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Transactions of the American Mathematical Society

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The Bergman spaces, the Bloch space, and Gleason's problem


Author: Ke He Zhu
Journal: Trans. Amer. Math. Soc. 309 (1988), 253-268
MSC: Primary 46E15; Secondary 32A35, 32H10, 46J15
MathSciNet review: 931533
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Abstract: Suppose $ f$ is a holomorphic function on the open unit ball $ {B_n}$ of $ {{\mathbf{C}}^n}$. For $ 1 \leqslant p < \infty $ and $ m > 0$ an integer, we show that $ f$ is in $ {L^p}({B_n},\,dV)$ (with $ dV$ the volume measure) iff all the functions $ {\partial ^m}f/\partial {z^{\alpha \,}}\;(\vert\alpha \vert\, = m)$ are in $ {L^p}({B_n},\,dV)$. We also prove that $ f$ is in the Bloch space of $ {B_n}$ iff all the functions $ {\partial ^m}f/\partial {z^\alpha }\;(\vert\alpha \vert\, = m)$ are bounded on $ {B_n}$. The corresponding result for the little Bloch space of $ {B_n}$ is established as well. We will solve Gleason's problem for the Bergman spaces and the Bloch space of $ {B_n}$ before proving the results stated above. The approach here is functional analytic. We make extensive use of the reproducing kernels of $ {B_n}$. The corresponding results for the polydisc in $ {{\mathbf{C}}^n}$ are indicated without detailed proof.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0931533-6
Article copyright: © Copyright 1988 American Mathematical Society