数学物理学报, 2023, 43(1): 1-13

$ \boldsymbol{H^{p,q,s}}({\Bbb D})$上的复合算子

陈洪欣,, 张学军,*, 周敏,

湖南师范大学数学与统计学院 长沙410006

Composition Operators on $ \boldsymbol{H^{p,q,s}}({\Bbb D})$

Chen Hongxin,, Zhang Xuejun,*, Zhou Min,

College of Mathematics and Statistics, Hunan Normal University, Changsha 410006

通讯作者: *张学军, E-mail: xuejunttt@263.net

收稿日期: 2022-05-2   修回日期: 2022-08-5  

基金资助: 国家自然科学基金(11942109)
湖南省自然科学基金(2022JJ30369)

Received: 2022-05-2   Revised: 2022-08-5  

Fund supported: The NSFC(11942109)
The NSFH(2022JJ30369)

作者简介 About authors

陈洪欣,E-mail:1755310775@qq.com

周敏,E-mail:1479898554@qq.com

摘要

设 $\varphi$ 是复平面${\Bbb C}$中单位圆盘 ${\Bbb D}$上的解析自映射. 该文刻画了一般Hardy型空间$H^{p,q,s}({\Bbb D})$上使得复合算子$C_{\varphi}$有界或者紧时的符号函数 $\varphi$.

关键词： 复合算子; 有界性; 紧性; 一般 Hardy 型空间; 单位圆盘

Abstract

Let $\varphi$ be an analytic self-map of the unit disc ${\Bbb D}$ in the complex plane ${\Bbb C}$. In this paper, the authors characterize those symbols $\varphi$ such that composition operators $C_{\varphi}$ are bounded or compact on the general Hardy type space $H^{p,q,s}({\Bbb D})$.

Keywords： Composition operator; Boundedness; Compactness; General Hardy type space; Unit disc

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本文引用格式

陈洪欣, 张学军, 周敏. $ \boldsymbol{H^{p,q,s}}({\Bbb D})$上的复合算子[J]. 数学物理学报, 2023, 43(1): 1-13

Chen Hongxin, Zhang Xuejun, Zhou Min. Composition Operators on $ \boldsymbol{H^{p,q,s}}({\Bbb D})$[J]. Acta Mathematica Scientia, 2023, 43(1): 1-13

1 引言

设 ${\Bbb D}$ 是复平面 ${\Bbb C}$上的单位圆盘, $H({\Bbb D})$ 表示 ${\Bbb D}$上的解析函数类, $H^{\infty}({\Bbb D})$ 表示 ${\Bbb D}$ 上的有界解析函数空间. 首先, 我们介绍几个本文要涉及的函数空间.

定义 1.1 设 $p>0$, 若 $h\in H({\Bbb D})$且满足

$ \| h\| _{p}=\sup_{u\in {\Bbb D}}(1-|u|^{2})^{p}\ | h(u)|<\infty,$

则称$h$属于增长空间 $H^{\infty}_{p}({\Bbb D})$.

定义 1.2 设 $p>0$ 和 $q> 0$, 若 $h\in H({\Bbb D})$且满足

$ \| h\| _{p,q}=\sup_{0\leq r<1}\left\{(1-r^{2})^{q}\int_{0}^{2\pi}|h(r{\rm e}^{{\rm i}\theta})|^{p}\ \frac{{\rm d}\theta}{2\pi}\right\}^{\frac{1}{p}}<\infty, $

则称$h$属于加权 Hardy 空间 $H_{q}^{p}({\Bbb D})$.

定义 1.3 设 $\beta>-1$ 和 $p>0$, 若 $h\in H({\Bbb D})$且满足

$\| h\| _{A_{\beta}^{p}}=\left(\int_{{\Bbb D}}|h(u)|^{p}\ {\rm d}v_{\beta}(u)\right)^{\frac{1}{p}}<\infty, $

则称$h$属于加权 Bergman 空间 $A_{\beta}^{p}({\Bbb D})$, 其中 ${\rm d}v$ 是${\Bbb D}$上正规化 Lebesgue 测度且 ${\rm d}v_{\beta}(u)=(\beta+1)(1-|u|^{2})^{\beta}\ {\rm d}v(u)$.

给定 $w\in {\Bbb D}$, 设 ${\Bbb D}$上的解析自同构 $\varphi_{w}(z)=\frac{w-z}{1-\overline{w}z}$, 很容易验证 $\varphi_{w}$ 满足 $\varphi_{w}(0)$ = $w$ 和 $\varphi_{w}(w)=0$以及 $\varphi_{w}^{-1}=\varphi_{w}$.

定义 1.4 设 $p>0$, $q+1\geq 0$, $s\geq 0$, $q+s\geq 0$, 若 $h\in H({\Bbb D})$ 且满足

$\| h\| _{p,q,s}=\sup_{0\leq \rho<1}M_{p,q,s}(\rho,h)<\infty,$

则称 $h$ 属于一般Hardy型空间 $H^{p,q,s}({\Bbb D})$, 其中

$M_{p,q,s}^{p}(\rho,h)=\sup_{w\in {\Bbb D}}(1-\rho^{2})^{q}\int_{0}^{2\pi}|h(\rho {\rm e}^{{\rm i}\theta})|^{p}\ (1-|\varphi_{w}(\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}\ \frac{{\rm d}\theta}{2\pi}. $

当 $p\geq 1$ 时, $H^{p,q,s}({\Bbb D})$ 依范数 $\|\cdot \| _{p,q,s}$ 构成一个Banach空间; 当 $0<p<1$ 时, $H^{p,q,s}({\Bbb D})$ 依距离 $r(h,g)=\| h-g\| ^{p}_{p,q,s}$ 构成一个完备的距离空间. 特别地, 当 $q=s=0$ 时, $H^{p,q,s}({\Bbb D})$ 恰好是 Hardy 空间$H^{p}({\Bbb D})$. 因此, $H^{p,q,s}({\Bbb D})$ 是 Hardy 空间 $H^{p}({\Bbb D})$ 的一种推广. 空间 $H^{p,q,s}({\Bbb D})$ 来源于一些实际应用[1⇓-3], 并在文献[4]中被正式命名为单位球上一般 Hardy 型空间. 最近, 我们在文献[5,6]中进一步讨论了该空间的一些性质. 实际上, $H^{p,q,s}({\Bbb D})$ 含有几个经典函数空间, 例如, 当 $s\geq 1=-q$ 时, $H^{p,q,s}({\Bbb D})= H^{\infty}({\Bbb D})$, 当 $s\geq 1>-q$ 时, $H^{p,q,s}({\Bbb D})=H^{\infty}_{\frac{q+1}{p}}({\Bbb D})$, 当 $q>0=s$ 时, $H^{p,q,s}({\Bbb D})=H^{p}_{q}({\Bbb D})$. 当 $0<s<1$ 时, $H^{p,q,s}({\Bbb D})$ 是一个不同于Hardy 空间 $H^{k}({\Bbb D})$ ( $0<k\leq \infty$) 和加权Hardy 空间 $H^{k}_{t}({\Bbb D})$ ($0<t<\infty$) 的新函数空间 (可参见文献[5]).

定义 1.5 设 $\varphi: {\Bbb D}\rightarrow {\Bbb D}$ 是一个解析映射, $\varphi$ 诱导$H({\Bbb D})$ 上一个算子$C_{\varphi}$: $C_{\varphi}g=g\circ\varphi$, 称该算子 $C_{\varphi}$ 为复合算子.

单位圆盘或单位球上各种解析函数之间复合型算子的研究已经具有相当长的历史, 并且有了大量的研究成果, 其中与本文有直接关系的文献如 [1⇓-3,7⇓⇓⇓⇓⇓⇓⇓⇓⇓-17] 等. 本文的主要目的就是讨论 $H^{p,q,s}({\Bbb D})$上复合算子 $C_{\varphi}$ 有界或紧时符号函数 $\varphi$ 满足的条件.

本文中,若存在常数 $c>0$ 使得 $ E_{1} \geq cE_{2}$ (或 $ E_{1} \leq cE_{2}$), 我们记为 $"E_{1} \gtrsim E_{2}" $ (或 $"E_{1}\lesssim E_{2}"$). 若 $"E_{1} \gtrsim E_{2}" $ 且 $"E_{1}\lesssim E_{2}"$, 称 $E_{1}$和$E_{2}$ 等价, 记为 $"E_{1}\asymp E_{2}"$.

2 一些引理

引理 2.1[5] 若 $h\in H^{p,q,s}({\Bbb D})$, 则

$|h(z)|\lesssim\frac{\| h\| _{p,q,s}}{(1-|z|^{2})^{\frac{q+1}{p}}} \ \ \mbox{对所有 $z\in {\Bbb D}$ 成立}. $

引理 2.2[9] 设 $\beta>-1$, 若 $h\in A_{\beta}^{1}({\Bbb D})$, 则可由积分表示

$h(z)=\int_{{\Bbb D}}\frac{h(w) \ {\rm d}v_{\beta}(w)}{(1-z \overline{w})^{2+\beta}} \ \ (z\in {\Bbb D}). $

引理 2.3[10] 若 $c>0$ 和 $\delta>-1$, 则

$\int_{0}^{2\pi}\frac{{\rm d}\theta}{|1-\overline{w} {\rm e}^{{\rm i}\theta}|^{1+c}}\asymp \int_{{\Bbb D}}\frac{(1-|u|^{2})^{\delta}\ {\rm d}v(u)}{|1-\overline{w}u|^{2+\delta+c}}\asymp\frac{1}{(1-|w|^{2})^{c}} \ \ \mbox{对所有 $w\in {\Bbb D}$ 成立}. $

下列引理是 Littlewood 从属定理 (可参见文献[12]).

