)\right|^{2}\right)^{\alpha-1}}. \end{equation} $

于是

$ \begin{align} \frac{(1-|z|^{2})^{\beta}|\varphi^{\prime}(z)||\varphi(z)|}{(1-|\varphi(z)|^{2})^{\alpha-1}}&\leq\sup\limits_{z\in \mathbb{D} }\frac{(1-|z|^{2})^{\beta}|\varphi^{\prime}(z)||\varphi(z)|}{(1-|\varphi(z)|^{2})^{\alpha-1}} \\ &\lesssim \sup\limits_{z\in\mathbb{D}}(1-|z|^2)^\beta\left ( \left | \frac{\partial\left[C_{\varphi}F_{\varphi(z)}^{\alpha}\right]}{\partial z}\left(z\right) \right | +\left | \frac{\partial\left[C_{\varphi}F_{\varphi(z)}^{\alpha}\right]}{\partial\overline{z}}\left(z\right) \right|\right ) \\ &\leq\|C_{\varphi}F_{\varphi(z)}^{\alpha}\|_{B_{\mathcal{H}}^{\beta }}\lesssim L. \end{align} $

(3.5)

由于$ |\varphi(z)|>s, $对任意$ z\in \mathbb{D}, $

$ \begin{eqnarray} \begin{split} \frac{(1-|z|^{2})^{\beta}|\varphi^{\prime}(z)|}{(1-|\varphi(z)|^{2})^{\alpha-1}}\lesssim \frac{L}{|\varphi(z)|}< \frac{L}{s} < \infty , \nonumber \end{split} \end{eqnarray} $

综上所述, $ \sup _{z \in \mathbb{D}}|\varphi_{\#}^{\alpha, \beta}(z)|<\infty. $由此证明了$ (2)\Rightarrow (3) $.

下面讨论$ C_{\varphi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $的紧性.

定理3.2    设$ \alpha>1 $, $ \beta>0 $且$ \varphi \in S\left( \mathbb{D} \right) $, 则下列条件等价：

$ (1) $$ C_{\varphi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $是紧的.

$ (2) $$ \lim\limits_{j\rightarrow \infty}\left\|C_{\varphi}P_{j}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}=0, P_{j}=(z^{j}+\overline{z}^{j} )j^{\alpha}, j\in \mathbb{N}. $

$ (3) $$ \lim\limits_{|\varphi(z)|\rightarrow1}|\varphi_{\#}^{\alpha, \beta}(z)| =0. $

证   先证$ (1)\Rightarrow (2) $.有界集$ \{P_j\}\subseteq \mathcal{Z}_{\mathcal{H}}^{\alpha} $在$ \mathbb{D} $的紧子集上一致收敛到0. 因为$ C_{\varphi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $是紧的, 根据引理2.2, 有$ \lim\limits_{j\rightarrow \infty}\left\|C_{\varphi}P_{j}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}=0. $

现假设$ (2) $成立, 则$ L:=\sup\limits_{j\in \mathbb{N}}\|C_{\varphi}P_{j}\|_\mathcal{B_{\mathcal{H}}^{\beta}}<\infty. $且对任意$ \varepsilon>0, $存在$ N\in\mathbb{N} $使得$ \left\|C_\varphi P_j\right\|_{\mathcal{B}_{\mathcal{H}}^\beta}<\varepsilon, j\geq N. $利用(3.3)中的检验函数$ F_{b}^\alpha(z) $, 对任意$ z \in \mathbb{D} $, 有

$ \begin{eqnarray} \begin{split} \left\|C_{\varphi}F_{\varphi(z)}^{\alpha}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}&\lesssim (1-|\varphi(z)|^{2})^{3}\left[\left(\sum\limits_{j=0}^{N-1}+\sum\limits_{j=N}^{\infty}\right)|\varphi(z)|^{j}\|C_{\varphi}P_{j}\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}\right]\\ &\le (1-|\varphi(z)|^{2})^{3}\sum\limits_{j=0}^{N-1}\|C_{\varphi}P_{j}\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}+\varepsilon \sum\limits_{j=N}^{\infty}|\varphi(z)|^{j}\\ &\lesssim (1-|\varphi(z)|^{2})^{3}NL+\varepsilon. \nonumber \end{split} \end{eqnarray} $

取$ s=\left [ 1-\left ( \frac{\varepsilon}{NL} \right )^{\frac{1}{3}} \right ]^{\frac{1}{2}}\in(0, 1), $对任意$ \left | \varphi (z) \right | > s $有$ \left\|C_{\varphi}F_{\varphi(z)}^{\alpha}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}< \ 2\varepsilon. $由(3.5)知,

$ \frac{(1-|z|^{2})^{\beta}|\varphi^{\prime}(z)|}{(1-|\varphi(z)|^{2})^{\alpha-1}}\lesssim \frac{\left\|C_{\varphi}F_{\varphi(z)}^{\alpha}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}}{|\varphi(z)|}< \frac{2\varepsilon}{s}\lesssim \varepsilon, $

从而

$ \lim\limits_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2})^{\beta}|\varphi^{\prime}(z)|}{(1-|\varphi(z)|^{2})^{\alpha-1}}=0. $

因此, $ (2)\Rightarrow (3) $成立.

下证$ (3)\Rightarrow (1) $. 假设$ \lim\limits_{|\varphi(z)|\rightarrow1}|\varphi_{\#}^{\alpha, \beta}(z)| =0, $则对任意$ \varepsilon>0, $存在$ s\in(0, 1), $当$ s<|\varphi(z)|<1 $有$ |\varphi_{\#}^{\alpha, \beta}(z)|=\frac{(1-|z|^{2})^{\beta}|\varphi^{\prime}(z)|}{(1-|\varphi(z)|^{2})^{\alpha-1}}<\varepsilon. $取函数列 $ \{f_{j}\}\subseteq \mathcal{Z}_{\mathcal{H}}^{\alpha } $满足$ M:=\sup_{j\in \mathbb{N}}||f_{j}||_{\mathcal{Z}_{\mathcal{H}}^{\alpha }}<\infty $, 且在$ \mathbb{D} $的紧子集上一致收敛到0, 只需证$ \lim\limits_{j\to\infty}\|C_{\varphi}f_{j}\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}=0. $由柯西估计知, 函数列$ \left \{ \frac{\partial f_{j}}{\partial z} \right \} $与$ \left \{ \frac{\partial f_{j}}{\partial \bar{z} } \right \} $也在$ \mathbb{D} $的紧子集上一致收敛到0.于是在$ \mathbb{D} $的紧子集上, 对上述$ \varepsilon>0, $存在$ N\in \mathbb{N}^{+}, $对任意$ j>N, $有

