用$B_n$表示$C^n$中的单位球; d$v$和d$\sigma$分别表示$B_n$和 $\partial B_n$的正规Lebesgue测度; 对$ z=(z_1,\cdots,z_n)$, $ w=(w_1,\cdots,w_n)$,我们定义$ \langle z,w\rangle = \sum\limits_{j=1}^nz_j\overline{w}_j$; $H(B_n)$表示$B_n$上全纯函数的全体; $H^\infty$表示$B_n$上的有界 全纯函数的全体. 对$\alpha>-1$, 定义${\rm d}v_\alpha(z)=c_\alpha(1-|z|^2)^\alpha {\rm d}v(z)$, 其中$ c_\alpha=\frac{\Gamma(n+1+\alpha)}{n!\Gamma(\alpha+1)}$.
若$f\in H(B_n)$,用${\nabla f(z)=\left(\frac{\partial f}{\partial z_{1}}(z),\cdots, \frac{\partial f}{\partial z_{n}}(z)\right)}$表示$f$的梯度, 用$ Rf(z)=\langle \nabla f(z),\overline{z}\rangle =\sum\limits_{j=1}^{n}z_{j}\frac{\partial f} {\partial z_{j}}(z)$ 表示$f$的径向导数.
设$0<p<\infty$, $\alpha>-1$,$B_n$上的加权Bergman空间$A_\alpha^p$是$H(B_n)$中满足 $$\|f\|_{A_\alpha^p}^p=\int_{B_n}|f(z)|^p{\rm d}v_\alpha(z)<\infty$$的函数$f$的全体; $B_n$上的加权Dirichlet空间$D_\alpha^p$是$H(B_n)$中满足 $Rf\in A_\alpha^p$的函数全体. 因此$f\in D_\alpha^p$当且仅当 $$ \|f\|_{D_\alpha^p}^p=|f(0)|^p+\|Rf\|_{A_\alpha^p}^p<\infty. $$
在单复变中,Cesàro算子定义如下 $$ C[f](z)=\sum_{j=0}^\infty \bigg(\frac{1}{j+1}\sum_{k=0}^ja_k\bigg)z^j,\ \ \ \ \ \ \ f(z)=\sum_{j=0}^\infty a_jz^j\in H(B_1). $$ 给定$g\in H(B_1)$,$H(B_1)$上的广义Cesàro算子定义为 $$ T_g(f)(z)=\int_0^zf(t)g'(t){\rm d}t \ \ \ \ (f\in H(B_1),\ z\in B_1). $$ 容易验证,当$ g(z)=\log\frac{1}{1-z}$时, $T_g(f)(z)=zC[f](z)$.
在多复变中,给定 $\psi\in H(B_n)$,$H(B_n)$上广义 Cesàro算子 $T_{\psi}$ 定义为 $$ T_{\psi}(f)(z)=\int_{0}^{1}f(tz)R\psi(tz)\frac{{\rm d}t}{t} \ \ \ \ (f\in H(B_n),\ z\in B_n).$$ 在复平面上，Cesàro 算子和广义 Cesàro算子已经被研究了相当长的时间,在 Hardy 空间、BMOA空间、 Bloch型空间、Bergman空间、 混合模空间等经典函数空间上获得了一系列成果(见文献[1, 2, 3, 4, 5, 6, 7]); 对于多复变情形,最近文献[8, 9, 10, 11, 12]分别讨论了单位球上 Bloch空间,混合模空间,Besov-Sobolev空间以及Bloch型空间与Dirichlet型空间之间的广义Cesàro算子的有界性和紧性问题. 本文将在上述基础之上讨论从Dirichlet型空间$D_\alpha^p$ 到 Dirichlet型空间$D_\beta^q$ 的广义Cesàro算子$T_{\psi}$, 给出了当$0<p\leq1$、$1<p<\alpha+1$或$p>n+1+\alpha$时$T_{\psi}$为有界算子或紧算子的充要条件. 同时,也给出了$p$取其它值时$T_{\psi}$为有界算子或紧算子的充分 条件或必要条件.
本文中$C,C',C''$等表示与变量$z$,$w$,$a$等无关但可以与某些有界量或参数$p$,$q$等 有关的正数,不同的地方可以代表不同的数.
$\bf 引理2.1$ ([13,定理2.10]) 设$a$,$b$为实数,定义积分算子$S$如下 $$Sf(z)=(1-|z|^2)^a \int_{B_n}\frac{(1-|w|^2)^b} {|1-\langle z,w \rangle|^{n+1+a+b}}f(w){\rm d}v(w),$$ 则对$- \infty < {\rm{ }}t{\rm{ }} < \infty $和$1\leq p<\infty$ 以下两条等价
(1) $S$是$L^p(B_n,{\rm d}v_t)$的有界算子($L^p(B_n,{\rm d}v_t)$表示Lebesgue空间);
(2) $- pa{\rm{ }} < {\rm{ }}t + {\rm{ }}1 < {\rm{ }}p(b + {\rm{ }}1).$
$\bf引理 2.2$ ([13,定理2.15]) 设$0<p\leq1$, $\alpha>-1$. 则 $$\int_{B_n}|f(z)|(1-|z|^2)^{(n+1+\alpha)/p-(n+1)}{\rm d}v(z)\leq C\|f\|_{A_\alpha^p}$$ 对所有的$f\in A_\alpha^p$成立.