引理 2.4[12] 设 $p>0$ 以及 $\psi:{\Bbb D}\rightarrow {\Bbb D}$ 是一个解析映射且满足 $\psi(0)=0$, 则

$\int_{0}^{2\pi}|f[\psi(\rho {\rm e}^{{\rm i}\theta})]|^{p}\ {\rm d}\theta\leq \int_{0}^{2\pi}|f(\rho {\rm e}^{{\rm i}\theta})|^{p}\ {\rm d}\theta $

对所有 $f\in H({\Bbb D})$ 和 $0\leq \rho<1$ 成立.

对 ${\Bbb D}$内两点有下列积分双向估计. 这些结果来自文献[4]中命题3.1 (情形 $n=1$).

引理 2.5[4] 设 $w$ 和 $a$ 是 $ {\Bbb D}$ 中两点.对 $r> 0$ 和 $t> 0$, 记

$I_{w,a}=\int_{0}^{2\pi}\frac{{\rm d}\theta}{|1-\overline{w}{\rm e}^{{\rm i}\theta}|^{ t}\ |1-\overline{a}{\rm e}^{{\rm i}\theta}|^{ r}}, $

则有下列双向估计

(1) 当 $t+r>1$ 和 $\max\{t,r\}<1$ 时, $ I_{w,a}\asymp \frac{1}{ |1-w\overline{a}|^{t+r-1}}$.

(2) 当 $t=1>r$ 时, $ I_{w,a}\asymp\frac{1}{|1-w\overline{a}|^{r}}\log\frac{e}{|1-w\overline{\varphi_{w}(a)}|}$.

(3) 当 $t=1=r$ 时, $ I_{w,a}\asymp \frac{1}{|1-w\overline{a}|}\log\frac{e}{1-|\varphi_{w}(a)|^{2}}$.

(4) 当 $t>1>r$ 时, $ I_{w,a}\asymp \frac{1}{(1-|w|^{2})^{ t-1}\ |1-w\overline{a}|^{ r}}$.

(5) 当 $t>1$ 且 $r>1$ 时, $I_{w,a} \asymp\frac{1}{(1-|w|^{2})^{t-1}|1-w\overline{a}|^{ r}} + \frac{1}{(1-|a|^{2})^{ r-1}|1-w\overline{a}|^{ t}}$.

(6) 当 $t>1=r$ 时, $I_{w,a} \asymp \frac{1}{(1-|w|^{2})^{t-1}|1-w\overline{a}|} + \frac{1}{|1-w\overline{a}|^{ t}}\log\frac{e}{1-|\varphi_{a}(w)|^{2}}$.

引理 2.6[6] 设 $p>0$ 和 $0<p_{0}\leq 1$, 则

$(1-\rho^{2})^{\frac{1-p_{0}}{p_{0}}}\int_{0}^{2\pi}|h(\rho^{2}{\rm e}^{{\rm i}\theta})|^{p}\ {\rm d}\theta\leq \left\{\int_{0}^{2\pi}|h(\rho {\rm e}^{{\rm i}\tau})|^{pp_{0}}\ {\rm d}\tau\right\}^{\frac{1}{p_{0}}} $

对所有 $h\in H({\Bbb D})$ 和 $0\leq \rho<1$ 成立.

3 主要结果

定理 3.1 设 $\varphi: \ {\Bbb D}\rightarrow {\Bbb D}$ 是一个解析映射.

(1) 若 $\varphi$ 是一个自同构, 则 $C_{\varphi}$ 在 $H^{p,q,s}({\Bbb D})$ 上有界.

(2) 若 $s\geq 1\geq -q$ 或 $s=0\leq q$, 则 $C_{\varphi}$ 在 $H^{p,q,s}({\Bbb D})$ 上总是有界的.

(3) 若 $q+1>0$ 且 $C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$ 上的紧算子, 则

$ \lim_{|w|\rightarrow 1^{-}}\frac{1-|w|^{2}}{1-|\varphi(w)|^{2}}=0.$

(4) 若 $s\geq 1> -q$ 或 $s=0< q$, 则 $C_{\varphi}$ 在 $H^{p,q,s}({\Bbb D})$ 上紧当且仅当(3.1)式成立.

(5) 若 $q=0=s$, 则 $C_{\varphi}$ 为 $H^{p,q,s}({\Bbb D})$ 上的紧算子当且仅当

$\lim_{|w|\rightarrow 1^{-}}\frac{N_{\varphi}(w)}{1-|w|^{2}}=0, $

其中 $N_{\varphi}(w)$ 是计数函数.

(6) 若 $\| \varphi\| _{\infty}<1$, 则 $C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$上的紧算子.

(7) 若 $q+1=0$, 则 $C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$ 上的紧算子当且仅当 $\| \varphi\| _{\infty}<1$.

(8) 当 $0<s<1$ 时, 若 $C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$ 上的有界算子, 则对任意 $t>0$ 有

$\sup_{0\leq \rho<1, a, w\in {\Bbb D}}(1-\rho^{2})^{q}\int_{0}^{2\pi}\frac{(1-|w|^{2})^{t}(1-|\varphi_{a}(\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}}{|1-\overline{w}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{q+1+t}}\ {\rm d}\theta<\infty.$

(9) 当 $0<s<1$ 且 $q+s>0$ 时, 如果 $p>1$ 取任意 $\beta >(q+1)/p-1$ 或者 $0<p\leq 1$ 取任意 $\beta>q+\max\{s,p\}-1$, 若

$\sup_{0\leq\tau\leq 2\pi}\int_{0}^{2\pi}\frac{(1-|a|^{2})^{s}|1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s}\ {\rm d}\theta}{(1-|\varphi(a)|^{2})^{s}|1-r {\rm e}^{-{\rm i}\tau}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2+\beta}|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}}\lesssim \frac{1}{(1-r\rho)^{\beta+1}}$

对所有 $a\in {\Bbb D}$ 和 $0\leq r,\rho<1$ 成立, 则 $C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$ 上的有界算子.

(10) 当 $0<s<1$ 且 $q+s>0$ 时, 若(3.4)式成立, 则 $C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$ 上的紧算子当且仅当(3.1)式成立.

证 为了下面的证明, 我们首先选取参数

$\alpha>\max\left\{\frac{q+1}{p}-1, \ \frac{q+s}{p}-1, \ \frac{q+s+1}{p}-2\right\}.$

(1) 若 $\varphi$ 是一个自同构, 则存在酉变换 $U$ 和对合自同构 $\varphi_{a_{0}}$ 使得 $\varphi=\varphi_{a_{0}}\circ U$, 其中 $a_{0}=\varphi(0)$. 因此, 我们仅需证明 $\varphi$ 是酉变换或者对合自同构时结论成立.

当 $\varphi$ 是一个酉变换时, 设 $\varphi(z)=\lambda z$ ($z\in {\Bbb D}$), 其中 $\lambda$ 是一个复常数满足 $|\lambda|=1$. 对任意 $g\in H^{p,q,s}({\Bbb D})$, 我们有

$ \begin{eqnarray*} \| C_{\varphi}g\| _{p,q,s}^{p}&=&\sup_{0\leq \rho<1}\sup_{a\in {\Bbb D}}(1-\rho^{2})^{q}\int_{0}^{2\pi}|g[\varphi(\rho {\rm e}^{{\rm i}\theta})]|^{p}(1-|\varphi_{a}(\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}\ \frac{{\rm d}\theta}{2\pi} \\ &=&\sup_{0\leq \rho<1}\sup_{ \lambda a\in {\Bbb D}}(1-|\lambda\rho|^{2})^{q}\int_{0}^{2\pi}|g(\lambda\rho {\rm e}^{{\rm i}\theta})|^{p} (1-|\varphi_{\lambda a}(\lambda\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}\ \frac{{\rm d}\theta}{2\pi}\\ &=&\| g\| _{p,q,s}^{p}. \end{eqnarray*}$

这表明 $C_{\varphi}$ 在 $H^{p,q,s}({\Bbb D})$ 上有界且 $\| \cdot\| _{p,q,s}$ 是酉不变的.

