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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Characterizations of Bergman spaces and Bloch space in the unit ball of $ {\bf C}\sp n$

Authors: Cai Heng Ouyang, Wei Sheng Yang and Ru Han Zhao
Journal: Trans. Amer. Math. Soc. 347 (1995), 4301-4313
MSC: Primary 32A37; Secondary 46E15
MathSciNet review: 1311908
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Abstract: In this paper we prove that, in the unit ball $ B$ of $ {{\mathbf{C}}^n}$, a holomorphic function $ f$ is in the Bergman space $ L_a^p(B),\;0 < p < \infty $, if and only if

$\displaystyle \int_B {\vert\tilde \nabla } f(z){\vert^2}\vert f(z){\vert^{p - 2}}{(1 - \vert z{\vert^2})^{n + 1}}d\lambda (z) < \infty ,$

where $ \tilde \nabla $ and $ \lambda $ denote the invariant gradient and invariant measure on $ B$, respectively. Further, we give some characterizations of Bloch functions in the unit ball $ B$, including an exponential decay characterization of Bloch functions. We also give the analogous results for $ \operatorname{BMOA} (\partial B)$ functions in the unit ball.

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Article copyright: © Copyright 1995 American Mathematical Society

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