关于从混合模空间到小 Zygmund 空间的 Volterra 型复合算


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本文作者相关文章
	刘永民

	于燕燕

刘永民¹, 于燕燕²

1.江苏师范大学数学与统计学院江苏徐州 221116;
2.徐州工程学院数学与物理科学学院江苏徐州 221111)

收稿日期:2014-01-08;修订日期:2014-10-23

基金项目: 国家自然科学基金(11171285) 和江苏高校优势学科建设工程资助项目资助

作者简介：刘永民,E-mail:minliu@jsnu.edu.cn;于燕燕, yuyanyan@xzit.edu.cn

摘要：利用混合模空间$H(p,\, q,\, \phi)$中函数的高阶导数的估计、解析函数的性质与算子理论, 给出了从混合模空间 $H(p,\, q,\, \phi)$到小Zygmund空间的Volterra型复合算子的有界性和紧性的特征, 获得了几个充要条件.

关键词：有界性紧性混合模空间小 Zygmund 空间 Volterra 型复合算子

On Volterra-type Composition Operators from Mixed Norm Spaces to Little Zygmund Spaces

Liu Yongmin¹, Yu Yanyan²

1.School of Mathematics and Statistics, Jiangsu Normal University, Jiangsu Xuzhou 221116;
2. School of Mathematics and Physics Science, Xuzhou Institute of Technology, Jiangsu Xuzhou 221111

Abstract : In this paper, using the estimate of the higher order derivative of the function in mixed norm spaces, the properties of the analytic function and operator theory, the authors characterize the boundedness and compactness of the Volterra-type composition operator from mixed norm spaces $H(p,\, q,\, \phi)$ to little Zygmund spaces. Some necessary and sufficient conditions of the Volterra-type composition operators to be bounded and compact are obtained.

Key words: Boundedness Compactness Mixed norm space Little Zygmund space Volterra-type composition operators

1 引言

本文采用文献^[1]中的符号. 设$D=\{z\in C: |z|<1\}$是复平面上的单位圆盘,$\varphi$是 $D\rightarrow D$ 的一个解析自映射. $H(D)$表示$D$上的所有解析函数的集合. 设 $0 < {\rm{ }}p,{\rm{ }}q{\rm{ }} < \infty ,\varphi $是一个正规函数,$H(p,{\rm{ }}q,{\rm{ }}\varphi ){\rm{ }} = L(p,{\rm{ }}q,{\rm{ }}\varphi ) \cap H(D)$表示混合模空间. ${\cal Z}$表示Zygmund空间. ${\cal Z}_{0}$ 是小Zygmund空间. 最近,刘永民和刘浩^[1]研究了以下两种Volterra型复合算子 $$(I_{g,\varphi}f)(z)=\int^{z}_{0}(f\circ\varphi)'(\xi)(g\circ\varphi)(\xi){\rm d}\xi,$$ $$(J_{g,\varphi}f)(z)=\int^{z}_{0}(f\circ\varphi)(\xi) (g\circ\varphi)'(\xi){\rm d}\xi, $$ 其中 $f,\,g \in H(D)$,给出了从混合模空间$H(p,\,q,\,\phi)$到Zygmund空间(小 Zygmund 空间)的Volterra型复合算子的有界性和紧性的特征刻画. 经过仔细的研究,我们发现文献^[1]中讨论的算子 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 及 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 的有界性和紧性的特征刻画有问题. 其中定理3、定理7 的证明存在瑕疵, 这是因为 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 及 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 的有界性不足以保证文献^[1]中公式$(31)$及$(56)$的成立. 其中定理4、定理8的证明依赖定理3、定理7 的证明,从而他们不成立. 经过改进,我们给出了算子 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 及 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 的一些有界性与紧性的特征. 各种函数空间之间的算子$I_{g,\varphi}$,$J_{g,\varphi}$ 的有界性和紧性的研究可以在文献^{[2, 3, 4, 5]}中找到. 一些函数空间之间的积分型算子的有界性和紧性的研究可见文献^{[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]}.文中字母 $C$ 是一个正常数,不同的地方可以不同.