$ \left|\frac{\partial f_j}{\partial z}(\varphi(z))\right|<\varepsilon, \quad \left|\frac{\partial f_j}{\partial\bar{z}}(\varphi(z))\right|<\varepsilon. $

根据(1.1)与(3.2),

$ \begin{equation} \begin{split} &\sup\limits_{z\in \mathbb{D} }(1-|z|^2)^\beta\left(\left|\frac{\partial[C_\varphi f_j(z)]}{\partial z}\right|+\left|\frac{\partial[C_\varphi f_j(z)]}{\partial\bar{z}}\right|\right)\\ \leq&\left(\sup\limits_{1>|\varphi(z)|>s}+\sup\limits_{|\varphi(z)|\le s}\right)(1-|z|^2)^\beta|\varphi^{\prime}(z)|\left(\left|\frac{\partial f_j}{\partial z}(\varphi(z))\right|+\left|\frac{\partial f_j}{\partial\bar{z}}(\varphi(z))\right|\right)\\ % \le&\sup\limits_{1>|\varphi(z)|>s} \frac{(1-|z|^{2})^{\beta}|\varphi^{\prime}(z)|}{(1-|\varphi(z)|^{2} )^{\alpha-1}}(1-|\varphi(z)|^{2} )^{\alpha-1}\left(\left|\frac{\partial f_j}{\partial z}(\varphi(z))\right|+\left|\frac{\partial f_j}{\partial\bar{z}}(\varphi(z))\right|\right)\\ \le &\|f_{j}\|_{\mathcal{B} _{\mathcal{H}}^{\alpha -1}}\sup\limits_{1>|\varphi(z)|>s} \frac{(1-|z|^{2})^{\beta}|\varphi^{\prime}(z)|}{(1-|\varphi(z)|^{2} )^{\alpha-1}}+\frac{L}{2} \sup\limits_{|\varphi(z)|\le s}\left(\left|\frac{\partial f_j}{\partial z}(\varphi(z))\right|+\left|\frac{\partial f_j}{\partial\bar{z}}(\varphi(z))\right|\right)\\ \lesssim & \|f_{j}\|_{\mathcal{Z} _{\mathcal{H}}^{\alpha}}\cdot \varepsilon +\frac{L}{2}\cdot 2\varepsilon \le (M+L)\varepsilon. \nonumber %取消自动编号 \end{split} \end{equation} $

这意味着,

$ \lim\limits_{j\to\infty}\sup\limits_{z\in \mathbb{D} }(1-|z|^2)^\beta\left(\left|\frac{\partial[C_\varphi f_j(z)]}{\partial z}\right|+\left|\frac{\partial[C_\varphi f_j(z)]}{\partial\bar{z}}\right|\right)=0. $

又$ \lim\limits_{j\to\infty}\left | C_\varphi f_j(0) \right |=\lim\limits_{j\to\infty}\left | f_j(\varphi (0)) \right |=0 $, 于是$ \lim\limits_{j\to\infty}\|C_{\varphi}f_{j}\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}=0. $由此证明了$ (3)\Rightarrow (1) $.

基于定理3.1和定理3.2, 本节利用文献[25]中的类似方法研究调和空间上复合算子的差分$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $的有界性和紧性.

定理4.1    设$ \alpha>1, \beta>0 $且$ \varphi $, $ \psi \in S\left( \mathbb{D} \right) $, 则下列条件等价：

$ (1) $$ C_{\varphi}-C_{\psi} : \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $有界.

$ (2) $$ C_{\varphi}-C_{\psi} : \mathcal{Z}_{\mathcal{H}, 0}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $有界.

$ (3) $下列不等式成立:

$ \begin{eqnarray} \sup _{z \in \mathbb{D}}\left(\left|\varphi_{\#}^{\alpha, \beta}(z)\right| \rho(z)+\left|\varphi_{\#}^{\alpha, \beta}(z)-\psi_{\#}^{\alpha, \beta}(z)\right|\right)<\infty. \;\;\; \end{eqnarray} $

(4.1)

证   $ (1)\Rightarrow (2) $是显然的. 现假设(4.1)成立. 设$ f \in \mathcal{Z}_{\mathcal{H}}^{\alpha} $且$ \|f\|_{\mathcal{Z}_{\mathcal{H}}^{\alpha}} \leqslant 1. $令$ f=g+\bar{h}, $其中$ g, h\in H(\mathbb{D}), h(0)=0, $则$ g, h\in\mathcal{Z}^{\alpha} $且$ \|g\|_{\mathcal{Z}^{\alpha}} \leqslant 1, \|h\|_{\mathcal{Z}^{\alpha}} \leqslant 1. $于是

$ \begin{eqnarray} &&\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta} \left(\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) f \right ) }{\partial z}(z)\right|+\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) f \right ) }{\partial\overline{z}}(z)\right|\right)\\ &=&\sup\limits_{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left (\left| g^{\prime}(\varphi(z)) \varphi^{\prime}(z)-g^{\prime}(\psi(z)) \psi^{\prime}(z)\right| +\left| \overline{h^{\prime}(\varphi(z))} \overline{ \varphi^{\prime}(z)} -\overline{h^{\prime}(\psi(z)) } \overline{\psi^{\prime}(z)} \right| \right )\\ &\le& \sup\limits_{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left|g^{\prime}(\varphi(z)) \varphi^{\prime}(z)-g^{\prime}(\psi(z)) \psi^{\prime}(z)\right|\\ &&+\sup\limits_{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left| h^{\prime}(\varphi(z)) \varphi^{\prime}(z) -h^{\prime}(\psi(z)) \psi^{\prime}(z) \right|\\ &=&L_{1} +L_{2} , \end{eqnarray} $