当 $\varphi$ 是一个对合自同构时, 根据 $\| \cdot\| _{p,q,s}$ 酉不变性可设 $\varphi=\varphi_{a_{0}}$ 满足 $a_{0}=\varphi(0)\geq 0$. 对任意 $0\leq\rho<1$, $0\leq \theta\leq 2\pi$, $a\in {\Bbb D}$, 根据变量替换 $\xi=\varphi_{a_{0}}(\rho {\rm e}^{{\rm i}\theta})$ 可得

$\xi=c_{0}+r{\rm e}^{{\rm i}\tau}=\frac{a_{0}-\rho {\rm e}^{{\rm i}\theta}}{1-a_{0}\rho {\rm e}^{{\rm i}\theta}}, \ \ c_{0}=\frac{(1-\rho^{2})a_{0}}{1-\rho^{2}a_{0}^{2}}, \ \ r=\frac{\rho(1-a_{0}^{2})}{1-\rho^{2}a_{0}^{2}}, \ \ c_{0}+r=\frac{a_{0}+\rho}{1+\rho a_{0}}, 1-c_{0}a_{0}=\frac{r}{\rho}, \ \ {\rm d}\theta=\frac{\varphi'_{a_{0}}(c_{0}+r{\rm e}^{{\rm i}\tau})r{\rm e}^{{\rm i}\tau}\ {\rm d}\tau}{\varphi_{a_{0}}(c_{0}+r{\rm e}^{{\rm i}\tau})}=\frac{1-\rho^{2}a_{0}^{2}}{|1-\rho a_{0}{\rm e}^{{\rm i}\tau}|^{2}}\ {\rm d}\tau.$

根据(3.5)式和 $a=\varphi_{a_{0}}[\varphi_{a_{0}}(a)],$ 结合文献[9]中引理 1.3 可得

$\begin{matrix} &\;& (1-\rho^{2})^{q}\int_{0}^{2\pi}|g[\varphi_{a_{0}}(\rho {\rm e}^{{\rm i}\theta})]|^{p}(1-|\varphi_{a}(\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}\ {\rm d}\theta \\ &\; =&(1-\rho^{2})^{q+s}(1-|a|^{2})^{s}\int_{0}^{2\pi}\frac{|g(c_{0}+r{\rm e}^{{\rm i}\tau})|^{p}(1-\rho^{2}a_{0}^{2})\ {\rm d}\tau}{|1-\rho a_{0}{\rm e}^{{\rm i}\tau}|^{2}|1-\overline{a}\varphi_{a_{0}}(c_{0}+r{\rm e}^{{\rm i}\tau})|^{2s}} \\ &\; =&\frac{(1-\rho^{2})^{q+s}(1-|a|^{2})^{s}(1-a_{0}^{2})^{2s}}{(1-\rho^{2}a_{0}^{2})^{2s-1}|1-a_{0}\overline{a}|^{2s}}\int_{0}^{2\pi}\frac{|g(c_{0}+r{\rm e}^{{\rm i}\tau})|^{p} \ {\rm d}\tau}{|1-\rho a_{0}{\rm e}^{{\rm i}\tau}|^{2-2s}|1-\overline{a}\varphi_{a_{0}}(c_{0}+r{\rm e}^{{\rm i}\tau})|^{2s}} \\ &\asymp&(1-\rho^{2})^{q+s}(1-|a|^{2})^{s}\int_{0}^{2\pi}\frac{|g(c_{0}+r{\rm e}^{{\rm i}\tau})|^{p}\ {\rm d}\tau}{|1-\overline{\varphi_{a_{0}}(a)}(c_{0}+r{\rm e}^{{\rm i}\tau})|^{2s}}. \end{matrix}$

我们考虑函数

$F(u)=\frac{g(u)}{(1-\overline{\varphi_{a_{0}}(a)}\ u)^{\frac{2s}{p}}} \ \ \ (u\in {\Bbb D}).$

根据条件 $\alpha>(q+1)/p-1$ 和引理 2.1, 2.2可得

$\begin{equation} F(w)=\int_{{\Bbb D}}\frac{F(u)}{(1-w\overline{u})^{2+\alpha}}\ {\rm d}v_{\alpha}(u) \ \ (w\in {\Bbb D}). \end{equation}$

(i) 情形 $q+s>0$.

当 $p>1$ 时, 条件 $\alpha> (q+s)/p-1$ 和 $q+s>0$表明我们可以取 $\max\{\alpha, q+s-1\}<\alpha_{1}<\min\{p(\alpha+1)-1, q+s+\alpha\}$, 使得 $(p\alpha-\alpha_{1})/(p-1)>-1, \ \alpha_{1}-\alpha>0, \ \alpha_{1}-q-s>-1, $$ q+s-\alpha_{1}+\alpha>0$. 由(3.7)式和 Hölder 不等式以及(3.5)式可得

$\begin{matrix} |F(c_{0}+r{\rm e}^{{\rm i}\tau})|^{p}&\leq&\left(\int_{{\Bbb D}}\frac{|F(u)|\ {\rm d}v_{\alpha}(u)}{|1-(c_{0}+r{\rm e}^{{\rm i}\tau})\overline{u}|^{2+\alpha}}\right)^{p} \\ & \lesssim& \int_{{\Bbb D}}\frac{|F(u)|^{p}(1-|u|^{2})^{\alpha_{1}} {\rm d}v(u)}{|1- (c_{0}+r{\rm e}^{{\rm i}\tau})\overline{u}|^{2+\alpha}}\left(\int_{{\Bbb D}}\frac{(1-|u|^{2})^{\frac{p\alpha-\alpha_{1}}{p-1}}\ {\rm d}v(u)}{|1- (c_{0}+r{\rm e}^{{\rm i}\tau})\overline{u}|^{2+\alpha}}\right)^{p-1} \\ & \asymp&\frac{1}{(1-|c_{0}+r{\rm e}^{{\rm i}\tau}|^{2})^{\alpha_{1}-\alpha}}\int_{{\Bbb D}}\frac{|F(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\ {\rm d}v(u)}{|1- (c_{0}+r{\rm e}^{{\rm i}\tau})\overline{u}|^{2+\alpha}} \\ & \lesssim&\frac{1}{(1-\rho^{2})^{\alpha_{1}-\alpha}}\int_{{\Bbb D}}\frac{|F(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\ {\rm d}v(u)}{|1-(c_{0}+r{\rm e}^{{\rm i}\tau})\overline{u}|^{2+\alpha}}. \end{matrix}$

因 $c_{0}+r=(a_{0}+\rho)/(1+\rho a_{0})<1$, 故 $r|\overline{u}|<1-c_{0}|\overline{u}|\leq |1-c_{0}\overline{u}|$. 因此,

$\left|\frac{r\overline{u}}{1-c_{0}\overline{u}}\right|<1 \ \ \mbox{对任意 $u\in D$ 成立}. $

由(3.6),(3.8)式, Fubini 定理, 引理 2.3, 文献[9,引理 1.8], $\alpha_{1}-q-s>-1$, $q+s-\alpha_{1}+\alpha>0$, 文献[引理 6], 文献[9,引理1.2], $c_{0}+r=(a_{0}+\rho)/(1+\rho a_{0})$, 我们可得

$\begin{eqnarray*} &\;& (1-\rho^{2})^{q}\int_{0}^{2\pi}|g[\varphi_{a_{0}}(\rho {\rm e}^{{\rm i}\theta})]|^{p} (1-|\varphi_{a}(\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}\ {\rm d}\theta \\ &\;\lesssim & \int_{{\Bbb D}}|F(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\left\{\int_{0}^{2\pi}\frac{(1-\rho^{2})^{q+s+\alpha-\alpha_{1}}(1-|a|^{2})^{s}\ {\rm d}\tau}{|1-(c_{0}+r{\rm e}^{{\rm i}\tau})\overline{u}|^{2+\alpha}}\right\}{\rm d}v(u)\\ &\; =&\int_{{\Bbb D}}\frac{|F(u)|^{p}(1-|u|^{2})^{\alpha_{1}}}{|1-c_{0}\overline{u}|^{2+\alpha}}\left\{\int_{0}^{2\pi}\frac{(1-\rho^{2})^{q+s+\alpha-\alpha_{1}}(1-|a|^{2})^{s}\ {\rm d}\tau}{|1-\frac{r\overline{u}}{1-c_{0}\overline{u}}\ {\rm e}^{{\rm i}\tau}|^{2+\alpha}}\right\}{\rm d}v(u)\\ &\; \asymp &(1-\rho^{2})^{q+s+\alpha-\alpha_{1}}(1-|a|^{2})^{s}\int_{{\Bbb D}}\frac{|1-c_{0}u|^{-1}|g(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\ {\rm d}v(u)}{|1-\overline{\varphi_{a_{0}}(a)}\ u|^{2s}\{|1-c_{0}u|-r|u|\}^{\alpha+1}}\\ &\; \lesssim &(1-\rho^{2})^{q+s+\alpha-\alpha_{1}}(1-|a|^{2})^{s}\int_{{\Bbb D}}\frac{|g(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\ {\rm d}v(u)}{|1-\overline{\varphi_{a_{0}}(a)}\ u|^{2s}\{1-(c_{0}+r)|u|\}^{\alpha+1}} \\ &\; =&(1-\rho^{2})^{q+s+\alpha-\alpha_{1}}(1-|a|^{2})^{s}\int_{0}^{1}\frac{2t(1-t^{2})^{\alpha_{1}}}{\{1-(c_{0}+r)t\}^{\alpha+1}}\frac{(1-t^{2})^{-s-q}}{(1-|\varphi_{a_{0}}(a)|^{2})^{s}} \\ &\;& \times \left\{(1-t^{2})^{q}\int_{0}^{2\pi}|g(t{\rm e}^{{\rm i}\theta})|^{p}(1-|\varphi_{\varphi_{a_{0}}(a)}(t{\rm e}^{{\rm i}\theta})|^{2})^{s}\ \frac{{\rm d}\theta}{2\pi}\right\}{\rm d}t \\ &\; \lesssim & \frac{(1-\rho)^{q+s-\alpha_{1}+\alpha}(1-|a|^{2})^{s}}{(1-|\varphi_{a_{0}}(a)|^{2})^{s}}\| g\| ^{p}_{p,q,s}\int_{0}^{1}\frac{(1-t)^{\alpha_{1}-q-s}\ {\rm d}t}{\{1-(c_{0}+r)t\}^{\alpha+1}}\\ &\; \asymp &\frac{(1-\rho)^{q+s-\alpha_{1}+\alpha}}{\{1-(c_{0}+r)\}^{q+s-\alpha_{1}+\alpha}}\| g\| ^{p}_{p,q,s}\asymp\| g\| ^{p}_{p,q,s}. \end{eqnarray*}$