2 预备引理

为了证明我们的结论,我们需要下面的引理.

$ 引理 2.1$^[18]设 $0<p,\,q<\infty$, $\phi$ 是正规函数,$f\in H(p,\,q,\,\phi)$,则对于任意 $n\in N_0$ (非负整数的集合), 存在一个与 $f$无关的正数$C$,满足 $$\left|f^{(n)}(z)\right|\leq C\frac{\|f\|_{p,\,q,\,\phi}}{\phi(|z|)\left(1-|z|^{2}\right)^{n+\frac{1}{q}}},\,z\in D.$$

$ 引理 2.2$ 设 $\varphi$ 是 $D\rightarrow D$ 的一个解析自映射,$\phi$ 是正规函数, $0<p$,$q<\infty$ 和 $g\in H(D)$. $I_{g,\varphi}$ (或 $J_{g,\varphi}$): $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是紧算子的充分必要条件是 $I_{g,\varphi}$ (或 $J_{g,\varphi}$): $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是有界算子,且对于 $H(p,\,q,\,\phi)$中在 $D$ 上内闭一致收敛于 0 的有界函数列 $\{f_{n}\}$, 有 $\|I_{g,\varphi} f_{n}\|_{{\cal Z}}\rightarrow 0,n\rightarrow \infty; (\|J_{g,\varphi} f_{n}\|_{\cal Z}\rightarrow 0, n\rightarrow \infty).$

可以用标准的方法证明引理 2.2,此处略去.

$ 引理 2.3$^[9] ${\cal Z}_{0}$中的闭子集 $K$ 是紧子集的充要条件是 $K$ 是有界集且满足 $$\lim_{|z|\rightarrow1}\sup_{f\in K}(1-|z|^{2})\left|f''(z)\right|=0.$$

$ 引理 2.4$^[1] 设 $\varphi$ 是 $D\rightarrow D$ 的一个解析自映射,$\phi$ 是正规函数,$0<p,\,q<\infty$ 和 $g\in H(D)$. 则 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是有界算子的充分必要条件是

\begin{equation} \label{e1} \sup_{z\in D}\frac{(1-|z|^{2})\left|\varphi'(z)\right|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}}}< \infty \end{equation}

(2.1)

且

\begin{equation} \label{e2} \sup_{z\in D}\frac{(1-|z|^{2})\left|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))\right|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }< \infty. \end{equation} %\end{theorem}

(2.2)

$ 引理 2.5$^[1] 设 $\varphi$ 是 $D\rightarrow D$ 的一个解析自映射,$\phi$ 是正规函数,$0<p,q<\infty$ 和 $g\in H(D)$. 则 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是有界算子的充分必要条件是

\begin{equation} \label{e32}\sup_{z\in D}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }< \infty \end{equation}

(2.3)

且

\begin{equation} \label{e33}\sup_{z\in D}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g''(\varphi(z))+\varphi''(z)g'(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{\frac{1}{q}} }< \infty. \end{equation}

(2.4)

3 $I_{g,\varphi}: H(p,\,q,\,\phi) \rightarrow {\cal Z}_0$}的有界性和紧性

本节讨论算子 $I_{g,\varphi}: H(p,\,q,\,\phi) \rightarrow {\cal Z}_0$ 的有界性和紧性.