$\bf 引理 2.3$ ([13,定理 2.2]) 设$\alpha>-1$, $f\in A_\alpha^1$. 则对所有的$z\in B_n$ $$ f(z)=\int_{B_n}\frac{f(w){\rm d}v_\alpha(w)}{(1-\langle z,w\rangle )^{n+1+\alpha}}. $$
$ \bf 引理 2.4$ ([13,定理 1.12]) 设$c$是实数,$t>-1$. 则积分 $$J_{c,t}=\int_{B_n}\frac{(1-|w|^2)^t{\rm d}v(w)}{|1-\langle z,w\rangle |^{n+1+t+c}},\ \ \ \ z\in B_n$$ 有如下性质:
(1) 当 $c<0$时,$ J_{c,t}$有界;
(2) 当 $c=0$时,$ J_{c,t}\sim \log\frac{1}{1-|z|^2}$ $(|z|\rightarrow 1^-$);
(3)~ 当$c>0$时,$ J_{c,t}\sim (1-|z|^2)^{-c}$ $(|z|\rightarrow 1^-)$.
$ \bf 引理 2.5$ 设$\alpha>-1$,$p>0$,$f\in D^p_\alpha$且$f(0)=0$,则存在常数$C>0$以及充分大的$\gamma$使得
(i) 当$0<p\leq 1$时,有 $ |f(z)|^p\leq C\int_{B_n}\frac{|Rf(w)|^p{\rm d}v_\gamma(w)}{|1-\langle z,w\rangle |^{n+1+\gamma-p}}$;
(ii) 当$p>1$时,对任给的$s>0$,有 $ |f(z)|^p\leq C\int_{B_n}\frac{|Rf(w)|^p{\rm d}v_\gamma(w)}{|1-\langle z,w\rangle |^{n+1+\gamma+s-p}}$.
$\bf 证$ 若$f\in D_\alpha^p$,则当$0<p\leq 1$时,取充分大的$ \beta\geq\frac{n+1+\alpha}{p}-n-1$, 当$p>1$时,取充分大的$ \beta>\frac{1+\alpha}{p}$就有$Rf\in A_\beta^1$,由引理2.3知, $$ Rf(z)=\int_{B_n}\frac{Rf(w){\rm d}v_\beta(w)}{(1-\langle z,w\rangle )^{n+1+\beta}},\ \ \ \ z\in B_n. $$ 由于 $Rf(0)=0$,所以 $$Rf(z)=\int_{B_n}{Rf(w)} \bigg(\frac{1}{(1-\langle z,w\rangle )^{n+1+\beta}}-1\bigg){\rm d}v_\beta(w),\ \ \ \ z\in B_n.$$ 又因为 $$f(z)-f(0)=\int_0^1\frac{Rf(tz)}{t}{\rm d}t=\int_{B_n}Rf(w)L(z,w){\rm d}v_\beta(w),$$ 其中 \begin{eqnarray*} |L(z,w)|&=& \bigg|\int_0^1(\frac{1}{(1-t\langle z,w\rangle )^{n+1+\beta}}-1)\frac{{\rm d}t}{t}\bigg|\\ &=&\bigg|\sum_{k=1}^\infty\frac{\Gamma(k+n+1+\beta)}{kk!\Gamma(n+1+\beta)}\langle z,w\rangle ^k\bigg|\\ &\leq& \frac{C}{|1-\langle z,w\rangle |^{n+\beta}}. \end{eqnarray*} 所以 $$ |f(z)|\leq C\int_{B_n}\frac{|Rf(w)|{\rm d}v_\beta(w)}{|1-\langle z,w\rangle |^{n+\beta}}. (2.1) $$
当$0<p\leq 1$时,令 $$ \beta=\frac{n+1+\gamma}{p}-(n+1),$$ 且$\beta$充分大使得 $\gamma > \alpha+p>-1$,对于固定的$z\in B_n$,由于 $$ \int_{B_n}|Rf(w)/(1-\langle w,z\rangle )^{n+\beta}|^p{\rm d}v_\gamma(w)\leq \frac{C}{(1-|z|^2)^{(n+\beta)p+\alpha-\gamma}}\int_{B_n}|Rf(w)|^p{\rm d}v_\alpha(w), $$ 所以$Rf(w)/(1-\langle w,z\rangle )^{n+\beta}\in A_{\gamma}^p$. 由引理2.2 和(2.1)式得 \begin{eqnarray*} |f(z)|^p&\leq& C \left(\int_{B_n} \bigg|\frac{Rf(w)}{(1-\langle z,w\rangle )^{n+\beta}}\bigg|(1-|w|^2)^{\frac{n+1+\gamma}{p}-(n+1)}{\rm d}v(w)\right)^p\\ &\leq& C\int_{B_n}\bigg|\frac{Rf(w)}{(1-\langle z,w\rangle )^{n+\beta}}\bigg|^p{\rm d}v_\gamma(w)\leq C\int_{B_n}\frac{|Rf(w)|^p{\rm d}v_\gamma(w)}{|1-\langle z,w\rangle |^{n+1+\gamma-p}}. \end{eqnarray*} 当$p>1$时,让$1/p+1/q=1$,任取非常小的$s>0$,由H$\ddot{o}$lder 不等式以及引理2.4得 \begin{eqnarray*} |f(z)|^p&\leq& C\left(\int_{B_n}\frac{|Rf(w)|{\rm d}v_\beta(w)}{|1-\langle z,w\rangle |^{n+\beta}}\right)^p\\ &\leq&C\int_{B_n}\frac{|Rf(w)|^p(1-|w|^2)^p{\rm d}v_{\beta-1}(w)}{|1-\langle z,w\rangle |^{n+s+\beta}} \left(\int_{B_n}\frac{{\rm d}v_{\beta-1}(w)}{|1-\langle z,w\rangle |^{n-\frac{qs}{p}+\beta}}\right)^{\frac{p}{q}}\\ &\leq&C\int_{B_n}\frac{|Rf(w)|^p(1-|w|^2)^p{\rm d}v_{\beta-1}(w)}{|1-\langle z,w\rangle |^{n+s+\beta}} \leq C\int_{B_n}\frac{|Rf(w)|^p{\rm d}v_{\beta-1}(w)}{|1-\langle z,w\rangle |^{n+s+\beta-p}}. \end{eqnarray*} 取$\gamma=\beta-1$,我们有 \begin{eqnarray*} |f(z)|^p\leq C\int_{B_n}\frac{|Rf(w)|^p{\rm d}v_\gamma(w)}{|1-\langle z,w\rangle |^{n+1+\gamma+s-p}}. \end{eqnarray*} 证毕.