当 $0<p\leq 1$ 时, 设 $\alpha=(2+\alpha')/p-2$. 条件 $\alpha>(q+s+1)/p-2$ 意味着 $\alpha'>q+s-1$. 由(3.7)式结合文献[9,引理 2.15] 可得

$\begin{equation} |F(c_{0}+r{\rm e}^{{\rm i}\tau})|^{p}\leq \left(\int_{{\Bbb D}}\frac{|F(u)|\ {\rm d}v_{\alpha}(u)}{|1-(c_{0}+r{\rm e}^{{\rm i}\tau})\overline{u}|^{2+\alpha}}\right)^{p} \lesssim \int_{{\Bbb D}}\frac{|F(u)|^{p}\ (1-|u|^{2})^{\alpha'} {\rm d}v(u)}{|1-(c_{0}+r{\rm e}^{{\rm i}\tau})\overline{u}|^{2+\alpha'}}. \end{equation}$

由(3.6)和(3.9)式, Fubini 定理, $r|\overline{u}|/|1-c_{0}\overline{u}|<1$, 引理 2.3, 文献[9,引理 1.8], $\alpha'-q-s>-1$, 文献[引理 6], $q+s>0$, $c_{0}+r=(a_{0}+\rho)/(1+\rho a_{0})$, 我们可得

$\begin{eqnarray*} &\;& (1-\rho^{2})^{q}\int_{0}^{2\pi}|g[\varphi_{a_{0}}(\rho {\rm e}^{{\rm i}\theta})]|^{p} (1-|\varphi_{a}(\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}\ {\rm d}\theta \\ &\;\lesssim &\int_{{\Bbb D}}|F(u)|^{p}(1-|u|^{2})^{\alpha'}\left\{\int_{0}^{2\pi}\frac{(1-\rho^{2})^{q+s}(1-|a|^{2})^{s}\ {\rm d}\tau}{|1-(c_{0}+r{\rm e}^{{\rm i}\tau})\overline{u}|^{2+\alpha'}}\right\}{\rm d}v(u) \\ &\; \lesssim &(1-\rho^{2})^{q+s}(1-|a|^{2})^{s}\int_{{\Bbb D}}\frac{|g(u)|^{p}(1-|u|^{2})^{\alpha'}\ {\rm d}v(u)}{|1-\overline{\varphi_{a_{0}}(a)}\ u|^{2s}\{1-(c_{0}+r)|u|\}^{\alpha'+1}}\\ &\; \lesssim &\frac{(1-\rho)^{q+s}(1-|a|^{2})^{s}}{(1-|\varphi_{a_{0}}(a)|^{2})^{s}}\| g\| ^{p}_{p,q,s}\int_{0}^{1}\frac{(1-t)^{\alpha'-q-s}\ {\rm d}t}{\{1-(c_{0}+r)t\}^{\alpha'+1}}\asymp\| g\| ^{p}_{p,q,s}. \end{eqnarray*}$

(ii) 情形 $q+s=0$.

在这种情形下, $H^{p,q,s}({\Bbb D})\subseteq H^{p}({\Bbb D})$, 因而 $g({\rm e}^{{\rm i}\tau})$ 在$ [2\pi]$ 上几乎处处有定义. 根据解析函数积分平均的递增性和文献[9,引理1.3]结合(3.5)式, 我们可得

$\begin{eqnarray*} (1-|a|^{2})^{s}\int_{0}^{2\pi}\left|\frac{g[\varphi_{a_{0}}(\rho {\rm e}^{{\rm i}\theta})]}{(1-\overline{a}\rho {\rm e}^{{\rm i}\theta})^{\frac{2s}{p}}}\right|^{p} {\rm d}\theta & \leq&(1-|a|^{2})^{s}\int_{0}^{2\pi}\left|\frac{g[\varphi_{a_{0}}( {\rm e}^{{\rm i}\theta})]}{(1-\overline{a} {\rm e}^{{\rm i}\theta})^{\frac{2s}{p}}}\right|^{p} {\rm d}\theta\\ &\; =&(1-|a|^{2})^{s}\int_{0}^{2\pi}\left|\frac{g({\rm e}^{{\rm i}\tau})}{\{1-\overline{a}\varphi_{a_{0}}({\rm e}^{{\rm i}\tau})\}^{\frac{2s}{p}}}\right|^{p} \frac{1-a_{0}^{2}}{|1-a_{0}{\rm e}^{{\rm i}\tau}|^{2}}\ {\rm d}\tau\\ &\; =&(1-|a|^{2})^{s}\int_{0}^{2\pi}\frac{|g({\rm e}^{{\rm i}\tau})|^{p}|1-a_{0}{\rm e}^{{\rm i}\tau}|^{2s-2}(1-a_{0}^{2})}{|1-a_{0}\overline{a}|^{2s}|1-\overline{\varphi_{a_{0}}(a)}{\rm e}^{{\rm i}\tau}|^{2s}}\ {\rm d}\tau\\ &\; \asymp &\int_{0}^{2\pi}\frac{|g({\rm e}^{{\rm i}\tau})|^{p}\ (1-|\varphi_{a_{0}}(a)|^{2})^{s}}{|1-\overline{\varphi_{a_{0}}(a)}{\rm e}^{{\rm i}\tau}|^{2s}}\ \frac{{\rm d}\tau}{2\pi}\leq\| g\| _{p,q,s}^{p}. \end{eqnarray*}$

总之, 对所有 $q+s\geq 0$, 我们都有 $\| C_{\varphi}g\| _{p,q,s}\lesssim \| g\| _{p,q,s}$.

(2) 根据文献[5,命题3.3], 当 $s\geq 1> -q$ 时, $H^{p,q,s}({\Bbb D})=H^{\infty}_{\frac{q+1}{p}}({\Bbb D})$;当 $s\geq 1=-q$ 时, $H^{p,q,s}({\Bbb D})=H^{\infty}({\Bbb D})$. 因此, 当 $s\geq 1\geq -q$ 时, $C_{\varphi}$ 在 $H^{p,q,s}({\Bbb D})$ 上总是有界的. 我们仅仅需要证明情形 $s=0\leq q$ 时的结果. 实际上, $\varphi=\varphi_{a_{0}}\circ\psi$, 其中 $a_{0}=\varphi(0)$ 和 $\psi: {\Bbb D}\rightarrow {\Bbb D}$ 是一个解析映射且满足 $\psi(0)=0$. 对任意 $g\in H^{p,q,s}({\Bbb D})$ 和 $0\leq\rho<1$, 根据引理 2.4 可得

$ (1-\rho^{2})^{q}\int_{0}^{2\pi}|g[\varphi(\rho {\rm e}^{{\rm i}\theta})]|^{p}\ {\rm d}\theta \leq (1-\rho^{2})^{q}\int_{0}^{2\pi}|g[\varphi_{a_{0}}(\rho {\rm e}^{{\rm i}\theta})]|^{p} {\rm d}\theta. $

余下的证明和(1)中情形 $s=0$ 时相同.

(3) 当 $\| \varphi\| _{\infty}=\sup \{ |\varphi(z)|: z\in D \}<1$ 时,(3.1)式是显然的. 当 $\| \varphi\| _{\infty}=1$ 时, 设 $\{w^{j}\}\subset {\Bbb D}$ 是任一使得 $|\varphi(w^{j})|\rightarrow 1$ ($j\rightarrow\infty$) 的序列. 我们取

$g_{j}(w)=\frac{(1-|\varphi(w^{j})|^{2})^{\frac{|q|}{p}+1}}{(1- w\overline{\varphi(w^{j})})^{\frac{q+1+|q|}{p}+1}} \ \ \ (w\in {\Bbb D}). $

则 $\{g_{j}\}$ 在 ${\Bbb D}$ 的任一紧子集上一致趋于0. 根据引理 2.5, 分三种情况 $0\leq 2s<1$, $2s=1$, $2s>1$, 容易证明$\sup \{\| g_{j}\| _{p,q,s}: j=1,2,\cdots \}\lesssim 1$ (可参见文献[5]). 因此, 由引理2.1可得

$\begin{equation} \left(\frac{1-|w^{j }|^{2}}{1-|\varphi(w^{j})|^{2}}\right)^{\frac{q+1}{p}}=(1-|w^{j}|^{2})^{\frac{q+1}{p}}|C_{\varphi}g_{j}(w^{j})|\lesssim \| C_{\varphi}g_{j}\| _{p,q,s}\rightarrow 0 \ \ (j\rightarrow\infty). \end{equation}$

这表明(3.1)式成立.

