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AN INTEGRAL REPRESENTATION FOR BESOV AND LIPSCHITZ SPACES

Part of: Holomorphic functions of several complex variables Spaces and algebras of analytic functions

Published online by Cambridge University Press:  17 October 2014

KEHE ZHU
KEHE ZHU*
Affiliation:
Department of Mathematics, SUNY at Albany, Albany, NY 12222, USA email kzhu@math.albany.edu
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Abstract

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It is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation

$$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$
where ${\it\mu}$ is a complex Borel measure with $|{\it\mu}|(\mathbb{D})<\infty$. We generalize this result to all Besov spaces $B_{p}$ with $0<p\leq 1$ and all Lipschitz spaces ${\rm\Lambda}_{t}$ with $t>1$. We also obtain a version for Bergman and Fock spaces.

Keywords

Bergman spacesBesov spacesLipschitz spacesFock spacesCarleson measuresatomic decompositionBerezin transform

MSC classification

Primary: 30H20: Bergman spaces, Fock spaces
Secondary: 30H25: Besov spaces and $Q_p$-spaces 32A36: Bergman spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 1 , February 2015 , pp. 129 - 144
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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