(4.2)

其中

$ \begin{eqnarray} L_{1}:=\sup\limits_{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left|g^{\prime}(\varphi(z)) \varphi^{\prime}(z)-g^{\prime}(\psi(z)) \psi^{\prime}(z)\right|, \\ L_{2}:=\sup\limits_{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left| h^{\prime}(\varphi(z)) \varphi^{\prime}(z)-h^{\prime}(\psi(z)) \psi^{\prime}(z) \right|. \end{eqnarray} $

根据引理2.1与引理2.3, 有

$ \begin{equation} \begin{split} L_{1}&=\sup\limits_{z \in \mathbb{D}}\left|\varphi_{\#}^{\alpha, \beta}(z) \left[\left(1-|\varphi(z)|^{2}\right)^{\alpha-1} g^{\prime}(\varphi(z))-\left(1-|\psi(z)|^{2}\right)^{\alpha-1} g^{\prime}(\psi(z))\right]\right.\\ &\quad+\left.\left(1-|\psi(z)|^{2}\right)^{\alpha-1} g^{\prime}(\psi(z)) \left(\varphi_{\#}^{\alpha, \beta}(z)-\psi_{\#}^{\alpha, \beta}(z)\right)\right| \\ &\lesssim\sup\limits_{z \in \mathbb{D}}\left(\left|\varphi_{\#}^{\alpha, \beta}(z)\right|\rho(z) +\left(1-|\psi(z)|^{2}\right)^{\alpha-1}|g^{\prime}(\psi(z))| \left|\varphi_{\#}^{\alpha, \beta}(z)-\psi_{\#}^{\alpha, \beta}(z)\right|\right)\\ &\lesssim \sup\limits_{z \in \mathbb{D}}\left(\left|\varphi_{\#}^{\alpha, \beta}(z)\right| \rho(z)+\left|\varphi_{\#}^{\alpha, \beta}(z)-\psi_{\#}^{\alpha, \beta}(z)\right|\right)<\infty. \nonumber %取消自动编号 \end{split} \end{equation} $

同理可证

$ \begin{equation} \begin{split} L_{2} \lesssim \sup\limits_{z \in \mathbb{D}}\left(\left|\varphi_{\#}^{\alpha, \beta}(z)\right| \rho(z)+\left|\varphi_{\#}^{\alpha, \beta}(z)-\psi_{\#}^{\alpha, \beta}(z)\right|\right)<\infty. \nonumber %取消自动编号 \end{split} \end{equation} $

故对于任意的$ f\in \mathcal{Z}_{\mathcal{H}}^{\alpha} $, 有

$ \begin{eqnarray*} \|\left(C_{\varphi}-C_{\psi}\right) f \|_{\mathcal{B}_{\mathcal{H}}^{\beta}} \lesssim |f(\varphi(0))-f(\psi(0))|+\sup\limits_{z \in \mathbb{D}}\left(\left|\varphi_{\#}^{\alpha, \beta}(z)\right| \rho(z)+\left|\varphi_{\#}^{\alpha, \beta}(z) -\psi_{\#}^{\alpha, \beta}(z)\right|\right)<\infty, \end{eqnarray*} $

这意味着$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $有界. 因此$ (3) \Rightarrow (1) $成立.

接下来证明$ (2) \Rightarrow (3) $. 假设$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}, 0}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $有界. 设$ \mathbb{D}_1=\{w\in \mathbb{D}:\;\varphi(w)= 0\} $且$ \mathbb{D}_2=\{w\in \mathbb{D}:\;\psi(w)= 0\} $. 对于固定的$ a \in \mathbb{D}\setminus\{0\} $, 定义

$ \begin{eqnarray} F_{a}(z):=f_{a}(z)+\overline{f_{a}(z)}, z \in \mathbb{D}, \;\;\; \end{eqnarray} $

(4.3)

其中

$ \begin{eqnarray} f_{a}(z)= \begin{cases}\frac{\left(1-|a|^{2}\right)^{\alpha-1}}{(2 \alpha-3) \bar{a}(1-\bar{a} z)^{2\alpha-3}}, &\alpha \neq \frac{3}{2} , \\ -\frac{\left(1-|a|^{2}\right)^{\frac{1}{2}} \ln (1-\bar{a} z)}{\bar{a}}, &\alpha=\frac{3}{2}.\end{cases} \end{eqnarray} $

利用文献[25]中$ f_{a} \in \mathcal{Z}_{\mathrm{0}}^{\alpha} $可得$ F_{a} \in \mathcal{Z}_{\mathcal{H}, 0}^{\alpha}. $进而对于$ w \notin \mathbb{D}_1 $成立

$ \begin{eqnarray} \begin{split} \infty&>\left\|\left(C_{\varphi}-C_{\psi}\right) F_{\varphi(w)}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}\nonumber\\ &\geqslant\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left(\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) F_{\varphi(w)} \right ) }{\partial z}(z)\right|+\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) F_{\varphi(w)} \right ) }{\partial\overline{z}}(z)\right|\right)\nonumber\\ &=\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}{ \left(\left|f_{\varphi(w)}^{\prime}(\varphi(z)) \varphi^{\prime}(z)-f_{\varphi(w)}^{\prime}(\psi(z)) \psi^{\prime}(z)\right| +\left|\overline{f_{\varphi(w)}^{\prime}(\varphi(z)) \varphi^{\prime}(z)}-\overline{f_{\varphi(w)}^{\prime}(\psi(z)) \psi^{\prime}(z)}\right|\right)}\nonumber\\ &=2\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left|f_{\varphi(w)}^{\prime}(\varphi(z)) \varphi^{\prime}(z)-f_{\varphi(w)}^{\prime}(\psi(z)) \psi^{\prime}(z)\right|.\;\;\; \end{split} \end{eqnarray} $

类似于[25, 定理3.3]的证明,

$ \begin{eqnarray} \sup _{w \in \mathbb{D} \backslash \mathbb{D}_{1}}\left|\varphi_{\#}^{\alpha, \beta}(w)-\psi_{\#}^{\alpha, \beta}(w)\left(\frac{\left(1-|\varphi(w)|^{2}\right)\left(1-|\psi(w)|^{2}\right)}{(1-\overline{\varphi(w)} \psi(w))^{2}}\right)^{\alpha-1}\right|<\infty. \end{eqnarray} $

