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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.

Keywords:  Bergman projection, Bergman spaces, and Orlicz spaces

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