$\bf 定理 3.1$ 设 $\varphi$ 是 $D\rightarrow D$ 的一个解析自映射,$\phi$ 是正规函数, $0<p,\,q<\infty$ 和 $g\in H(D)$. 则 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是有界算子的充分必要条件是 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是有界算子且

\begin{equation} \label{e20} \lim_{|z|\rightarrow1}(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|=0,\end{equation}

(3.1)

\begin{equation} \label{e21} \lim_{|z|\rightarrow1}(1-|z|^{2})|(\varphi'(z))^{2}g(\varphi(z))|=0. \end{equation}

(3.2)

$\bf 证$ 必要性. 设 $I_{g,\varphi}$: $H(p,\,q,\, \phi)\rightarrow {\cal Z}_{0}$是有界算子,则对任意的 $f\in H(p,\,q,\,\phi)$, $I_{g,\varphi}f\in {\cal Z}_{0}$. 取检验函数$f(z)=z \in H(p,\,q,\,\phi)$,得到 $$ \lim_{|z|\rightarrow 1}(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|=0. $$ 即(3.1)式成立.

取检验函数 $f(z)=z^2\in H(p,\,q,\,\phi)$, 由三角不等式及 $\|\varphi\|_{\infty}\leq 1$,得到

\begin{eqnarray} &&2(1-|z|^{2})|\varphi'(z)|^2|g(\varphi(z))|\nonumber\\ &\leq &(1-|z|^{2})|(I_{g,\varphi}f)''(z)|+2(1-|z|^{2})|\varphi(z)||(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|\nonumber\\ &\leq &(1-|z|^{2})|(I_{g,\varphi}f)''(z)|+2(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|, \end{eqnarray}

(3.3)

由此及 (3.1)式,我们得到 $$\lim_{|z|\rightarrow1}(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|=0.$$ 即 (3.2)式成立.

充分性. 对任意的多项式 $p$,由于 \begin{eqnarray*} \label{x1} &&(1-|z|^{2})|(I_{g,\varphi}p)''(z)|\\ &\leq& (1-|z|^{2})|p''(\varphi(z))||\varphi'(z)|^2\cdot|g(\varphi(z))|\\ &&+(1-|z|^{2})|p'(\varphi(z))|\cdot|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|\\ &\leq& \|p''\|_\infty(1-|z|^{2})|\varphi'(z)|^2|g(\varphi(z))|\\ &&+\|p'\|_\infty(1-|z|^{2})\cdot|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|, \end{eqnarray*} 利用 (3.1) 和 (3.2)式,得出 $I_{g, \varphi}p\in {\cal Z}_{0}$. 由于多项式的全体在 $H(p,\,q,\,\phi)$ 中稠密, 所以对任意的 $f\in H(p,\,q,\,\phi)$, 存在多项式序列 $p_k$,使得 $$\lim\limits_{k\rightarrow \infty}\|p_k-f\|_{p,\,q,\,\phi}=0.$$ $I_{g,\varphi}: H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 的有界性给出 $$ \|I_{g,\varphi}f-I_{g,\varphi}p_k\|_{{\cal Z}} \leq\|I_{g,\varphi}\|_{H(p,\,q,\,\phi)\rightarrow{\cal Z}}\|p_k-f\|_{p,\,q,\,\phi} \rightarrow 0,k\rightarrow \infty, $$ 由于 ${\cal Z}_0$ 是 ${\cal Z}$ 的闭子空间,因此 $I_{g,\varphi}(H(p,\,q,\,\phi))\subseteq{\cal Z}_0$, 于是 $I_{g,\varphi}: H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 是有界算子.

$\bf 定理 3.2$ 设 $\varphi$ 是 $D\rightarrow D$ 的一个解析自映射,$\phi$ 是正规函数, $0<p,\,q<\infty$ 和 $g\in H(D)$. 则下列条件等价.

(a)~ $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是紧算子.