(4.4)

进一步得到

$ \begin{eqnarray} \sup _{w \in \mathbb{D} \backslash \mathbb{D}_{1}}\left(\left|\varphi_{\#}^{\alpha, \beta}(w)-\psi_{\#}^{\alpha, \beta}(w)\right|-\left|\psi_{\#}^{\alpha, \beta}(w)\right| \rho(w)\right)<\infty.\;\;\; \end{eqnarray} $

(4.5)

另一方面, 对于固定的$ a \in \mathbb{D}\setminus\{0\} $, 定义

$ \begin{eqnarray} K_{a}(z):=k_{a}(z)+\overline{k_{a}(z)}, z \in \mathbb{D}, \;\;\; \end{eqnarray} $

(4.6)

其中

$ \begin{eqnarray} k_a(z)=\left\{\begin{array}{l} \frac{(a-z)\left(1-|a|^{2}\right)^{\alpha-1}}{(2 \alpha-2) \bar{a}(1-\bar{a} z)^{2\alpha-2}}+\frac{\left(1-|a|^{2}\right)^{\alpha-1}}{(2 \alpha-2)(2 \alpha-3){\bar{a}^{2}}(1-\bar{a} z)^{2 \alpha-3}}, \;\; \alpha \neq \frac{3}{2}, \\ \frac{(a-z)\left(1-|a|^{2}\right)^{\frac{1}{2}}}{\bar{a}(1-\bar{a} z)}-\frac{\left(1-|a|^{2}\right)^{\frac{1}{2}} \ln (1-\bar{a} z)}{\bar{a}^{2}}, \;\;\alpha=\frac{3}{2}. \end{array}\right. \end{eqnarray} $

利用文献[25]中$ k_{a} \in \mathcal{Z}^{\alpha}_{0} $可得$ K_{a} \in \mathcal{Z}_{\mathcal{H}, 0}^{\alpha}. $进而对于$ w \notin \mathbb{D}_1 $, 有

$ \begin{eqnarray*} \begin{split} \infty&>\left\|\left(C_{\varphi}-C_{\psi}\right) K_{\varphi(w)}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}\nonumber\\ &\geqslant\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left(\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) K_{\varphi(w)} \right ) }{\partial z}(z)\right|+\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) K_{\varphi(w)} \right ) }{\partial\overline{z}}(z)\right|\right)\nonumber\\ &=2\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left|k_{\varphi(w)}^{\prime}(\varphi(z)) \varphi^{\prime}(z)-k_{\varphi(w)}^{\prime}(\psi(z)) \psi^{\prime}(z)\right|\nonumber\\ &\geqslant 2\left|\psi_{\#}^{\alpha, \beta}(w) \left(\frac{\left(1-|\varphi(w)|^{2}\right)\left(1-|\psi(w)|^{2}\right)} {(1-\overline{\varphi(w)} \psi(w))^{2}}\right)^{\alpha-1}\right|\rho(w).\end{split} \end{eqnarray*} $

这意味着

$ \begin{eqnarray} \sup _{w \in \mathbb{D} \backslash \mathbb{D}_{1}}\left|\psi_{\#}^{\alpha, \beta}(w) \left(\frac{\left(1-|\varphi(w)|^{2}\right)\left(1-|\psi(w)|^{2}\right)}{(1-\overline{\varphi(w)} \psi(w))^{2}}\right)^{\alpha-1}\right|\rho(w)<\infty.\;\;\; \end{eqnarray} $

(4.7)

将(4.7)代入(4.4) 可得

$ \begin{eqnarray} \sup _{w \in \mathbb{D} \backslash \mathbb{D}_{1}}\left|\varphi_{\#}^{\alpha, \beta}(w)\right|\rho(w)<\infty.\;\;\; \end{eqnarray} $

(4.8)

同理可得

$ \begin{eqnarray} \sup _{w \in \mathbb{D} \backslash \mathbb{D}_{2}}\left|\psi_{\#}^{\alpha, \beta}(w)\right|\rho(w)<\infty. \end{eqnarray} $

(4.9)

结合(4.5)与(4.9), 我们便可得出

$ \begin{eqnarray} \sup _{w \in \mathbb{D} \backslash (\mathbb{D}_{1} \cup \mathbb{D}_{2})}\left|\varphi_{\#}^{\alpha, \beta}(w)-\psi_{\#}^{\alpha, \beta}(w)\right|<\infty.\;\;\; \end{eqnarray} $

(4.10)

故将(4.8)与(4.10) 结合即可得到如下结论

$ \begin{eqnarray} \sup _{w \in \mathbb{D} \backslash (\mathbb{D}_{1} \cup \mathbb{D}_{2})}\left(\left|\varphi_{\#}^{\alpha, \beta}(w)\right| \rho(w)+\left|\varphi_{\#}^{\alpha, \beta}(w)-\psi_{\#}^{\alpha, \beta}(w)\right|\right)<\infty.\;\;\; \end{eqnarray} $

(4.11)

若$ w\in \mathbb{D}_1\cap\mathbb{D}_2 $, 则$ \rho(w)=0 $. 我们取$ H(z):=z+\bar{z}\in \mathcal{Z}_{\mathcal{H}, 0}^{\alpha} $, 因此得到

$ \begin{eqnarray*} \begin{split} \infty&>\left\|\left(C_{\varphi}-C_{\psi}\right) H\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}\nonumber\\ &\geqslant\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left(\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) H \right ) }{\partial z}(z)\right|+\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) H \right ) }{\partial\overline{z}}(z)\right|\right)\nonumber\\ &=\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left(\left|\varphi^{\prime}(z)- \psi^{\prime}(z)\right|+\left|\overline{\varphi^{\prime}(z)}- \overline{\psi^{\prime}(z)}\right| \right)\\ &=2 \left|\left(1-|w|^{2}\right)^{\beta} \varphi^{\prime}(w)-\left(1-|w|^{2}\right)^{\beta} \psi^{\prime}(w)\right|\\ &=\left|\varphi_{\#}^{\alpha, \beta}(w)\right|\cdot\rho(w)+\left|\varphi_{\#}^{\alpha, \beta}(w)-\psi_{\#}^{\alpha, \beta}(w)\right|, \end{split} \end{eqnarray*} $