(4) 必要性在结论 (3)中已经证明. 下面我们证充分性.

设序列 $\{h_{k}\}$ 在 ${\Bbb D}$ 的任一紧子集上一致趋于0, 且对所有 $k\in \{1,2,\cdots\}$满足 $\| h_{k}\| _{p,q,s}\leq 1$. 当 $s\geq 1>-q$ 时, $H^{p,q,s}({\Bbb D})=H^{\infty}_{\frac{q+1}{p}}({\Bbb D})$. 若 $(3.1)$ 式成立, 容易证明 $\| C_{\varphi}h_{k}\| _{p,q,s}\rightarrow0 \ (k\rightarrow\infty)$. 当 $s=0<q$ 时, 对任意 $k\in \{1,2,\cdots\}$ 和 $\rho\in (1/2,1)$, 由(3.7)式, 我们可以取

$ T_{\rho}h_{k}(w)=\int_{{\Bbb D}}H_{\rho}(w,u)h_{k}(u)\ {\rm d}v_{\alpha}(u), $

其中

$H_{\rho}(w,u)=\frac{\chi_{\rho}(w)}{ |1-\varphi(w)\overline{u}|^{2+\alpha}} \ \ (w,\ u \in {\Bbb D}), $

$\chi_{\rho}$ 表示集合 $\{z: \rho<|z|<1\}$ 的特征函数. 设 $a_{0}=\varphi(0)$ 和

$ M_{\rho}=\sup_{\rho<|z|<1}\frac{1-|z|^{2}}{1-|\varphi(z)|^{2}}. $

当 $p>1$ 时, 条件 $\alpha> q/p-1$ 和 $q>0$ 意味着我们可以取 $\max\{\alpha, q-1\}<\alpha_{1}<\min\{p(\alpha+1)-1, q+\alpha\}$ 使得 $\left(\alpha-\frac{\alpha_{1}}{p}\right)\frac{p}{p-1}>-1, \ \alpha_{1}-\alpha>0, \ \alpha_{1}-q>-1, \ q-\alpha_{1}+\alpha>0.$ 当 $|w|>\rho$ 时, 根据 Hölder 不等式和引理2.3可得

$ \begin{matrix} \{T_{\rho}(|h_{k}|)(w)\}^{p}&\lesssim& \frac{\chi_{\rho}(w)}{(1-|\varphi(w)|^{2})^{\alpha_{1}-\alpha}}\int_{{\Bbb D}}\frac{|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\ {\rm d}v(u)}{|1-\varphi(w)\overline{u}|^{2+\alpha}} \\ &\leq &\frac{M_{\rho}^{\alpha_{1}-\alpha}}{(1-|w|^{2})^{\alpha_{1}-\alpha}}\int_{{\Bbb D}}\frac{|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\ {\rm d}v(u)}{|1-\varphi(w)\overline{u}|^{2+\alpha}}. \end{matrix}$

如果 $1/2<r<1$ 且 $u\in {\Bbb D}$, 我们有

$\begin{matrix} 1-r^{2}|\varphi_{a_{0}}(u)|^{2} & = & 1-r^{2}+\frac{r^{2}(1-|a_{0}|^{2})(1-|u|^{2})}{|1-\overline{a_{0}}u|^{2}} \\ & \geq& 1-r^{2}+\frac{1-|a_{0}|}{4(1+|a_{0}|)}(1-|u|^{2}) \\ &\geq& \frac{1-|a_{0}|}{4(1+|a_{0}|)}(2-r-|u|) \\ &\geq& \frac{1-|a_{0}|}{4(1+|a_{0}|)}(1-r|u|). \end{matrix}$

如果 $0\leq r\leq 1/2$ 和 $u\in {\Bbb D}$, 我们有

$\begin{equation} 1-r^{2}|\varphi_{a_{0}}(u)|^{2}\geq \frac{3}{4}\geq \frac{3(2-r-|u|)}{8}\geq \frac{3(1-r|u|)}{8}. \end{equation}$

设 $\varphi=\varphi_{a_{0}}\circ\psi$ 满足 $\psi(0)=0$. 当$\rho<r<1$时, 根据 Fubini 定理, 引理 2.4, 文献[9,引理1.3和引理1.8],(3.11)-(3.13)式, $\alpha_{1}-q>-1$, $q-\alpha_{1}+\alpha>0$, 文献[引理6], 可得

$\begin{matrix} &\;& (1-r^{2})^{q}\int_{0}^{2\pi}\{T_{\rho}(|h_{k}|)(r{\rm e}^{{\rm i}\theta})\}^{p}\ {\rm d}\theta \\ &\lesssim& M_{\rho}^{\alpha_{1}-\alpha}(1-r^{2})^{q+\alpha-\alpha_{1}} \int_{{\Bbb D}}|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\left\{\int_{0}^{2\pi}\frac{ {\rm d}\theta}{|1-\varphi_{a_{0}}[\psi(r{\rm e}^{{\rm i}\theta})]\overline{u}|^{2+\alpha}}\right\}{\rm d}v(u) \\ &\; \leq & \int_{{\Bbb D}}|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\left\{\int_{0}^{2\pi}\frac{ M_{\rho}^{\alpha_{1}-\alpha}(1-r^{2})^{q+\alpha-\alpha_{1}}\ {\rm d}\theta}{|1-\varphi_{a_{0}}(r{\rm e}^{{\rm i}\theta})\overline{u}|^{2+\alpha}}\right\}{\rm d}v(u) \\ &\; \asymp & \int_{{\Bbb D}}|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\left\{\int_{0}^{2\pi}\frac{ M_{\rho}^{\alpha_{1}-\alpha}(1-r^{2})^{q+\alpha-\alpha_{1}}\ {\rm d}\theta}{|1- r{\rm e}^{{\rm i}\theta}\overline{\varphi_{a_{0}}(u)}|^{2+\alpha}}\right\}{\rm d}v(u) \\ &\; \asymp & M_{\rho}^{\alpha_{1}-\alpha}(1-r^{2})^{q+\alpha-\alpha_{1}}\int_{{\Bbb D}}\frac{|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\ {\rm d}v(u)}{(1-r^{2}|\varphi_{a_{0}}(u)|^{2})^{1+\alpha}} \\ &\; \lesssim & M_{\rho}^{\alpha_{1}-\alpha}(1-r^{2})^{q+\alpha-\alpha_{1}}\int_{{\Bbb D}}\frac{|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha_{1}}\ {\rm d}v(u)}{(1-r|u|)^{1+\alpha}} \\ &\lesssim& M_{\rho}^{\alpha_{1}-\alpha}(1-r)^{q+\alpha-\alpha_{1}}\| h_{k}\| _{p,q,s}^{p}\int_{0}^{1}\frac{(1-t)^{\alpha_{1}-q}\ {\rm d}t}{(1-rt)^{1+\alpha}} \lesssim M_{\rho}^{\alpha_{1}-\alpha}. \end{matrix}$

根据(3.7)(情形 $s=0$) 和(3.14)式, 有

$\begin{eqnarray*} &\;& \sup_{0\leq r<1}(1-r^{2})^{q}\int_{0}^{2\pi}|h_{k}[\varphi(r {\rm e}^{{\rm i}\theta})]|^{p}\ {\rm d}\theta\\ &\lesssim& \sup_{0\leq r\leq \rho}(1-r^{2})^{q}\int_{0}^{2\pi}|h_{k}[\varphi(r {\rm e}^{{\rm i}\theta})]|^{p}\ {\rm d}\theta +\sup_{\rho< r<1}(1-r^{2})^{q}\int_{0}^{2\pi}\{T_{\rho}(|h_{k}|)(r{\rm e}^{{\rm i}\theta})\}^{p}\ {\rm d}\theta\\ & \lesssim &\sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p}+M_{\rho}^{\alpha_{1}-\alpha}\rightarrow M_{\rho}^{\alpha_{1}-\alpha} \ \ (k\rightarrow\infty). \end{eqnarray*}$

因而当(3.1)式成立时我们有 $\| C_{\varphi}h_{k}\| _{p,q,s}\rightarrow0 \ (k\rightarrow\infty)$. 这表明 $C_{\varphi}$是 $H^{p,q,s}({\Bbb D})$上的紧算子.