(b)~ 当 $\|\varphi\|_\infty <1$ 时,(3.1),(3.2) 式成立;

当 $\|\varphi\|_\infty =1$ 时,(3.1),(3.2) 式成立以及

\begin{equation} \label{e10}\lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }=0 , \end{equation}

(3.4)

\begin{equation} \label{e11}\lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }=0. \end{equation}

(3.5)

(c)

\begin{equation} \label{e27} \lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }=0, \end{equation}

(3.6)

\begin{equation} \label{e28}\lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }=0 .\end{equation}

(3.7)

$\bf 证$ (b)$\Rightarrow$(c). 如果 $\|\varphi\|_\infty <1$,利用 $\phi$ 的正规性,存在数 $s>0$,使得 $\frac{\phi(r)}{(1-r)^{s}}$ 在 $[0,1)$ 上单调减少,于是 $$\phi(\|\varphi\|_\infty)\leq \frac{\phi(|\varphi(z)|)(1-\|\varphi\|_\infty)^{s}}{(1-|\varphi(z)|)^{s}}\leq \phi(|\varphi(z)|),z\in D.$$ 利用 (3.1) 和 (3.2)式,我们得到 $$ \lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} } \leq \lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(\|\varphi\|_\infty)\left(1-\|\varphi\|_\infty^{2}\right)^{2+\frac{1}{q}} }=0 $$ 和 \begin{eqnarray*} &&\lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }\nonumber\\ &\leq& \lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|}{\phi(\|\varphi\|_\infty)\left(1-\|\varphi\|_\infty^{2}\right)^{1+\frac{1}{q}} }=0, \end{eqnarray*} 由此知 (3.6) 和 (3.7)式成立.

如果 $\|\varphi\|_\infty =1$,由(3.4)式可得, 对于任意 $\varepsilon> 0$,存在 $r\in (0,1),$使当 $r<|\varphi(z)|<1$时, 有 $$\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }<\varepsilon,$$ 由 (3.2)式可得,对于 $\varepsilon> 0,$存在 $\rho\in (0,1),$ 使当 $\rho<|z|<1$ 时,有

\begin{equation} \label{e24}(1-|z|^{2})|(\varphi'(z))^{2}g(\varphi(z))|<\varepsilon\inf_{t\in [0,r]}\phi(t)(1-t^{2})^{2+\frac{1}{q}}. \end{equation}

(3.8)

于是,当 $\rho<|z|<1,r<|\varphi(z)|<1$时,有

\begin{equation} \label{e25}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }<\varepsilon. \end{equation}

(3.9)

另一方面,若 $\rho<|z|<1$,$|\varphi(z)|\leq r,$ 由(3.8)式得

\begin{equation} \label{e26}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }\leq \frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\inf_{t\in [0,r]}\phi(t)(1-t^{2})^{2+\frac{1}{q}}}<\varepsilon. \end{equation}

(3.10)

由(3.9),(3.10)式得(3.6)式成立.

同样,由(3.5)和(3.1)式得(3.7)式成立.

(c)$\Rightarrow$(a). 由(3.6) 和 (3.7)式得 (2.1) 和 (2.2) 式成立, 从而由引理2.4知 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是有界算子. 对于任意 $f\in H(p,\,q,\,\phi),$在引理2.1中分别取 $n=1$,$n=2,$ 得

$\begin{array}{l} (1 - |z{|^2})|({I_{g,\phi }}f)\prime \prime (z)|\\ \le (1 - |z{|^2})|f\prime \prime (\varphi (z))\left| {{\rm{ }}\cdot{\rm{ }}} \right|\varphi \prime (z){|^2}\cdot{\rm{ }}|g(\varphi (z))|\\ + (1 - |z{|^2})|f\prime (\varphi (z))\left| {{\rm{ }}\cdot{\rm{ }}} \right|{(\varphi \prime (z))^2}g\prime (\phi (z)){\rm{ }} + \varphi \prime \prime (z)g(\varphi (z))|\\ \le C\frac{{(1 - |z{|^2})|\varphi \prime (z){|^2}\cdot{\rm{ }}|g(\varphi (z))|}}{{\phi (|\varphi (z)|){\rm{ }}{{(1 - |\varphi (z){|^2})}^{2 + \frac{1}{q}}}}}||f|{|_{p,{\rm{ }}q,{\rm{ }}\varphi }}\\ + C\frac{{(1 - |z{|^2})|{{(\varphi \prime (z))}^2}g\prime (\varphi (z)){\rm{ }} + \varphi \prime \prime (z)g(\varphi (z))|}}{{\phi (|\varphi (z)|){\rm{ }}{{(1 - |\varphi (z){|^2})}^{1 + \frac{1}{q}}}}}||f|{|_{p,{\rm{ }}q,{\rm{ }}\varphi }} \end{array}$