其意味着

$ \begin{eqnarray} \sup _{w \in \mathbb{D}_{1} \cap \mathbb{D}_{2}}\left( \left|\varphi_{\#}^{\alpha, \beta}(w)\right|\cdot\rho(w)+\left|\varphi_{\#}^{\alpha, \beta}(w)-\psi_{\#}^{\alpha, \beta}(w)\right|\right)<\infty.\;\;\; \end{eqnarray} $

(4.12)

若$ w\in \mathbb{D}_2\setminus \mathbb{D}_1 $, 则$ \rho(w)=|\varphi(w)| $. 对于固定的$ a \in \mathbb{D} \setminus \{0\} $, 选取$ P_{a}(z):=p_{a}(z)+\overline{p_{a}(z)}, $其中

$ \begin{eqnarray*} p_a(z)=\left\{\begin{array}{l} -\frac{1}{\bar{a}^2}[\ln (1-\bar{a} z)+\bar{a} z], \; \ \alpha=2, \\ \frac{1}{\bar{a}^2}\left[\frac{1}{1-\bar{a} z}+\ln (1-\bar{a} z)\right], \; \ \alpha=3, \\ \frac{1}{\bar{a}^2}\left[\frac{1}{(\alpha-2)(1-\bar{a} z)^{\alpha-2}}-\frac{1}{(\alpha-3)(1-\bar{a} z)^{\alpha-3}}\right], \; \ \alpha \neq 2, \alpha \neq 3. \end{array}\right. \end{eqnarray*} $

利用文献[25]中$ p_{a} \in \mathcal{Z}_{\mathrm{0}}^{\alpha} $可得$ P_{a} \in \mathcal{Z}_{\mathcal{H}, 0}^{\alpha}. $由此可得

$ \begin{eqnarray*} \begin{split} \infty&>\left\|\left(C_{\varphi}-C_{\psi}\right) P_{\varphi(w)}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}\\ &\geqslant\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left(\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) P_{\varphi(w)} \right ) }{\partial z}(z)\right|+\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) P_{\varphi(w)} \right ) }{\partial\overline{z}}(z)\right|\right)\\ &\geqslant\left|\left(1-|w|^{2}\right)^{\beta} \varphi^{\prime}(w) \frac{\varphi(w)}{\left(1-|\varphi(w)|^{2}\right)^{\alpha-1}}- 0\right|+\left|\left(1-|w|^{2}\right)^{\beta} \overline{\varphi^{\prime}(w)} \frac{\overline{\varphi(w)}}{\left(1-|\varphi(w)|^{2}\right)^{\alpha-1}}- 0\right|\\ &=2\frac{\left(1-|w|^{2}\right)^{\beta} |\varphi^{\prime}(w)|}{\left(1-|\varphi(w)|^{2}\right)^{\alpha-1}} \rho(w)=2\left|\varphi_{\#}^{\alpha, \beta}(w)\right| \rho(w).\end{split} \end{eqnarray*} $

这意味着$ \sup\limits _{w \in \mathbb{D}_{2} \backslash \mathbb{D}_{1}}\left|\varphi_{\#}^{\alpha, \beta}(w)\right| \rho(w)<\infty. $

类似地, 对于固定的$ a \in \mathbb{D} $, 我们定义$ q_{a}(z)=az-\frac{1}{2} z^{2} \in \mathcal{Z}_{\mathrm{0}}^{\alpha}, $并取$ Q_{a}(z):=q_{a}(z)+\overline{q_{a}(z)}\in \mathcal{Z}_{\mathcal{H}, 0}^{\alpha}. $于是有

$ \begin{eqnarray*} \begin{split} \infty&>\left\|\left(C_{\varphi}-C_{\psi}\right) Q_{\varphi(w)}\right\|_{\mathcal{B}_{\mathcal{H}}^{\beta}}\nonumber\\ &\geqslant\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left(\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) Q_{\varphi(w)} \right ) }{\partial z}(z)\right|+\left|\frac{\partial \left ( \left ( C_\varphi-C_{\psi } \right ) Q_{\varphi(w)} \right ) }{\partial\overline{z}}(z)\right|\right)\nonumber\\ &\geqslant\left|0-\left(1-|w|^{2}\right)^{\beta}\psi^{\prime}(w) \varphi(w)\right|+\left|0-\left(1-|w|^{2}\right)^{\beta}\overline{\psi^{\prime}(w)} \overline{\varphi(w)}\right|\nonumber\\ &=2\left|\psi_{\#}^{\alpha, \beta}(w)\right| \rho(w).\;\;\; \label{SP} \end{split} \end{eqnarray*} $

故

$ \begin{eqnarray} \sup _{w \in \mathbb{D}_{2} \backslash \mathbb{D}_{1}}\left|\psi_{\#}^{\alpha, \beta}(w)\right| \rho(w)<\infty.\;\;\; \end{eqnarray} $

(4.13)

将(4.5)与(4.13)结合便可得到

$ \begin{eqnarray*} \sup _{w \in \mathbb{D}_{2} \backslash \mathbb{D}_{1}}\left|\varphi_{\#}^{\alpha, \beta}(w)-\psi_{\#}^{\alpha, \beta}(w)\right|<\infty. \end{eqnarray*} $

因此推导出

$ \begin{eqnarray} \sup _{w \in \mathbb{D}_{2} \backslash \mathbb{D}_{1}}\left(\left|\varphi_{\#}^{\alpha, \beta}(w)\right| \cdot \rho(w)+\left|\varphi_{\#}^{\alpha, \beta}(w)-\psi_{\#}^{\alpha, \beta}(w)\right|\right)<\infty.\;\;\; \end{eqnarray} $

(4.14)

类似于(4.14)的推导步骤, 又可证

$ \begin{eqnarray} \sup _{w \in \mathbb{D}_{1} \backslash \mathbb{D}_{2}}\left(\left|\varphi_{\#}^{\alpha, \beta}(w)\right| \rho(w)+\left|\varphi_{\#}^{\alpha, \beta}(w)-\psi_{\#}^{\alpha, \beta}(w)\right|\right)<\infty.\;\;\; \end{eqnarray} $

(4.15)

综合(4.11), (4.12), (4.14)和(4.15)可证得(4.1), 因此完成了$ (2) \Rightarrow (3) $的证明.