$\bf 引理 2.6$ 设$\alpha>-1$,$p>0$,$f\in D^p_\alpha$, 则存在常数$C>0$使得 $$ |f(z)|\leq C\|f\|_{D_\alpha^p}\left\{ \begin{array}{ll} 1,& n+1+\alpha<p; \\ \log\frac{2}{1-|z|^2},\ \ & n+1+\alpha=p; \\[2mm] (1-|z|^2)^{\frac{n+1+\alpha-p}{p}},& n+1+\alpha>p. \end{array} \right. (2.2) $$
$\bf 证$ 若$f\in D_\alpha^p$,类似引理2.5的证明, 对充分大的 $\beta>\max\{\frac{n+1+\alpha}{p}-n-1, \frac{1+\alpha}{p}\}$ 有 $$ Rf(z)=\int_{B_n}{Rf(w)} \bigg(\frac{1}{(1-\langle z,w\rangle )^{n+1+\beta}}-1\bigg){\rm d}v_\beta(w),\ \ \ \ z\in B_n. (2.3) $$ 因此,当$p>1$时,任给$\sigma>0$,由引理2.4得 \begin{eqnarray*} |Rf(z)|^p&\leq & C \left(\int_{B_n}|Rf(w)|^p\frac{|1-(1-\langle z,w\rangle )^{n+1+\beta}|^p}{|1-\langle z,w\rangle |^{n+1+\beta-p\sigma}}{\rm d}v_\beta(w)\right)\\ && \times\left(\int_{B_n}\frac{1}{|1-\langle z,w\rangle |^{n+1+\beta+\frac{p}{p-1}\sigma}}{\rm d}v_\beta(w)\right)^{p-1}\\ &\leq&C \left(\frac{|z|^p}{(1-|z|^2)^{n+1+\alpha-p\sigma}}\int_{B_n}|Rf(w)|^p{\rm d}v_\alpha(w)\right)\left((1-|z|^2)^{-\frac{p}{p-1}\sigma}\right)^{p-1}\\ &\leq&C \frac{|z|^p}{(1-|z|^2)^{n+1+\alpha}}\int_{B_n}|Rf(w)|^p{\rm d}v_\alpha(w). \end{eqnarray*} 当$0<p< 1$时,令 $$ \beta=\frac{n+1+\gamma}{p}-(n+1),$$ 且$\beta$充分大使得 $\gamma > \alpha+p>-1$,对于固定的$z\in B_n$,$Rf(w)\langle w,z\rangle /(1-\langle w,z\rangle )^{n+1+\beta}\in A_{\gamma}^p$. 由引理2.2和(2.3)式得 \begin{eqnarray*} |Rf(z)|^p &\leq& C \left(\int_{B_n} \bigg|\frac{Rf(w)(1-(1-\langle z,w\rangle )^{n+1+\beta})} {(1-\langle z,w\rangle )^{n+1+\beta}}\bigg|(1-|w|^2)^{\frac{n+1+\gamma}{p}-(n+1)}{\rm d}v(w)\right)^p\\ &\leq& C\int_{B_n}\bigg|\frac{Rf(w)\langle z,w\rangle } {(1-\langle z,w\rangle )^{n+1+\beta}}\bigg|^p{\rm d}v_\gamma(w)\leq C |z|^p\int_{B_n}\frac{|Rf(w)|^p{\rm d}v_\gamma(w)}{|1-\langle z,w\rangle |^{n+1+\gamma}}\\ &\leq& C\frac{|z|^p}{(1-|z|^2)^{n+1+\alpha}}\int_{B_n}|Rf(w)|^p{\rm d}v_\alpha(w). \end{eqnarray*} 因此,对所有的$0<p<\infty$有 $$|Rf(z)| \leq C\frac{|z|}{(1-|z|^2)^{\frac{n+1+\alpha}{p}}}\|f\|_{D_\alpha^p},$$ 所以 \begin{eqnarray*} |f(z)|&=&\bigg|f(0)+\int_0^1\frac{Rf(tz)}{t}{\rm d}t\bigg| \leq |f(0)|+C\|f\|_{D_\alpha^p}\int_0^1\frac{|tz|} {t(1-|tz|^2)^{\frac{n+1+\alpha}{p}}}{\rm d}t\\ &\leq& |f(0)|+C\|f\|_{D_\alpha^p}\int_0^1 \frac{|z|{\rm d}t}{(1-|tz|^2)^{\frac{n+1+\alpha}{p}}}\\ &\leq& |f(0)|+C\|f\|_{D_\alpha^p}\int_0^{|z|} \frac{{\rm d}t}{(1-t)^{\frac{n+1+\alpha}{p}}}\\ &\leq & C\|f\|_{D_\alpha^p} \left\{ \begin{array}{ll} 1,& n+1+\alpha<p; \\ \log\frac{2}{1-|z|^2},\ \ & n+1+\alpha=p; \\[2mm] (1-|z|^2)^{\frac{n+1+\alpha-p}{p}},~~ & n+1+\alpha>p. \end{array} \right. \end{eqnarray*} 证毕.