当 $0<p\leq 1$时, 设 $\alpha=(2+\alpha')/p-2$, 这意味着 $\alpha'>q-1$. 我们选取 $0<\varepsilon<\min\{q,1+\alpha'\}$. 由(3.7)式, 文献[9,引理2.15], Fubini 定理, 文献[9,引理 1.8], 引理 2.4, 文献[9,引理 1.3], 文献[引理 6], 我们可得

$\begin{matrix} &\;& \sup_{0\leq t<1}(1-t^{2})^{q}\int_{0}^{2\pi}|h_{k}[\varphi(t {\rm e}^{{\rm i}\theta})]|^{p}\ {\rm d}\theta \\ &\lesssim& \sup_{0\leq t\leq \rho}(1-t^{2})^{q}\int_{0}^{2\pi}|h_{k}[\varphi(t {\rm e}^{{\rm i}\theta})]|^{p}\ {\rm d}\theta \\ &&+\sup_{\rho< t<1} \int_{{\Bbb D}}|h_{k}(u)|^{p}\left\{\int_{0}^{2\pi}\frac{ (1-t^{2})^{q}(1-|u|^{2})^{\alpha'} \ {\rm d}\theta}{|1- \varphi(t{\rm e}^{{\rm i}\theta})\overline{u}|^{2+\alpha'}}\right\}{\rm d}v(u) \\ &\lesssim& \sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p} + M_{\rho}^{\varepsilon}\sup_{\rho< t<1} \int_{{\Bbb D}}|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha'}\left\{\int_{0}^{2\pi}\frac{ (1-t^{2})^{q-\varepsilon} \ {\rm d}\theta}{|1- u\overline{\varphi(t{\rm e}^{{\rm i}\theta})}|^{2+\alpha'-\varepsilon}}\right\}{\rm d}v(u) \\ & \leq&\sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p}+\sup_{\rho< t<1} \int_{{\Bbb D}}|h_{k}(u)|^{p}\left\{\int_{0}^{2\pi}\frac{M_{\rho}^{\varepsilon}(1-|u|^{2})^{\alpha'} (1-t^{2})^{q-\varepsilon} {\rm d}\theta}{|1- u\overline{\varphi_{a_{0}}(t{\rm e}^{{\rm i}\theta})}|^{2+\alpha'-\varepsilon}}\right\}{\rm d}v(u) \\ & \asymp&\sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p}+\sup_{\rho< t<1} \int_{{\Bbb D}}|h_{k}(u)|^{p}\left\{\int_{0}^{2\pi}\frac{M_{\rho}^{\varepsilon}(1-|u|^{2})^{\alpha'} (1-t^{2})^{q-\varepsilon} {\rm d}\theta}{|1- \overline{\varphi_{a_{0}}(u)}t{\rm e}^{{\rm i}\theta}|^{2+\alpha'-\varepsilon}}\right\}{\rm d}v(u) \\ & \asymp&\sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p}+\sup_{\rho< t<1} \int_{{\Bbb D}}|h_{k}(u)|^{p}\left\{\frac{M_{\rho}^{\varepsilon}(1-|u|^{2})^{\alpha'} (1-t^{2})^{q-\varepsilon}}{(1-t^{2}|\varphi_{a_{0}}(u)|^{2})^{1+\alpha'-\varepsilon}}\right\}{\rm d}v(u) \\ & \lesssim &\sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p}+M_{\rho}^{\varepsilon}\sup_{\rho< t<1} (1-t^{2})^{q-\varepsilon}\int_{{\Bbb D}}\frac{|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha'}\ {\rm d}v(u)}{(1-t|u|)^{1+\alpha'-\varepsilon}} \\ & \lesssim& \sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p}+M_{\rho}^{\varepsilon}\sup_{\rho< t<1} (1-t)^{q-\varepsilon}\| h_{k}\| _{p,q,s}^{p}\int_{0}^{1}\frac{(1-\varrho)^{\alpha'-q}\ {\rm d}\varrho}{(1-t\varrho)^{1+\alpha'-\varepsilon}} \\ &\lesssim &\sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p}+M_{\rho }^{\varepsilon}\rightarrow M_{\rho}^{\varepsilon} \ \ (k\rightarrow\infty). \end{matrix}$

根据(3.1)和(3.15)式可得 $C_{\varphi}$ 是$H^{p,q,s}({\Bbb D})$上的紧算子.

(5) 该结果来自文献[7,定理 3.20].

(6) 如果 $\| \varphi\| _{\infty}<1$, 则任意序列$\{h_{k}\}$满足在${\Bbb D}$的任一紧子集上一致趋于0且 $\| h_{k}\| _{p,q,s}\leq 1$ 对所有 $k$ 成立时有

$\begin{eqnarray*} &&\sup_{0\leq \rho<1}\sup_{a\in {\Bbb D}}(1-\rho^{2})^{q}\int_{0}^{2\pi}|h_{k}[\varphi(\rho {\rm e}^{{\rm i}\theta})]|^{p}(1-|\varphi_{a}(\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}\ {\rm d}\theta \\ &\leq &\max_{|w|\leq \| \varphi\| _{\infty}}|h_{k}(w)|^{p}\sup_{0\leq \rho<1}\sup_{a\in {\Bbb D}}\int_{0}^{2\pi}\frac{(1-\rho^{2})^{q+s}(1-|a|^{2})^{s}\ {\rm d}\theta}{|1-\rho\overline{a} {\rm e}^{{\rm i}\theta}|^{2s}} \\ &\lesssim& \max_{|w|\leq \| \varphi\| _{\infty}}|h_{k}(w)|^{p} \ \rightarrow 0 \ \ (k\rightarrow\infty) \ \Rightarrow \ \mbox{$C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$上的紧算子.} \end{eqnarray*}$

(7) 当 $q+1=0$ 和 $\| \varphi\| _{\infty}=1$时, 假如 $C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$上的紧算子, 我们可导出与(3.10)式矛盾.

(8) 对任意 $w\in {\Bbb D}$ 和 $t>0$, 设

$f_{w}(z)=\frac{(1-|w|^{2})^{\frac{t}{p}}}{(1-z\overline{w})^{\frac{q+1+t}{p}}} \ \ (z\in {\Bbb D}). $

根据引理 2.5(1)-(6), 容易证明 $\sup \{\| f_{w}\| _{p,q,s}: w\in {\Bbb D}\}\lesssim 1$. 根据$C_{\varphi}$在$H^{p,q,s}({\Bbb D})$上的有界性可得(3.3)式成立.

(9) 对任意 $g\in H^{p,q,s}({\Bbb D})$ 和$a\in {\Bbb D}$, 取

$G_{a,g}(w)=\frac{g(w)}{(1-\overline{\varphi(a)}w)^{\frac{2s}{p}}} \ \ (w\in {\Bbb D}). $

根据引理 2.1, 2.2 可得

$G_{a,g}(w)=\int_{{\Bbb D}}\frac{G_{a,g}(u)\ {\rm d}v_{\alpha}(u)}{(1-w\overline{u})^{2+\alpha}} \ \ (w\in {\Bbb D}). $

因此,

$|G_{a,g}[\varphi(z)]|\leq \int_{{\Bbb D}}\frac{|G_{a,g}(u)|\ {\rm d}v_{\alpha}(u)}{|1-\varphi(z) \overline{u}|^{2+\alpha}} \ \ (z\in {\Bbb D}). $

对任意 $0\leq \rho<1$, 若 $p>1$ 和 $\max\{0, -q\}<s<1$, 由条件 $\alpha>(q+1)/p-1$ 可选择

$\max\{\alpha, \ q+s-1\}<\alpha_{1}<\min\{p(\alpha+1)-1, \ q+s+\alpha\}. $

当(3.4)式成立时, 根据(3.16)式, Hölder 不等式,

$(1-\rho^{2})/\{1-|\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2}\}\leq \{1+|\varphi(0)|\}/\{1-|\varphi(0)|\}, $