接下来继续证明$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $的紧性. 记

$ \begin{eqnarray*} &&\Gamma(\varphi):=\left\{\left\{z_{n}\right\} \subset \mathbb{D}:\left|\varphi\left(z_{n}\right)\right| \rightarrow 1\right\};\; \;\;\Gamma(\psi):=\left\{\left\{z_{n}\right\} \subset \mathbb{D}:\left|\psi\left(z_{n}\right)\right| \rightarrow 1\right\};\\ &&D(\varphi):=\left\{\left\{z_{n}\right\} \subset \mathbb{D}:\left|\varphi\left(z_{n}\right)\right| \rightarrow 1, \left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right| \nrightarrow 0\right\};\\ &&D(\psi):=\left\{\left\{z_{n}\right\} \subset \mathbb{D}:\left|\psi\left(z_{n}\right)\right| \rightarrow 1, \left|\psi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right| \nrightarrow 0\right\}. \end{eqnarray*} $

定理4.2    设$ \alpha>1, \beta>0 $且$ \varphi $, $ \psi \in S\left( \mathbb{D} \right) $. 设$ C_{\varphi} $与$ C_{\psi} $均非紧但都有界, 则下列条件成立：

$ (1) $$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $是紧的.

$ (2) $$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}, 0}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $是紧的.

$ (3) $下列条件成立：

$ (i) $$ D(\varphi)=D(\psi), D(\varphi) \subseteq \Gamma(\psi) $.

$ (ii) $对于$ \{z_{n}\} \in \Gamma(\varphi) \cap \Gamma(\psi) $,

$ \begin{eqnarray*} \lim _{n \rightarrow \infty}\left[\left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right| \rho\left(z_{n}\right)+\left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)-\psi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right| \right]=0. \end{eqnarray*} $

证   $ (1)\Rightarrow (2) $是显然的. 下证$ (3) \Rightarrow (1) $. 取$ \left\{f_{n}\right\}\subset \mathcal{Z}_{\mathcal{H}}^{\alpha} $使得$ \sup\limits_{n\in \mathbb{N}} \|f_{n}\|_{\mathcal{Z}_{\mathcal{H}}^{\alpha}} \leqslant 1 $, 且在$ \mathbb{D} $上的紧子集上一致收敛到0. 应用反证法, 我们假设$ \lim\limits _{n \rightarrow \infty}\left\|\left(C_{\varphi}-C_{\psi}\right) f_{n}\right\| _{\mathcal{B_{\mathcal{H}}^{\beta}}}\ne 0, $则存在一个$ \varepsilon_{0}>0 $与一个$ \left\{f_{n}\right\} $的子序列$ \left\{f_{n_{k}}\right\} $, 使得$ \|\left(C_{\varphi}-C_{\psi}\right) f_{n_{k}} \|_{\mathcal{B_{\mathcal{H}}^{\beta}}}>\varepsilon_{0} $. 为方便起见, 我们仍用$ \left\{f_{n}\right\} $来表示子序列$ \left\{f_{n_{k}}\right\} $. 因此对于$ \varepsilon_{0}>0 $, 有$ \left\|\left(C_{\varphi}-C_{\psi}\right) f_{n}\right\| _{\mathcal{B_{\mathcal{H}}^{\beta}}} >\varepsilon_{0}, $对于所有的$ n $成立. 设$ f_{n} \left ( z \right ) =g_{n} \left ( z \right )+\overline{h_{n} \left ( z \right )}, $其中$ g_{n}, h_{n}\in H(\mathbb{D} ), h_{n}(0)=0, $则$ g_{n}, h_{n}\in \mathcal{Z} ^{\alpha }. $由$ \left\{f_{n}\right\} $的性质知$ \left\{g_{n}\right\} $与$ \left\{h_{n}\right\} $满足$ \|g_{n}\|_{\mathcal{Z}^{\alpha}} \leqslant 1, \|h_{n}\|_{\mathcal{Z}^{\alpha}} \leqslant 1 $且在$ \mathbb{D} $的紧子集上一致收敛到0. 进而可得到$ g_{n}^{\prime}(\varphi(z))=\frac{\partial f_{n} }{\partial z}(\varphi (z)), \; g_{n}^{\prime}(\psi (z))=\frac{\partial f_{n} }{\partial z}(\psi (z)), \; \overline{h_{n}^{\prime}(\varphi(z))} =\frac{\partial f_{n} }{\partial \bar{z} }(\varphi (z)), \;\overline{h_{n}^{\prime}(\psi (z))} =\frac{\partial f_{n} }{\partial \bar{z} }(\psi (z)), $于是

$ \begin{eqnarray*} &&\quad\left\|\left(C_{\varphi}-C_{\psi}\right) f_{n}\right\| _{\mathcal{B_{\mathcal{H}}^{\beta}}} \\ &&=\left | f_{n}(\varphi(0))-f_{n}(\psi(0)) \right | \\ &&\quad+\sup\limits_{z\in \mathbb{D} }(1-|z|^{2})^{\beta}\left ( \left | \frac{\partial f_{n} }{\partial z}(\varphi(z)) {\varphi}' (z) - \frac{\partial f_{n}}{\partial z}(\psi (z)) {\psi }' (z) \right | + \left | \frac{\partial f_{n} }{\partial \bar{z} }(\varphi(z)) \overline{{\varphi}' (z)} - \frac{\partial f_{n}}{\partial \bar{z} }(\psi (z)) \overline{{\psi }' (z)} \right | \right )\\ && =\left | f_{n}(\varphi(0))-f_{n}(\psi(0)) \right | \\ &&\quad+\sup\limits_{z\in \mathbb{D}}\left( \left| \varphi_{\#}^{\alpha, \beta}\left(z\right)\left(1-\left|\varphi\left(z\right)\right|^{2}\right)^{\alpha-1} g_{n}^{\prime}(\varphi (z)) -\psi_{\#}^{\alpha, \beta}\left(z\right)\left(1-\left|\psi\left(z\right)\right|^{2}\right)^{\alpha-1} g_{n}^{\prime}(\psi (z)) \right|\right.\\ && \quad \quad \qquad+\left.\left| \varphi_{\#}^{\alpha, \beta}\left(z\right) \left(1-\left|\varphi\left(z\right)\right|^{2}\right)^{\alpha-1} h_{n}^{\prime}(\varphi(z))-\psi_{\#}^{\alpha, \beta}\left(z\right) \left(1-\left|\psi\left(z\right)\right|^{2}\right)^{\alpha-1} h_{n}^{\prime}(\psi (z)) \right|\right). \end{eqnarray*} $