$\bf 引理 2.7$ $^{[16]}$\quad 设$q>-1$,则存在常数$C>0$,使得对一切$w\in B_n$,都有 $$ \int_{B_n}\bigg|\log\frac{1}{1-\langle z,w\rangle }\bigg|^2\frac{(1-|z|^2)^q}{|1-\langle z,w\rangle |^{n+1+q}}{\rm d}v(z)\leq C\left(\log\frac{1}{1-|w|^2}\right)^2. $$
$\bf 定理1$ 设 $\psi\in H(B_n)$,$0<p\leq q<\infty$,$\alpha,\ \beta>-1$. 则
(i) 当$0<p\leq 1$或$1<p<1+\alpha$时,$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子的充要条件为 $$ \sup_{a\in B_n}\int_{B_n}\frac{|R\psi(z)|^q(1-|a|^2)^{q}{\rm d}v_\beta(z)}{|1-\langle a,z\rangle |^{\frac{q(n+1+\alpha)}{p}}}=M_1<\infty; (3.1) $$
(ii) 当$p>n+1+\alpha$时,$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子的充要条件为$\psi\in D_{\beta}^q$;
(iii) 当$1<p<n+1+\alpha$且$\alpha+1\leq p$时,若$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子,则(3.1)式成立; 若存在$s{\rm{ }} > 0$,使得 $$ \sup_{a\in B_n}\int_{B_n}\frac{|R\psi(z)|^q(1-|a|^2)^{q}{\rm d}v_\beta(z)}{|1-\langle a,z\rangle |^{\frac{q(n+1+\alpha+s)}{p}}}=M_2<\infty,(3.2) $$ 则$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子; (iv)~ 当$p=n+1+\alpha$时,若$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子,则 $$ \sup_{a\in B_n}\int_{B_n}|R\psi(z)|^q \bigg(\log\frac{2}{1-|a|^2}\bigg)^{-\frac{2q}{p}} \bigg|\log\frac{2}{1-\langle a,z\rangle }\bigg|^{q+\frac{2q}{p}}{\rm d}v_\beta(z)=M_3<\infty;(3.3) $$ 若 $$ \int_{B_n}|R\psi(z)|^q \bigg(\log{\frac{2}{1-|z|^2}}\bigg)^q{\rm d}v_\beta(z)=M_4<\infty,(3.4) $$ 则$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子.
$\bf 证$ 我们先证充分性. 对任意的$f\in D_{\alpha}^p$, \begin{eqnarray*} \left(\|T_\psi(f)\|_{D_\beta^q}\right)^{p} &=& \left(|T_\psi(f)(0)|^q+ \int_{B_n}|R(T_\psi(f))(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}} \\ &=&\left(\int_{B_n}|R\psi(z)f(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}. \end{eqnarray*} (i) (a) 当$0 < {\rm{ }}p \le 1$时,若(3.1)式成立,由Minkowski不等式和引理2.5得,对充分大的$\gamma>\alpha+p>-1$ 有 \begin{eqnarray*} && \left(\int_{B_n}|R\psi(z)f(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq& C\left(\int_{B_n}\left(\int_{B_n} \frac{|Rf(w)|^p}{|1-\langle z,w\rangle |^{n+1+\gamma-p}}{\rm d}v_{\gamma}(w)\right)^{\frac{q}{p}}|R\psi(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq&C\int_{B_n}\left(\int_{B_n}\frac{|Rf(w)|^q|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^{\frac{q}{p}(n+1+\gamma-p)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v_{\gamma}(w)\\ &\leq & C\int_{B_n}|Rf(w)|^p(1-|w|^2)^{\alpha}\left(\int_{B_n}\frac{(1-|w|^2)^{\frac{q}{p}(\gamma-\alpha)}|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^ {\frac{q}{p}(n+1+\gamma-p)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v(w)\\ &\leq & C\int_{B_n}|Rf(w)|^p(1-|w|^2)^{\alpha}\left(\int_{B_n}\frac{(1-|w|^2)^{q}|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^ {\frac{q}{p}(n+1+\alpha)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v(w)\\ &\leq&C(M_1)^{\frac{p}{q}}\|f\|^p_{D_\alpha^p}. \end{eqnarray*} 所以 $T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子. (b) 当$1<p<1+\alpha$时,对充分大的 $\gamma>\max\{\alpha, \frac{1+\alpha}{p}\}>1$, 由文献[13]中定理2.2的证明,Hölder不等式以及(2.1)式和引理2.4得 \begin{eqnarray*} |f(z)|^p&=& \bigg|\int_{B_n}\frac{f(w)(1-|z|^2)^{n+1+\gamma}}{|1-\langle z,w\rangle |^{2(n+1+\gamma)}}{\rm d}v_\gamma(w)\bigg|^p \\ &\leq& \int_{B_n}\frac{|f(w)|^p(1-|z|^2)^{n+1+\gamma}}{|1-\langle z,w\rangle |^{2(n+1+\gamma)}}{\rm d}v_\gamma(w) \left(\int_{B_n}\frac{(1-|z|^2)^{n+1+\gamma}}{|1-\langle z,w\rangle |^{2(n+1+\gamma)}}{\rm d}v_\gamma(w)\right)^{p-1}\\ &\leq& C \int_{B_n}\frac{1}{|1-\langle z,w\rangle |^{n+1+\gamma}} \left(\int_{B_n}\frac{|Rf(\eta)|}{|1-\langle