Journal of Operator Theory

Volume 46, Issue 1, Summer 2001 pp. 3-24.

Bergman projection and Bergman spaces

Authors: Yaohua Deng (1), Li Huang (2), Tao Zhao (3), and Dechao Zheng (4)
Author institution: (1) Research Center of Applied Math., Xian Jiaotong University, Xian, 710049, P.R. China
(2) Siemens Business Service I&C, Siemens Ltd. China, Beijing, 100015, P.R. China
(3) Department of Mathematics, Tufts University, Medford, MA 02145, USA,
(4) Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

Summary: In this paper we study mapping properties of the Bergman projection

$P$ , i.e.\ w hich function spaces or classes are preserved by

$P.$ It is shown that the Bergman projection is of weak type

$(1,1)$ and bounded on the Orlicz space

$L^{\phi}({\bbb D},\dd A)$ iff

$L^{\phi}({\bbb D},\dd A)$ is reflexive. So the dual space of the Bergman space

$L^{\phi}_{a}$ is

$L^{\psi}_{a}$ if

$L^{\phi}({\bbb D},\dd A)$ is reflexive, where

$\phi$ and

$\psi$ are a pair of complementary Young functions. In addition, we also get that the Kolmogorov type inequality and the Zygmund type inequality hold for the Bergman projection.

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