根据$ C_{\varphi} $与$ C_{\psi} $的有界性, 则当$ n\rightarrow \infty $时$ f_{n}(\varphi(0))-f_{n}(\psi(0)) \rightarrow 0 $. 故存在$ \{z_{n}\} \subset \mathbb{D} $使得

$ \begin{eqnarray*} &&\left| \varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)\left(1-\left|\varphi\left(z_{n}\right)\right|^{2}\right)^{\alpha-1} g_{n}^{\prime}(\varphi (z_{n})) -\psi_{\#}^{\alpha, \beta}\left(z_{n}\right)\left(1-\left|\psi\left(z_{n}\right)\right|^{2}\right)^{\alpha-1} g_{n}^{\prime}(\psi (z_{n})) \right| \nonumber\\ &&+\left| \varphi_{\#}^{\alpha, \beta}\left(z_{n}\right) \left(1-\left|\varphi\left(z_{n}\right)\right|^{2}\right)^{\alpha-1} h_{n}^{\prime}(\varphi(z_{n}))-\psi_{\#}^{\alpha, \beta}\left(z_{n}\right) \left(1-\left|\psi\left(z_{n}\right)\right|^{2}\right)^{\alpha-1} h_{n}^{\prime}(\psi (z_{n})) \right|>\varepsilon_{0}.\nonumber \end{eqnarray*} $

上式表明, 至少有一项$ n\rightarrow \infty $时不趋于0. 不妨设

$ \begin{eqnarray} \left| \varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)\left(1-\left|\varphi\left(z_{n}\right)\right|^{2}\right)^{\alpha-1} g_{n}^{\prime}(\varphi (z_{n})) -\psi_{\#}^{\alpha, \beta}\left(z_{n}\right)\left(1-\left|\psi\left(z_{n}\right)\right|^{2}\right)^{\alpha-1} g_{n}^{\prime}(\psi (z_{n})) \right| >\frac{\varepsilon_{0}}{2}. \;\;\; \end{eqnarray} $

(4.16)

将[25, 定理2]的证明中$ (3)\Rightarrow (1) $对于$ {f_{n}} $的步骤应用于$ {g_{n}} $, 当$ n\rightarrow \infty $成立

$ \begin{eqnarray*} \begin{split} \left|\varphi_{\#}^{\alpha, \beta}(z_{n}) \cdot\left(1-|\varphi(z_{n})|^{2}\right)^{\alpha-1} g_{n}^{\prime}(\varphi(z_{n}))-\psi_{\#}^{\alpha, \beta}(z_{n}) \cdot\left(1-|\psi(z_{n})|^{2}\right)^{\alpha-1} g_{n}^{\prime}(\psi(z_{n}))\right|\rightarrow 0. \end{split} \end{eqnarray*} $

与(4.16) 矛盾. 综上所述, 即证$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}, 0}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $是紧的.

下面证明$ (2) \Rightarrow (3) $. 假设$ C_{\varphi}-C_{\psi}: \mathcal{Z}_{\mathcal{H}, 0}^{\alpha} \rightarrow \mathcal{B_{\mathcal{H}}^{\beta}} $是紧的, 但$ C_{\varphi} $与$ C_{\psi} $都非紧. 定理3.2意味着此时存在序列$ \left\{z_{n}\right\} \in D(\varphi) $即$ \left|\varphi\left(z_{n}\right)\right| \rightarrow 1 $时有$ \left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right| \nrightarrow 0 $. 对于上述$ \{\varphi(z_n)\} \subset \mathbb{D} $我们参考(4.3)与(4.7) 定义$ F_{\varphi(z_n)} $, $ K_{\varphi(z_n)} $, 它们在$ \mathcal{Z}_{\mathcal{H}, 0}^{\alpha} $中都有界且当$ n \rightarrow \infty $时在$ \mathbb{D} $上的每个紧子集中一致收敛到0. 根据引理2.2, 当$ n \rightarrow \infty $时, 有

$ \begin{align} 0&\leftarrow\left\|\left(C_{\varphi}-C_{\psi}\right) F_{\varphi(z_{n})}\right\|_{\mathcal{B_{\mathcal{H}}^{\beta}}}\\ &\geqslant\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left(\left|f_{\varphi(z_{n})}^{\prime}(\varphi(z)) \varphi^{\prime}(z)-f_{\varphi(z_{n})}^{\prime}(\psi(z)) \psi^{\prime}(z)\right|\right.\\ &\qquad \qquad \qquad \quad +\left.\left|\overline{f_{\varphi(z_{n})}^{\prime}(\varphi(z)) \varphi^{\prime}(z)}-\overline{f_{\varphi(z_{n})}^{\prime}(\psi(z)) \psi^{\prime}(z)}\right|\right)\\ &\geqslant\left|\varphi_{\#}^{\alpha, \beta}(z_{n})-\psi_{\#}^{\alpha, \beta}(z_{n}) \left(\frac{\left(1-|\varphi(z_{n})|^{2}\right)\left(1-|\psi(z_{n})|^{2} \right)}{(1-\overline{\varphi(z_{n})} \psi(z_{n}))^{2}}\right)^{\alpha-1}\right|, \;\;\; \end{align} $

(4.17)