w,\eta\rangle |^{n+\gamma}}{\rm d}v_\gamma(\eta)\right)^p{\rm d}v_\gamma(w). \end{eqnarray*} 由Minkowski不等式以及引理2.1得 \begin{eqnarray*} &&\left(\int_{B_n}|R\psi(z)f(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq& C \int_{B_n}\left(\int_{B_n}\frac{|Rf(\eta)|}{|1-\langle w,\eta\rangle |^{n+\gamma}}{\rm d}v_\gamma(\eta)\right)^p\left( \int_{B_n}\frac{(1-|w|^2)^q|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^{\frac{q}{p}(n+1+\alpha)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v_{\alpha-p}(w)\\ &\leq& C(M_1)^{\frac{p}{q}}\int_{B_n}\left(\int_{B_n}\frac{|Rf(\eta)(1-|\eta|^2)|(1-|\eta|^2)^{\gamma-1}}{|1-\langle w,\eta\rangle |^{n+\gamma}}{\rm d}v(\eta)\right)^p{\rm d}v_{\alpha-p}(w)\\ &\leq& C'(M_1)^{\frac{p}{q}}\int_{B_n}|Rf(w)(1-|w|^2)|^p{\rm d}v_{\alpha-p}(w)\leq C'(M_1)^{\frac{p}{q}}\|f\|^p_{D_\alpha^p}. \end{eqnarray*} 所以$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子.
(ii) 当$p>n+1+\alpha$时,若$\psi\in D_{\beta}^q$,则由引理2.6得 \begin{eqnarray*} \|T_\psi(f)\|_{D_\beta^q} &=&\left(\int_{B_n}|R\psi(z)f(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{1}{q}}\\ &\leq & C \|f\|_{D_\alpha^p}\left(\int_{B_n}|R\psi(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{1}{q}}\leq C \|\psi\|_{D_\beta^q}\|f\|_{D_\alpha^p}. \end{eqnarray*}
(iii) 当$1<p<n+1+\alpha$且$\alpha+1\leq p$时,若存在$s>0$,使得 $$ \sup_{a\in B_n}\int_{B_n}\frac{|R\psi(z)|^q(1-|a|^2)^{q}{\rm d}v_\beta(z)}{|1-\langle a,z\rangle |^{\frac{q(n+1+\alpha+s)}{p}}}=M_2<\infty,(3.5) $$ 则由Minkowski不等式和引理2.5类似(i)$_{(a)}$的证明得,对充分大的$\gamma>\frac{1+\alpha}{p}>-1$有 \begin{eqnarray*} && \left(\int_{B_n}|R\psi(z)f(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq& C\left(\int_{B_n}\left(\int_{B_n} \frac{|Rf(w)|^p}{|1-\langle z,w\rangle |^{n+1+\gamma+s-p}}{\rm d}v_{\gamma}(w)\right)^{\frac{q}{p}}|R\psi(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq & C\int_{B_n}|Rf(w)|^p(1-|w|^2)^{\alpha}\left(\int_{B_n}\frac{(1-|w|^2)^{q}|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^ {\frac{q}{p}(n+1+\alpha+s)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v(w)\\ &\leq&C(M_2)^{\frac{p}{q}}\|f\|_{D_\alpha^p}. \end{eqnarray*} 所以 $T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子.
(iv) 当$p=n+1+\alpha$时,若(3.4)式成立,则由引理2.6得 \begin{eqnarray*} \|T_\psi(f)\|_{D_\beta^q} &=&\left(\int_{B_n}|R\psi(z)f(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{1}{q}}\\ &\leq& C \|f\|_{D_\alpha^p}\left(\int_{B_n}|R\psi(z)|^{q}(\log{\frac{2}{1-|z|^2}})^q{\rm d}v_\beta(z)\right)^{\frac{1}{q}} \leq C (M_4)^{\frac{1}{q}}\|f\|_{D_\alpha^p}. \end{eqnarray*} 所以 $T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子.
必要性: 若$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子,则存在常数$C>0$使得对任意的$f\in D_{\alpha}^p$,有 $\|T_\psi(f)\|_{D^q_{\beta}}\leq C\|f\|_{D_{\alpha}^p}$,取$f=1$得 $\psi\in D_{\beta}^q$,从而(ii)的必要性得证.
任取$a\in B_n$,当$0<p<n+1+\alpha$时,令$ f_a(z)=\frac{(1-|a|^2)}{(1-\langle z,a\rangle )^\frac{n+1+\alpha}{p}}$,则 $ Rf_a(z)=\frac{n+1+\alpha}{p}\frac{\langle z,a\rangle (1-|a|^2)}{(1-\langle z,a\rangle )^\frac{n+1+\alpha+p}{p}}$. 由引理2.4得 \begin{eqnarray*} \|f_a\|^p_{D_{\alpha}^p}&=&|f_a(0)|^p+\int_{B_n}|Rf_a(z)|^p{\rm d}v_\alpha(z)\leq 1+\int_{B_n}|\frac{n+1+\alpha}{p}\frac{\langle z,a\rangle (1-|a|^2)}{(1-\langle z,a\rangle )^\frac{n+1+\alpha+p}{p}}|^p{\rm d}v_\alpha(z)\\ &\leq& C\left(1+\int_{B_n}\frac{(1-|a|^2)^p}{|1-\langle z,a\rangle |^{n+1+\alpha+p}}{\rm d}v_\alpha(z)\right)\leq C', \end{eqnarray*} 所以$f_a(z)\in D_{\alpha}^p$且$\|f_a\|_{D_{\alpha}^p}\leq C'$ \ ($C'$与$a$无关).