(3.11)

由 (3.6),(3.7)式和 (3.11) 式可知, 对于任意 $f\in H(p,\,q,\,\phi)$,$I_{g,\varphi}f\in {\cal Z}_{0}.$ 于是 $I_{g,\varphi}(H(p,\,q,\,\phi))\subseteq {\cal Z}_{0},$ 所以 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是有界算子. 故集合 $$I_{g,\varphi}\left(\{f\in H(p,\,q,\,\phi): \|f\|_{p,\,q,\,\phi}\leq 1\}\right)$$ 在 ${\cal Z}_0$ 中是有界的,由引理2.3可知,要证 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是紧算子,只要证

\begin{equation} \label{e30}\lim_{|z|\rightarrow1}\sup_{\|f\|_{p,\,q,\,\phi}\leq 1}(1-|z|^{2})|(I_{g,\varphi}f)''(z)|=0,\end{equation}

(3.12)

即可. 在不等式 (3.11) 中两边关于集合 $\{f\in H(p,\,q,\,\phi): \|f\|_{p,\,q,\,\phi}\leq 1\}$ 取上确界,然后令 $|z|\rightarrow 1$,由 (3.6),(3.7)式可得 (3.12) 式成立,所以 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是紧算子.

(a)$\Rightarrow$(b). 设 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$是紧算子,则 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$是有界算子,由定理3.1,我们得到 (3.1) 和 (3.2)式成立.

如果 $\|\varphi\|_\infty =1$,现在假设(3.4)式不成立, 则存在 $\varepsilon_{0}>0,$一序列 $\{z_k\}_{k\in N}\subset D$满足$\lim\limits_{k\rightarrow\infty}|\varphi(z_k)|=1, $及 $$\frac{(1-|z_k|^{2})|\varphi'(z_k)|^2\cdot|g(\varphi(z_k))|}{\phi(|\varphi(z_k)|)(1-|\varphi(z_k)|^{2})^{2+\frac{1}{q}}}\geq \varepsilon_{0}>0 . $$ 取文献[1,定理 2] 中检验函数列 $\{h_k\}_{k\in N}$,即 $$ h_k(z)=C_{1}\frac{(1-|\varphi(z_k)|^{2})^{t+2}}{\phi(|\varphi(z_k)|)\left(1-\overline{\varphi(z_k)}z\right)^{2+t+\frac{1}{q}}}- C_{2}\frac{(1-|\varphi(z_k)|^{2})^{t+1}}{\phi(|\varphi(z_k)|)\left(1-\overline{\varphi(z_k)}z\right)^{1+t+\frac{1}{q}}},k\in N, $$ 其中 $C_j=t+j+\frac{1}{q}$,$j=1,2$,$t$ 由正规函数 $\phi$ 确定. 那么 $h_k\in H(p,\,q,\,\phi)$,且 $\sup\limits_{k\in N}\|h_k\|_{p,\,q,\,\phi}\leq C. $又 $\{h_k\}$在 $D$的紧子集上一致收敛于0,且 $$h'_k(\varphi(z_k))=0,\,h''_k(\varphi(z_k))=\frac{C_{1}C_{2}\left(\overline{\varphi(z_k)}\right)^2}{\phi(|\varphi(z_k)|)\left(1-|\varphi(z_k)|^2\right)^{2+\frac{1}{q}}}.$$ 依引理2.2,算子 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 的紧性给出

\begin{equation} \label{e29}\lim_{k\rightarrow\infty}\|I_{g,\varphi}h_k\|_{\cal Z}=0.\end{equation}

(3.13)

