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Theory of Bergman Spaces in the Unit Ball of \(\mathbb{C}^n\)
Ruhan Zhao, SUNY, College at Brockport, NY, and Kehe Zhu, SUNY at Albany, NY
A publication of the Société Mathématique de France.
Mémoires de la Société Mathématique de France
2008; 103 pp; softcover
Number: 115
ISBN-10: 2-85629-267-4
ISBN-13: 978-2-85629-267-9
List Price: US$42
Individual Members: US$37.80
Order Code: SMFMEM/115
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There has been a great deal of work done in recent years on weighted Bergman spaces \(A^p_\alpha\) on the unit ball \({\mathbb B}_n\) of \({\mathbb C}^n\), where \(0 < p < \infty\) and \(\alpha>-1\). The authors extend this study in a very natural way to the case where \(\alpha\) is any real number and \(0 < p\le\infty\). This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space \(H^2\), and the so-called Arveson space. Some of the results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians interested in analysis.

Table of Contents

  • Introduction
  • Various special cases
  • Preliminaries
  • Isomorphism of Bergman spaces
  • Several characterizations of \({A^p_\alpha}\)
  • Holomorphic Lipschitz spaces
  • Pointwise estimates
  • Duality
  • Integral representations
  • Atomic decomposition
  • Complex interpolation
  • Reproducing kernels
  • Carleson type measures
  • Coefficient multipliers
  • Lacunary series
  • Inclusion relations
  • Further remarks
  • Bibliography
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