再由$T_\psi$的有界以性及$T_\psi(f)(0)=0$得 \begin{eqnarray*} \sup_{a\in B_n}\int_{B_n}\frac{|R\psi(z)|^q(1-|a|^2)^{q}{\rm d}v_\beta(z)}{|1-\langle a,z\rangle |^{\frac{q(n+1+\alpha)}{p}}}&=& \sup_{a\in B_n}\int_{B_n}|R\psi(z)|^q|f_a(z)|^q{\rm d}v_\beta(z)\\ &=&\sup_{a\in B_n}\|T_\psi(f_a)\|^q_{D_{\beta^q}}\\ &\leq & C\sup_{a\in B_n}\|f_a\|^{\frac{q}{p}}_{D_{\alpha}^p}<C''. \end{eqnarray*}
(i),(iii)的必要性得证.
当$p=n+1+\alpha$时,令$ f_a(z)=(\log\frac{2}{1-|a|^2})^{-\frac{2}{p}}(\log\frac{2}{1-\langle a,z\rangle })^{1+\frac{2}{p}}$,由引理 2.7 容易证得$f_a(z)\in D_{\alpha}^p$且$\|f_a\|_{D_{\alpha}^p}\leq C$ \ ($C$与$a$无关),类似前面我们可以证得(3.3)式成立.
$\bf 定理2$ 设 $\psi\in H(B_n)$,$0<p\leq q<\infty$,$\alpha,\ \beta>-1$. 则
(i) 当$0<p\leq 1$或当$1<p<1+\alpha$时,$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子的充要条件为$\psi\in D_{\beta}^q$且 $$ \lim_{|a|\rightarrow 1}\int_{B_n}\frac{|R\psi(z)|^q(1-|a|^2)^{q}{\rm d}v_\beta(z)}{|1-\langle a,z\rangle |^{\frac{q(n+1+\alpha)}{p}}}=0; o(3.6) $$
(ii) 当$p>n+1+\alpha$时,$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子的充要条件为$\psi\in D_{\beta}^q$;
(iii) 当$1<p<n+1+\alpha$且$\alpha+1\leq p$时,若 $T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子,则(3.6)式成立; 若$\psi\in D_{\beta}^q$且存在$s>0$,使得 $$ \lim_{|a|\rightarrow 1}\int_{B_n}\frac{|R\psi(z)|^q(1-|a|^2)^{q}{\rm d}v_\beta(z)}{|1-\langle a,z\rangle |^{\frac{q(n+1+\alpha+s)}{p}}}=0,(3.7) $$ 则 $T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子;
(iv) 当$p=n+1+\alpha$时,若$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子,则 $$ \lim_{|a|\rightarrow 1}\int_{B_n}|R\psi(z)|^q \bigg(\log\frac{2}{1-|a|^2}\bigg)^{-\frac{2q}{p}} \bigg|\log\frac{2}{1-\langle a,z\rangle }\bigg|^{q+\frac{2q}{p}}{\rm d}v_\beta(z)=0; (3.8) $$ 若$\psi\in D_{\beta}^q$且 $$ \int_{B_n}|R\psi(z)|^q\bigg(\log{\frac{2}{1-|z|^2}}\bigg)^q{\rm d}v_\beta(z)<\infty,(3.9) $$
则$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子.