另一方面,当 $k$ 充分大时, \begin{eqnarray*} \|I_{g,\varphi}h_k\|_{\cal Z} & \geq& \sup_{z\in D}(1-|z|^{2})|(I_{g,\varphi}h_k)''(z)|\\ &\geq& (1-|z_k|^{2})|\varphi'(z_k)|^2\cdot|g(\varphi(z_k))|\cdot|h_k''(\varphi(z_k))|\\ &&-(1-|z|^{2})|h'(\varphi(z_k))|\cdot|(\varphi'(z_k))^{2}g'(\varphi(z_k))+\varphi''(z_k)g(\varphi(z_k))|\\ &=&C_{1}C_{2}\frac{(1-|z_k|^{2})|\varphi'(z_k)|^2\cdot|\varphi(z_k)|^2\cdot|g(\varphi(z_k))|}{\phi(|\varphi(z_k)|)(1-|\varphi(z_k)|^{2})^{2+\frac{1}{q}}} \\ &\geq& \frac{C_{1}C_{2}}{2}\varepsilon_{0}. \end{eqnarray*} 可以看出上式与(3.13)式矛盾. 同理,取检验函数 $$f_k(z)=\frac{(1-|\varphi(z_k)|^{2})^{t+1}}{\phi(|\varphi(z_k)|)\left(1-\overline{\varphi(z_k)}z\right)^{1+t+\frac{1}{q}}},\ k\in N,$$ 可证(3.5)式成立

$\bf 注1$ 由 (a) 推出 (3.4),(3.5) 式成立也可直接证明. 这是因为 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是紧算子, 则 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是紧算子,根据文献[1,定理2]可知 (3.4),(3.5)式成立.

$\bf 注2$ 由定理3.1及定理3.2可以看出,当 $\|\varphi\|_\infty =1$ 时,算子$I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 的有界性和算子 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 的紧性并不等价.

4 $J_{g,\varphi}: H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$}的有界性和紧性

本节讨论算子 $J_{g,\varphi}: H(p,\,q,\,\phi) \rightarrow {\cal Z}_0$ 的有界性和紧性. $\bf 定理 4.1$ 设 $\varphi$ 是 $D\rightarrow D$ 的一个解析自映射,$\phi$ 是正规函数,$0<p,\,q<\infty$ 和 $g\in H(D)$. 则 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是有界算子的充分必要条件是 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是有界算子,

\begin{equation} \label{e51}\lim_{|z|\rightarrow1}(1-|z|^{2})\left|g''(\varphi(z))(\varphi'(z))^{2}+\varphi''(z)g'(\varphi(z))\right|=0, \end{equation}

(4.1)

\begin{equation} \label{e52}\lim_{|z|\rightarrow1}(1-|z|^{2})\left|(\varphi'(z))^{2}g'(\varphi(z))\right|=0. \end{equation}

(4.2)

$\bf 证$ 该定理的证明类似于定理3.1的证明, 此处略去

$\bf 定理 4.2$ 设 $\varphi$ 是 $D\rightarrow D$ 的一个解析自映射,$\phi$ 是正规函数,$0<p,\,q<\infty$ 和 $g\in H(D)$. 则下列条件等价.

(a)~ $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是紧算子.

(b)~ 当 $\|\varphi\|_\infty <1$ 时,(4.1),(4.2) 式成立;

当 $\|\varphi\|_\infty =1$ 时,(4.1),(4.2) 式成立以及

\begin{equation} \label{e41} \lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g'(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }=0,\end{equation}

(4.3)

\begin{equation} \label{e42}\lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g''(\varphi(z))+\varphi''(z)g'(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{\frac{1}{q}} }=0. \end{equation}

(4.4)

(c)

\begin{equation} \label{x42}\lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g''(\varphi(z))+\varphi''(z)g'(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{\frac{1}{q}} }=0,\end{equation}

(4.5)

\begin{equation} \label{x41}\lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }=0. \end{equation}

(4.6)

$\bf 证$ (b)$\Rightarrow$(c). 利用定理3.2 中同样的方法,由 (4.4) 和 (4.1)式可以得到 (4.5) 式成立. 由 (4.3) 和 (4.2) 式可以得到 (4.6) 式成立.