Fubini 定理, 文献[9,引理 1.8],(3.4)式, 文献[引理 6], 我们可得

$\begin{eqnarray*} &\;& (1-\rho^{2})^{q}\int_{0}^{2\pi}|g[\varphi(\rho {\rm e}^{{\rm i}\theta})]|^{p}(1-|\varphi_{a}(\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}\ {\rm d}\theta\\ &\; =&(1-\rho^{2})^{q+s}(1-|a|^{2})^{s}\int_{0}^{2\pi}|G_{a,g}[\varphi(\rho {\rm e}^{{\rm i}\theta})]|^{p}\frac{|1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s}}{|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}}\ {\rm d}\theta\\ &\;\lesssim & (1-\rho^{2})^{q+s}(1-|a|^{2})^{s}\int_{0}^{2\pi}\left\{\int_{{\Bbb D}}\frac{|G_{a,g}(u)|\ {\rm d}v_{\alpha}(u)}{|1-\varphi(\rho {\rm e}^{{\rm i}\theta}) \overline{u}|^{2+\alpha}}\right\}^{p}\frac{|1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s}}{|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}}\ {\rm d}\theta \\ &\; \lesssim & \int_{0}^{2\pi}\int_{{\Bbb D}}\frac{(1-\rho^{2})^{q+s}(1-|a|^{2})^{s}|g(u)|^{p}|1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s} (1-|u|^{2})^{\alpha_{1}}\ {\rm d}v(u){\rm d}\theta}{(1-|\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2})^{\alpha_{1}-\alpha}|1-\varphi(\rho {\rm e}^{{\rm i}\theta}) \overline{u}|^{2+\alpha}|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}|1-\overline{\varphi(a)}u|^{2s}} \\ &\; \lesssim & \int_{0}^{2\pi}\int_{{\Bbb D}}\frac{(1-\rho^{2})^{q+s+\alpha-\alpha_{1}}|g(u)|^{p}|1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s} (1-|u|^{2})^{\alpha_{1}}}{(1-|a|^{2})^{-s}|1-\varphi(\rho {\rm e}^{{\rm i}\theta}) \overline{u}|^{2+\alpha}|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}|1-\overline{\varphi(a)}u|^{2s}}\ {\rm d}v(u){\rm d}\theta \\ &\; =&\int_{{\Bbb D}}\frac{|g(u)|^{p}(1-|u|^{2})^{\alpha_{1}}}{(1-|a|^{2})^{-s}|1-\overline{\varphi(a)}u|^{2s}}\int_{0}^{2\pi}\frac{(1-\rho^{2})^{q+s+\alpha-\alpha_{1}} |1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s}\ {\rm d}\theta {\rm d}v(u)}{|1-\varphi(\rho {\rm e}^{{\rm i}\theta}) \overline{u}|^{2+\alpha}|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}}\\ &\; \lesssim & \int_{0}^{1}(1-r^{2})^{\alpha_{1}-q-s}\int_{0}^{2\pi}(1-r^{2})^{q}|g(r {\rm e}^{{\rm i}\tau})|^{p}(1-|\varphi_{\varphi(a)}(r {\rm e}^{{\rm i}\tau})|^{2})^{s}\\ &\;& \times \left\{\frac{(1-|a|^{2})^{s}}{(1-|\varphi(a)|^{2})^{s}}\int_{0}^{2\pi}\frac{(1-\rho^{2})^{q+s+\alpha-\alpha_{1}}|1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s}\ {\rm d}\theta}{|1-r {\rm e}^{-{\rm i}\tau}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2+\alpha}|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}}\right\} {\rm d}\tau {\rm d}r\\ &\; \lesssim &(1-\rho)^{q+s+\alpha-\alpha_{1}}\| g\| _{p,q,s}^{p}\int_{0}^{1}\frac{(1-r)^{\alpha_{1}-q-s}}{(1-r\rho)^{\alpha+1}}\ {\rm d}r \lesssim \| g\| _{p,q,s}^{p}. \end{eqnarray*}$

这意味着 $C_{\varphi}$是$H^{p,q,s}({\Bbb D})$上的有界算子.

当 $0<p\leq 1$ 和 $\max\{0, -q\}<s<1$ 时, 设 $\alpha=(\alpha'+2)/p-2$. 由 $\alpha>\max\{(q+1)/p-1,$$(q+s+1)/p-2\}$ 知, $\alpha'>q+\max\{s,p\}-1$. 若(3.4)式成立, 由(3.16)式, 文献[9,引理 2.15], Fubini 定理, 文献[9,引理 1.8],(3.4)式, 文献[引理 6], 有

$\begin{eqnarray*} &\;& (1-\rho^{2})^{q}\int_{0}^{2\pi}|g[\varphi(\rho {\rm e}^{{\rm i}\theta})]|^{p}(1-|\varphi_{a}(\rho {\rm e}^{{\rm i}\theta})|^{2})^{s}\ {\rm d}\theta \\ &\; \lesssim &\int_{0}^{2\pi}\int_{{\Bbb D}}\frac{(1-\rho^{2})^{q+s}(1-|a|^{2})^{s}|g(u)|^{p}|1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s}(1-|u|^{2})^{\alpha'}}{|1-\varphi(\rho {\rm e}^{{\rm i}\theta}) \overline{u}|^{2+\alpha'}|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}|1-\overline{\varphi(a)}u|^{2s}}\ {\rm d}\theta {\rm d}v(u) \\ &\; =&\int_{{\Bbb D}}\frac{(1-|a|^{2})^{s}|g(u)|^{p}(1-|u|^{2})^{\alpha'}}{|1-\overline{\varphi(a)}u|^{2s}}\int_{0}^{2\pi}\frac{(1-\rho^{2})^{q+s} |1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s}\ {\rm d}\theta {\rm d}v(u)}{|1-\varphi(\rho {\rm e}^{{\rm i}\theta}) \overline{u}|^{2+\alpha'}|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}}\\ &\; \lesssim &\int_{0}^{1}(1-r^{2})^{\alpha'-q-s}\int_{0}^{2\pi}(1-r^{2})^{q}|g(r {\rm e}^{{\rm i}\tau})|^{p}(1-|\varphi_{\varphi(a)}(r {\rm e}^{{\rm i}\tau})|^{2})^{s}\\ &\;& \times \left\{\int_{0}^{2\pi}\frac{(1-|a|^{2})^{s}(1-\rho^{2})^{q+s}|1-\overline{\varphi(a)}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2s}\ {\rm d}\theta}{(1-|\varphi(a)|^{2})^{s}|1-r {\rm e}^{-{\rm i}\tau}\varphi(\rho {\rm e}^{{\rm i}\theta})|^{2+\alpha'}|1-\overline{a}\rho {\rm e}^{{\rm i}\theta}|^{2s}}\right\}{\rm d}\tau {\rm d}r \\ &\; \lesssim &(1-\rho)^{q+s}\| g\| _{p,q,s}^{p}\int_{0}^{1}\frac{(1-r)^{\alpha'-q-s}}{(1-r\rho)^{\alpha'+1}}\ {\rm d}r \lesssim \| g\| _{p,q,s}^{p}. \end{eqnarray*}$

这表明 $C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$ 上的有界算子.

(10) 设 $\{h_{k}\}$ 是任一在 ${\Bbb D}$ 的任一紧子集上一致趋于0 且满足 $\| h_{k}\| _{p,q,s}\leq 1$ 对所有 $k\in \{1,2,\cdots\}$ 成立的序列. 设 $1/2<\rho<1$. 当 $p>1$ 时, 类似(3.14)式的证明, 根据(3.16)和(3.4)式可得

$\begin{eqnarray*} &\;& \sup_{0\leq r<1}(1-r^{2})^{q}\int_{0}^{2\pi}|h_{k}[\varphi(r{\rm e}^{{\rm i}\theta})]|^{p}(1-|\varphi_{a}(r{\rm e}^{{\rm i}\theta})|^{2})^{s}\ {\rm d}\theta\\ &\lesssim& \max_{|u|\leq \rho}|h_{k}[\varphi(u)]|^{p} +\sup_{\rho<r<1}(1-r^{2})^{q}\\ &&\times \int_{0}^{2\pi}(1-|\varphi_{a}(r{\rm e}^{{\rm i}\theta})|^{2})^{s}|G_{a,h_{k}}[\varphi(r{\rm e}^{{\rm i}\theta})]| ^{p}|1-\overline{\varphi(a)}\varphi(r{\rm e}^{{\rm i}\theta})|^{2s}\ {\rm d}\theta\\ & \lesssim& \max_{|u|\leq \rho}|h_{k}[\varphi(u)]|^{p}+M_{\rho}^{\alpha_{1}-\alpha}\sup_{\rho<r<1}(1-r^{2}) ^{q+s+\alpha-\alpha_{1}}(1-|a|^{2})^{s} \\ &\;& \times \int_{{\Bbb D}}\frac{|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha_{1}}}{|1-\varphi(a)\overline{u}|^{2s}} \left\{\int_{0}^{2\pi}\frac{|1-\overline{\varphi(a)}\varphi(r{\rm e}^{{\rm i}\theta})|^{2s} \ {\rm d}\theta}{|1-\varphi(r{\rm e}^{{\rm i}\theta})\overline{u}|^{2+\alpha}|1-r\overline{a}{\rm e}^{{\rm i}\theta}|^{2s}}\right\}{\rm d}v(u) \\ & \lesssim &\max_{|u|\leq \rho}|h_{k}[\varphi(u)]|^{p}+M_{\rho}^{\alpha_{1}-\alpha}\sup_{\rho<r<1}(1-r^{2})^{q+s+\alpha-\alpha_{1}}\\ \\ &\;& \times \int_{0}^{1}\frac{(1-t)^{\alpha_{1}-q-s}}{(1-rt)^{\alpha+1}}\left\{\int_{0}^{2\pi}(1-t^{2})^{q}|h_{k}(t{\rm e}^{{\rm i}\tau})|^{p}(1-|\varphi_{\varphi(a)}(t{\rm e}^{{\rm i}\tau})|^{2})^{s}\ {\rm d}\tau\right\}{\rm d}t \\ & \lesssim &\max_{|u|\leq \rho}|h_{k}[\varphi(u)]|^{p}+M_{\rho}^{\alpha_{1}-\alpha}\rightarrow M_{\rho}^{\alpha_{1}-\alpha} \ \ ( k\rightarrow\infty). \end{eqnarray*}$

当 $0<p\leq 1$ 时, 我们选取 $0<\varepsilon<\min\{q+s, 1+\alpha'-q-\max\{s,p\}\}$. 类似(3.15)式的证明, 根据(3.16)和(3.4)式可得