$\bf 证$ 我们先证充分性. 设$\{f_{j}\}$为在$B_n$的任一紧子集上一致收敛于0且满足 $\|f_{j}\|_{D_\alpha^p}\leq 1$的序列. 则 \begin{eqnarray*} \left(\|T_\psi(f_j)\|_{D_\beta^q}\right)^{p} &=& \left(|T_\psi(f_j)(0)|^q+ \int_{B_n}|R(T_\psi(f_j))(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}} \\ &=&\left(\int_{B_n}|R\psi(z)f_j(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}. \end{eqnarray*}
若(3.6)、(3.7)和(3.9)式成立,且$\psi\in D_{\beta}^q$,则容易证明存在常数$M>0$使得 $$ \sup_{a\in B_n}\int_{B_n}\frac{|R\psi(z)|^q(1-|a|^2)^{q}{\rm d}v_\beta(z)}{|1-\langle a,z\rangle |^{\frac{q(n+1+\alpha)}{p}}}< M; (3.10) $$ $$ \sup_{a\in B_n}\int_{B_n}\frac{|R\psi(z)|^q(1-|a|^2)^{q}{\rm d}v_\beta(z)}{|1-\langle a,z\rangle |^{\frac{q(n+1+\alpha+s)}{p}}}< M; (3.11) $$ 且对任给的$0<\varepsilon<1$,存在$0 < {\rm{ }}{r_0} < 1$,使得当$|a|>r_0$时有
(i) (a) 当$0<p\leq 1$时,由Minkowski不等式、引理2.5、(3.10)、(3.12)和(3.16)式得,对$j>J_0$以及充分大的 $\gamma>\alpha+p>-1$ 有 \begin{eqnarray*} && \left(\|T_\psi(f_j)\|_{D_\beta^q}\right)^{p}=\left(\int_{B_n}|R\psi(z)f_j(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq& C\left(\int_{B_n}\left(\int_{B_n}\frac{|Rf_j(w)|^p}{|1-\langle z,w\rangle |^{n+1+\gamma-p}}{\rm d}v_{\gamma}(w)\right)^{\frac{q}{p}}|R\psi(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq&C\int_{B_n}\left(\int_{B_n}\frac{|Rf_j(w)|^q|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^{\frac{q}{p}(n+1+\gamma-p)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v_{\gamma}(w)\\ &\leq & C\int_{B_n}|Rf_j(w)|^p(1-|w|^2)^{\alpha}\left(\int_{B_n}\frac{(1-|w|^2)^{\frac{q}{p}(\gamma-\alpha)}|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^ {\frac{q}{p}(n+1+\gamma-p)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v(w)\\ &\leq & C\bigg(\int_{r_0B_n}+\int_{B_n-r_0B_n}\bigg)|Rf_j(w)|^p(1-|w|^2)^{\alpha}\left(\int_{B_n}\frac{(1-|w|^2)^{q}|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^ {\frac{q}{p}(n+1+\alpha)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v(w)\\ &\leq &CM^{\frac{p}{q}}\int_{r_0B_n}|Rf_j(w)|^p(1-|w|^2)^{\alpha}{\rm d}v(z)+C\varepsilon^{\frac{p}{q}}\int_{B_n-r_0B_n}|Rf_j(w)|^p(1-|w|^2)^{\alpha}{\rm d}v(z)\\ &\leq&C(M^{\frac{p}{q}}\varepsilon^p+\varepsilon^{\frac{p}{q}}\|f_j\|^p_{D_\alpha^p}) \leq C(M^{\frac{p}{q}}\varepsilon^p+\varepsilon^{\frac{p}{q}}). \end{eqnarray*} 所以$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子.
(b) 当$1<p<1+\alpha$时,类似有界性的证明,对充分大的$\gamma>\max\{\alpha,\frac{1+\alpha}{p}\}>1$ 有 \begin{eqnarray*} |f_j(z)|^p&\leq& C\int_{B_n}\frac{|f_j(w)|^p}{|1-\langle z,w\rangle |^{(n+1+\gamma)}}{\rm d}v_\gamma(w)\\ &\leq& C \int_{B_n}\frac{1}{|1-\langle z,w\rangle |^{n+1+\gamma}} \left(\int_{B_n}\frac{|Rf_j(\eta)|}{|1-\langle w,\eta\rangle |^{n+\gamma}}{\rm d}v_\gamma(\eta)\right)^p{\rm d}v_\gamma(w). \end{eqnarray*} 由Minkowski不等式、引理2.1、(3.10)、(3.12)以及(3.16)式得,对$j>J_0$有 \begin{eqnarray*} &&\left(\|T_\psi(f_j)\|_{D_\beta^q}\right)^{p}=\left(\int_{B_n}|R\psi(z)f_j(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq& C\int_{B_n} |f_j(w)|^p\left(\int_{B_n}\frac{(1-|w|^2)^q|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^{\frac{q}{p}(n+1+\alpha)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v_{\alpha-p}(w)\\ &=& C\bigg(\int_{r_0B_n}+\int_{B_n-r_0B_n}\bigg) |f_j(w)|^p\left(\int_{B_n}\frac{(1-|w|^2)^q|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^{\frac{q}{p}(n+1+\alpha)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v_{\alpha-p}(w)\\ &\leq& C\left(M^{\frac{p}{q}}\varepsilon^p+\varepsilon^{\frac{p}{q}}\int_{B_n} |f_j(w)|^p{\rm d}v_{\alpha-p}(w)\right)\\ &\leq& C\left(M^{\frac{p}{q}}\varepsilon^p+\varepsilon^{\frac{p}{q}} \int_{B_n}\left(\int_{B_n}\frac{|Rf(\eta)(1-|\eta|^2)|(1-|\eta|^2)^{\gamma-1}}{|1-\langle w,\eta\rangle |^{n+\gamma}}{\rm d}v(\eta)\right)^p{\rm d}v_{\alpha-p}(w)\right)\\ &\leq& C(M^{\frac{p}{q}}\varepsilon^p+\varepsilon^{\frac{p}{q}}\|f_j\|^p_{D_\alpha^p})\leq C(M^{\frac{p}{q}}\varepsilon^p+\varepsilon^{\frac{p}{q}}). \end{eqnarray*} 所以$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子.
(ii) 当$p>n+1+\alpha$时,若$\psi\in D_{\beta}^q$,则 由引理2.6、(3.14)和(3.16)式得,当$j>J_0$时有 \begin{eqnarray*} \|T_\psi(f_j)\|^q_{D_\beta^q} &=&\int_{B_n}|R\psi(z)f_j(z)|^{q}{\rm d}v_\beta(z)= \bigg(\int_{r_0B_n}+\int_{B_n-r_0B_n}\bigg)|R\psi(z)f_j(z)|^{q}{\rm d}v_\beta(z)\\ &\leq& C\left(\varepsilon^q\|\psi\|_{D_\beta^q}+\|f\|_{D_\alpha^p}\int_{B_n-r_0B_n}|R\psi(z)|^{q}{\rm d}v_\beta(z)\right)\leq C(\varepsilon^q\|\psi\|_{D_\beta^q}+\varepsilon). \end{eqnarray*} 所以$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子.