(c)$\Rightarrow$(a). 对于任意 $f\in H(p,\,q,\,\phi),$ 在引理2.1 中分别取 $n=0$,$n=1,$得

\begin{eqnarray} \label{x43} &&(1-|z|^{2})|(J_{g,\varphi}f)''(z)|\nonumber\\ &\leq& (1-|z|^{2})|\varphi'(z)|^2\cdot|g'(\varphi(z))|\cdot|f'(\varphi(z))|\nonumber\\ &&+(1-|z|^{2})|f(\varphi(z))|\cdot|(\varphi'(z))^{2}g''(\varphi(z))+\varphi''(z)g'(\varphi(z))|\nonumber\\ &\leq& C\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g'(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}}}\|f\|_{p,\,q,\,\phi}\nonumber\\ &&+C\frac{(1-|z|^{2})|(\varphi'(z))^{2}g''(\varphi(z))+\varphi''(z)g'(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{\frac{1}{q}}}\|f\|_{p,\,q,\,\phi}. \end{eqnarray}

(4.7)

由 (4.5),(4.6) 和 (4.7)式, 对任意的 $f\in H(p,\,q,\,\phi)$, $J_{g,\varphi}f\in {\cal Z}_{0}$ 且 (2.3) 和 (2.4) 式成立. 利用引理2.5,$J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是有界算子,从而 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是有界算子. 如果 $|\varphi(z)|\rightarrow 1$,则 $|z|\rightarrow 1$. 于是若 (4.5) 和 (4.6) 式成立,则 (4.3) 和 (4.4) 式成立,根据文献[1,定理6], $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是紧算子, ${\cal Z}_0$ 是 ${\cal Z}$ 的闭子空间表明 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 是紧算子.

(a)$\Rightarrow$(b). 设 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$是紧算子,则 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$是有界算子,取检验函数 $f(z)=1\in H(p,\,q,\,\phi)$,得到 $$\lim_{|z|\rightarrow 1}(1-|z|^{2})|g''(\varphi(z))(\varphi'(z))^{2}+\varphi''(z)g'(\varphi(z))|=0.$$ 取检验函数 $f(z)=z\in H(p,\,q,\,\phi)$并利用上式,得到 $$\lim_{|z|\rightarrow1}(1-|z|^{2})|\varphi'(z)|^2\cdot|g'(\varphi(z))|=0.$$ 我们得到 (4.1) 和 (4.2)式成立.

若 $\|\varphi\|_\infty =1$,现在假设(4.3)不成立, 分别取文献[1,定理 6] 中检验函数 $\{q_k\}_{k\in N}$ 和 $\{p_k\}_{k\in N}$,即 $$ q_k(z)=\frac{\left(1-|\varphi(z_k)|^2\right)^{t}}{\phi(|\varphi(z_k)|)\left(1-\overline{\varphi(z_k)}z\right)^{t+\frac{1}{q}}}- \frac{1}{\phi(|\varphi(z_k)|)(1-|\varphi(z_k)|^{2})^{\frac{1}{q}}}, $$ $$ p_k(z)=\frac{(1-|\varphi(z_k)|^{2})^{t}}{\phi(|\varphi(z_k)|)\left(1-\overline{\varphi(z_k)}z\right)^{t+\frac{1}{q}}}, $$ 类似定理3.2的证明可知 (4.3) 和 (4.4)式成立. $bf 注3$ 由定理4.1及定理4.2可以看出, 当 $\|\varphi\|_\infty =1$ 时,算子$J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 的有界性和算子 $J_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 的紧性并不等价.

$\bf 致谢:$ 作者衷心感谢审稿人提出的修改意见.

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