$\begin{eqnarray*} &\;& \sup_{0\leq r<1,a\in D}(1-r^{2})^{q}\int_{0}^{2\pi}|h_{k}[\varphi(r{\rm e}^{{\rm i}\theta})]|^{p}(1-|\varphi_{a}(r{\rm e}^{{\rm i}\theta})|^{2})^{s}\ {\rm d}\theta\\ &\; \lesssim &\sup_{0\leq r\leq \rho,a\in D}(1-r^{2})^{q}\int_{0}^{2\pi}|h_{k}[\varphi(r {\rm e}^{{\rm i}\theta})]|^{p}(1-|\varphi_{a}(r{\rm e}^{{\rm i}\theta})|^{2})^{s}\ {\rm d}\theta+(1-|a|^{2})^{s}\\ &\;& \times\sup_{\rho< r<1,a\in D} \int_{{\Bbb D}}\frac{|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha'}}{|1-\overline{\varphi(a)}u|^{2s}}\int_{0}^{2\pi}\frac{ (1-r^{2})^{q+s}|1-\overline{\varphi(a)}\varphi(r{\rm e}^{{\rm i}\theta})|^{2s} \ {\rm d}\theta}{|1- u\overline{\varphi(r{\rm e}^{{\rm i}\theta})}|^{2+\alpha'}|1-r\overline{a}{\rm e}^{{\rm i}\theta}|^{2s}}\ {\rm d}v(u) \\ &\;\lesssim &\sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p}+\sup_{\rho<r<1,a\in D}M_{\rho}^{\varepsilon}(1-r^{2})^{q+s-\varepsilon}(1-|a|^{2})^{s}\\ &\;& \times \ \int_{{\Bbb D}}\frac{|h_{k}(u)|^{p}(1-|u|^{2})^{\alpha'}}{|1-\overline{\varphi(a)}u|^{2s}}\left\{\int_{0}^{2\pi}\frac{ |1-\overline{\varphi(a)}\varphi(r{\rm e}^{{\rm i}\theta})|^{2s} \ {\rm d}\theta}{|1- u\overline{\varphi(r{\rm e}^{{\rm i}\theta})}|^{2+\alpha'-\varepsilon}|1-r\overline{a}{\rm e}^{{\rm i}\theta}|^{2s}}\right\}{\rm d}v(u) \\ &\;\lesssim &\sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p} +M_{\rho}^{\varepsilon}\sup_{\rho<r<1}(1-r)^{q+s-\varepsilon}\| h_{k}\| _{p,q,s}^{p} \int_{0}^{1}\frac{(1-t)^{\alpha'-q-s}}{(1-rt)^{1+\alpha'-\varepsilon}} \ {\rm d}t \\ &\;\lesssim &\sup_{|w|\leq \rho}|h_{k}[\varphi(w)]|^{p}+M_{\rho}^{\varepsilon}\rightarrow M_{\rho}^{\varepsilon} \ \ (k\rightarrow\infty). \end{eqnarray*}$

根据(3.1)式可得 $C_{\varphi}$ 是 $H^{p,q,s}({\Bbb D})$ 上的紧算子.

本定理证完.

根据文献[5,命题3.3]可知, 如果 $0<s<1$, 则 $H^{\frac{p}{q+1}}({\Bbb D})\subsetneq H^{p,q,s}({\Bbb D})$. 进一步, 我们有下列结果.

定理 3.2 设 $0<s<1$ 以及 $\varphi: \ {\Bbb D}\rightarrow {\Bbb D}$ 是一个解析映射.

(1) $C_{\varphi}$ 总是 $H^{\frac{p}{q+1}}({\Bbb D})$ 到 $H^{p,q,s}({\Bbb D})$ 的有界算子.

(2) 若 $C_{\varphi}$ 是 $H^{\frac{p}{q+1}}({\Bbb D})$ 到 $H^{p,q,s}({\Bbb D})$ 的紧算子, 则(3.1)式成立.

(3) 若(3.2)式成立, 则 $C_{\varphi}$ 是 $H^{\frac{p}{q+1}}({\Bbb D})$ 到 $H^{p,q,s}({\Bbb D})$ 的紧算子.

证 (1) 对任意 $f\in H^{\frac{p}{q+1}}({\Bbb D}) $, 当$-1<q<0$ 时, 根据 H"{o}lder 不等式, 引理 2.3, $C_{\varphi}$ 在 $H^{\frac{p}{q+1}}({\Bbb D})$ 上的有界性, 我们可得

$\begin{eqnarray*} \| C_{\varphi}f\| _{p,q,s}^{p}&\leq& \sup_{0\leq r<1}\sup_{ a\in {\Bbb D}}(1-r^{2})^{q+s}(1-|a|^{2})^{s} \\ &&\times\left\{\int_{0}^{2\pi}|f[\varphi(r{\rm e}^{{\rm i}\theta})]|^{\frac{p}{q+1}}\ {\rm d}\theta\right\}^{q+1}\left\{\int_{0}^{2\pi}\frac{{\rm d}\theta}{|1- \overline{a}r{\rm e}^{{\rm i}\theta}|^{\frac{2s}{-q}}}\right\}^{-q} \\ &\lesssim &\sup_{0\leq r<1}\sup_{ a\in {\Bbb D}}\frac{(1-r^{2})^{q+s}(1-|a|^{2})^{s}\| f\| ^{p}_{\frac{p}{q+1},0,0}}{(1-r^{2}|a|^{2})^{q+2s}}\lesssim \| f\| ^{p}_{\frac{p}{q+1},0,0}. \end{eqnarray*}$

当 $q\geq 0$ 时, 设 $p_{0}=1/(q+1)$. 由引理 2.6 及 $C_{\varphi}$ 在 $H^{pp_{0}}({\Bbb D})$ 上的有界性可得

$\begin{eqnarray*} \| C_{\varphi}f\| _{p,q,s}^{p}&\leq& \sup_{0\leq r<1}(1-r^{2})^{q}\int_{0}^{2\pi}|f[\varphi(r{\rm e}^{{\rm i}\theta})]|^{p}\ {\rm d}\theta \\ &\leq& \sup_{0\leq r<1}\frac{(1-r^{2})^{q}}{(1-r^{2})^{\frac{1-p_{0}}{p_{0}}}}\| C_{\varphi}f\| _{pp_{0},0,0}^{p}\lesssim\| f\| _{\frac{p}{q+1},0,0}^{p}. \end{eqnarray*}$

这表明 $C_{\varphi}$ 总是$H^{\frac{p}{q+1}}({\Bbb D})$ 到 $H^{p,q,s}({\Bbb D})$的有界算子.

(2) 若 $C_{\varphi}$ 是$H^{\frac{p}{q+1}}({\Bbb D})$ 到 $H^{p,q,s}({\Bbb D})$ 的紧算子, 根据(3.10)式可得(3.1)式成立.

(3) 设 $\{h_{k}\}$ 是任一在 ${\Bbb D}$ 的任一紧子集上一致趋于0 且满足$\| h_{k}\| _{\frac{p}{q+1},0,0}\leq 1$ 对所有 $k\in \{1,2,\cdots\}$ 成立的序列. 当 $-1<q<0$ 时, 有

$\begin{eqnarray*} \| C_{\varphi}h_{k}\| _{p,q,s}^{p} &\lesssim& \sup_{0\leq r<1}\sup_{ a\in {\Bbb D}}\frac{(1-r^{2})^{q+s}(1-|a|^{2})^{s}}{(1-r^{2}|a|^{2})^{q+2s}}\left\{\int_{0}^{2\pi}|h_{k}[\varphi(r{\rm e}^{{\rm i}\theta})]|^{\frac{p}{q+1}}\ \frac{{\rm d}\theta}{2\pi}\right\}^{q+1} \\ &\leq&\sup_{0\leq r<1}\left\{\int_{0}^{2\pi}|h_{k}[\varphi(r{\rm e}^{{\rm i}\theta})]|^{\frac{p}{q+1}}\ \frac{{\rm d}\theta}{2\pi}\right\}^{q+1}= \| C_{\varphi}h_{k}\| _{\frac{p}{q+1},0,0}^{p}. \end{eqnarray*}$

当 $q\geq 0$ 时, 可得

$\| C_{\varphi}h_{k}\| _{p,q,s}^{p} \leq \sup_{0\leq r<1}\frac{(1-r^{2})^{q}}{(1-r^{2})^{\frac{1-p_{0}}{p_{0}}}}\left\{\int_{0}^{2\pi}|h_{k}[\varphi(\sqrt{r}{\rm e}^{{\rm i}\theta})]|^{\frac{p}{q+1}}\ \frac{{\rm d}\theta}{2\pi}\right\}^{q+1}= \| C_{\varphi}h_{k}\| _{\frac{p}{q+1},0,0}^{p}. $

如果(3.2)式成立, 根据 Hardy空间的结论有

$\lim_{k\rightarrow\infty}\| C_{\varphi}h_{k}\| _{\frac{p}{q+1},0,0}=0. \ \ \mbox{因此,} \ \ \lim_{k\rightarrow\infty}\| C_{\varphi}h_{k}\| _{p,q,s}=0.$

这表明 $C_{\varphi}$ 是 $H^{\frac{p}{q+1}}({\Bbb D})$ 到 $H^{p,q,s}({\Bbb D})$ 的紧算子. 本定理证完.

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