|
|
| |
| BERGMAN TYPE OPERATOR ON MIXED NORM SPACES WITH APPLICATIONS |
| |
Citation： |
Ren Guangbin,Shi Jihuai.BERGMAN TYPE OPERATOR ON MIXED NORM SPACES WITH APPLICATIONS[J].Chinese Annals of Mathematics B,1997,18(3):265~276 |
| Page view： 1128
Net amount： 709 |
Authors： |
Ren Guangbin; Shi Jihuai |
Foundation： |
the National Natural Science Foundation of China and the Doctoral Program Fundation of the State Education
Commission of China |
|
|
| Abstract： |
The authors investigate the conditions for the boundedness
of Bergman type operators $ P_{s,t} $ in mixed norm space $ L_{p,q}(\varphi) $ on the unit ball of $\bold C^n\ (n \geq 1)$,
and obtain a sufficient condition and a necessary condition for general normal function $\varphi$, and a sufficient and necessary condition for $$\varphi(r)=(1-r^2)^{\alpha}{\log^{\beta} (2(1-r)^{-1})}\ \ (\alpha>0,\beta\geq 0). $$This generalizes the result
of Forelli-Rudin$^{[3]}$ on Bergman operator in Bergman space. As applications, a more natural method is given to compute the duality of the mixed norm space, solve the Gleason's problem for mixed norm space and obtain the characterization of
mixed norm space in terms of partial derivatives. Moreover, it is proved that $ f\in L_{\infty,q}^{(0)}(\varphi) $ iff all the functions $ {(1-|z|^2)^{|\alpha|}}{\frac {\partial^{|\alpha|}f}{\partial z^{\alpha}}}(z)\in
L_{\infty,q}^{(0)}(\varphi) $ for holomorphic function $f$, $ 1\le q\le {\infty}. $ |
Keywords： |
Bergman projection, Normal function, Mixed norm space |
Classification： |
32A30, 47B38 |
|
|
Download PDF Full-Text
|
|
|
|