(iii) 当$1<p<n+1+\alpha$且$\alpha+1\leq p$时,若(3.7)式成立,则 由Minkowski不等式、引理2.5、(3.11)、(3.13)和(3.16)式类似(i)$_{(a)}$的证明得,对充分大的 $\gamma>\frac{1+\alpha}{p}>-1$ 有 \begin{eqnarray*} && \left(\|T_\psi(f_j)\|_{D_\beta^q}\right)^{p}=\left(\int_{B_n}|R\psi(z)f_j(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq& C\left(\int_{B_n}\left(\frac{|Rf_j(w)|^p}{|1-\langle z,w\rangle |^{n+1+\gamma+s-p}}{\rm d}v_{\gamma}(w)\right)^{\frac{q}{p}}|R\psi(z)|^{q}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}\\ &\leq & C\int_{B_n}|Rf_j(w)|^p(1-|w|^2)^{\alpha}\left(\int_{B_n}\frac{(1-|w|^2)^{q}|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^ {\frac{q}{p}(n+1+\alpha+s)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v(w)\\ &\leq&C\bigg(\int_{r_0B_n}+\int_{B_n-r_0B_n}\bigg)|Rf_j(w)|^p(1-|w|^2)^{\alpha}\left(\int_{B_n}\frac{(1-|w|^2)^{q}|R\psi(z)|^{q}}{|1-\langle z,w\rangle |^ {\frac{q}{p}(n+1+\alpha+s)}}{\rm d}v_\beta(z)\right)^{\frac{p}{q}}{\rm d}v(w)\\ &\leq& C(M^{\frac{p}{q}}\varepsilon^p+\varepsilon^{\frac{p}{q}}\|f_j\|^p_{D_\alpha^p})\leq C(M^{\frac{p}{q}}\varepsilon^p+\varepsilon^{\frac{p}{q}}). \end{eqnarray*} 所以$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子.
(iv) 当$p=n+1+\alpha$时,若$\psi\in D_{\beta}^q$且(3.9)式成立, 则由引理2.6、(3.15)和(3.16)式得 \begin{eqnarray*} \|T_\psi(f_j)\|^q_{D_\beta^q} &=&\int_{B_n}|R\psi(z)f_j(z)|^{q}{\rm d}v_\beta(z)\\ &\leq&C\bigg(\int_{r_0B_n}+\int_{B_n-r_0B_n}\bigg)|R\psi(z)f_j(z)|^{q}{\rm d}v_\beta(z)\\ &\leq& C\left(\varepsilon^q\|\psi\|^q_{D_\beta^q} +\|f_j\|_{D_\alpha^p}^{q}\int_{B_n-r_0B_n}|R\psi(z)|^q \bigg(\log{\frac{2}{1-|z|^2}}\bigg)^q{\rm d}v_\beta(z)\right)\\ &\leq& C(\varepsilon^q\|\psi\|^q_{D_\beta^q}+\varepsilon ) . \end{eqnarray*} 所以$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子.
必要性: 若$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是紧算子, 则$T_\psi:D_{\alpha}^p \rightarrow D_{\beta}^q$是有界算子. 取$f=1$得 $\psi\in D_{\beta}^q$,则(ii)的必要性得证.
任取$a_j\in B_n$且$|a_j|\rightarrow 1(j\rightarrow\infty)$, 当$0<p<n+1+\alpha$时,令$ f_j(z)=\frac{(1-|a_j|^2)} {(1-\langle z,a_j\rangle)^\frac{n+1+\alpha}{p}}$,则 $\{f_{j}\}$在$B_n$上内闭一致收敛于0且$\|f_j\|_{D_\alpha^p}\leq c$, 再根据$T_\psi$的紧性知 ${\lim\limits_{j\rightarrow \infty}\|T_{\psi}f_{j}\|_{D_\beta^q}=0}$.
由$T_\psi(f_j)(0)=0$得 $$ \int_{B_n}\frac{|R\psi(z)|^q(1-|a_j|^2)^{q}{\rm d}v_\beta(z)} {|1-\langle a_j,z\rangle |^{\frac{q(n+1+\alpha)}{p}}}= \int_{B_n}|R\psi(z)|^q|f_j(z)|^q{\rm d}v_\beta(z)=\|T_\psi(f_j)\|_{D_{\beta}^q}^q. $$ 所以 $$ \lim_{j\rightarrow\infty}\int_{B_n}\frac{|R\psi(z)|^q (1-|a_j|^2)^{q}{\rm d}v_\beta(z)}{|1-\langle a_j,z\rangle |^{\frac{q(n+1+\alpha)}{p}}}=0. $$ 由$a_j$的任意性得 $$ \lim_{|a|\rightarrow 1}\int_{B_n}\frac{|R\psi(z)|^q(1-|a|^2)^{q} {\rm d}v_\beta(z)}{|1-\langle a,z\rangle |^{\frac{q(n+1+\alpha)}{p}}}=0. $$ (i),(iii)的必要性得证.
当$p=n+1+\alpha$时,令$ f_j(z)= (\log\frac{2}{1-|a_j|^2})^{-\frac{2}{p}} (\log\frac{2}{1-\langle a_j,z\rangle })^{1+\frac{2}{p}}$ 类似可以证得(3.8)式成立.
2015, Vol. 35A Issue (1): 182-193
PDF (310 KB)
2015, Vol. 35A Issue (1): 182-193
PDF (310 KB)