且

$ \begin{align} 0&\leftarrow\left\|\left(C_{\varphi}-C_{\psi}\right) K_{\varphi(z_n)}\right\|_{\mathcal{B_{\mathcal{H}}^{\beta}}}\\ &\geqslant\sup _{z \in \mathbb{D}}\left(1-|z|^{2}\right)^{\beta}\left(\left|k_{\varphi(z_{n})}^{\prime}(\varphi(z)) \varphi^{\prime}(z)-k_{\varphi(z_{n})}^{\prime}(\psi(z)) \psi^{\prime}(z)\right|\right.\\ &=\left|\psi_{\#}^{\alpha, \beta}(z_n) \left(\frac{\left(1-|\varphi(z_{n})|^{2}\right)\left(1-|\psi(z_{n})|^{2}\right)}{(1-\overline{\varphi(z_{n})} \psi(z_{n}))^{2}}\right)^{\alpha-1}\right|\rho(z_n).\;\;\; \end{align} $

(4.18)

将(4.17)与(4.18)相结合, 我们得到

$ \begin{eqnarray} \lim _{n \rightarrow \infty}\left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right| \rho\left(z_{n}\right)=0.\;\;\; \end{eqnarray} $

(4.19)

由于$ \left\{z_{n}\right\} \in D(\varphi) $, 我们有$ \left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right| \nrightarrow 0, n \rightarrow \infty $, 又根据(4.19)可得

$ \begin{eqnarray*} \lim\limits_{n \rightarrow \infty}\left|\frac{\varphi\left(z_{n}\right)-\psi\left(z_{n}\right)}{1-\overline{\varphi\left(z_{n}\right)} \psi\left(z_{n}\right)}\right|=\lim\limits_{n \rightarrow \infty} \rho\left(z_{n}\right)=0. \end{eqnarray*} $

因此, 对于任意的$ \left\{z_{n}\right\} \in D(\varphi) $, 有

$ \begin{eqnarray} \lim _{n \rightarrow \infty}\left|\varphi{(z_n)}-\psi{(z_n)}\right|=0, \;\;\; \end{eqnarray} $

(4.20)

且$ \lim _{n \rightarrow \infty}\left||\varphi\left(z_{n}\right)|-|\psi\left(z_{n}\right)|\right| \leqslant\lim _{n \rightarrow \infty}\left|\varphi{(z_n)}-\psi{(z_n)}\right|=0. $故$ \left|\psi\left(z_{n}\right)\right| \rightarrow 1 $, $ n \rightarrow \infty $. 因此, 对于任意的$ \left\{z_{n}\right\} \in D(\varphi) $, 有$ \left\{z_{n}\right\} \in \Gamma(\psi) $, 则$ D(\varphi) \subseteq \Gamma(\psi) $. 进一步得到$ D(\varphi) \subseteq \Gamma(\varphi) \cap \Gamma(\psi) $. 又对于任意的$ \left\{z_{n}\right\} \in \Gamma(\varphi) \cap \Gamma(\psi) $, 由于(4.18), 当$ n \rightarrow \infty $时, 有

$ \begin{eqnarray*} \begin{split} 0&\leftarrow\left\|\left(C_{\varphi}-C_{\psi}\right) F_{\varphi(z_n)}\right\|_{\mathcal{B_{\mathcal{H}}^{\beta}}}\\ &\geqslant\left|\varphi_{\#}^{\alpha, \beta}(z_n)-\psi_{\#}^{\alpha, \beta}(z_n) \left(\frac{\left(1-|\varphi(z_{n})|^{2}\right) \left(1-|\psi(z_{n})|^{2}\right)}{(1-\overline{\varphi(z_{n})} \psi(z_{n}))^{2}}\right)^{\alpha-1}\right|\\ &\geqslant\left|\varphi_{\#}^{\alpha, \beta}(z_n)-\psi_{\#}^{\alpha, \beta}(z_n)\right|\\ &\quad- \left| \psi_{\#}^{\alpha, \beta}(z_n) \left[\left(1-\left|\varphi\left(z_{n}\right)\right|^{2}\right)^{\alpha-1} f_{\varphi\left(z_{n}\right)}^{\prime} \left(\varphi\left(z_{n}\right)\right)- \left(1-\left|\psi\left(z_{n}\right)\right|^{2}\right)^{\alpha-1} f_{\varphi\left(z_{n}\right)}^{\prime}\left(\psi\left(z_{n}\right)\right)\right]\right|\\ &\gtrsim\left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)-\psi_{\#}^{\alpha, \beta} \left(z_{n}\right)\right|-\left|\psi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right| \rho\left(z_{n}\right), \end{split} \end{eqnarray*} $

则根据$ C_{\psi} $的有界性与$ \rho\left(z_{n}\right) \rightarrow 0 $, $ n\rightarrow \infty $, 可以推得

$ \begin{eqnarray} \lim _{n \rightarrow \infty}\left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)-\psi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right|=0.\;\;\; \end{eqnarray} $

(4.21)

因此对于任意的$ \left\{z_{n}\right\} \in \Gamma(\varphi) \cap \Gamma(\psi) $, 将(4.19)与(4.21)相结合, 便可得到

$ \begin{eqnarray*} \lim _{n \rightarrow \infty}\left(\left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right| \rho\left(z_{n}\right)+\left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)-\psi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right|\right)=0. \end{eqnarray*} $

对于$ \left\{z_{n}\right\} \in D(\varphi) $, 根据(4.20)与(4.21)

$ \begin{eqnarray} &\lim\limits _{n \rightarrow \infty} \left|\psi\left(z_{n}\right)\right|=\lim _{n \rightarrow \infty} \left|\varphi\left(z_{n}\right)\right|=1, \\ &\lim\limits _{n \rightarrow \infty} \left|\psi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right|=\lim _{n \rightarrow \infty} \left|\varphi_{\#}^{\alpha, \beta}\left(z_{n}\right)\right|\ne 0. \end{eqnarray} $

于是$ D(\varphi) \subseteq D(\psi) $. 同理易证$ D(\psi) \subseteq D(\varphi) $. 因此$ D(\varphi) = D(\psi) $. 故$ (2) \Rightarrow (3) $成立.