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Area functions characterizations of weighted Bergman spaces. (English) Zbl 1352.32004

Summary: The aim of this paper is to find weights \(W\) in the unit ball of \(\mathbb{C}^n\) for which characterization of the area integrals of Bergman spaces \(A^p(W)\) holds. The area functions, related to those used to describe Hardy spaces, involve the radial derivative, the complex gradient and the invariant gradient. We extend to certain Bekollé weights the characterization by Z. Chen and W. Ouyang of the Bergman spaces with the classical weights \(W(z)=(1-|z|)^\alpha\) using area functions.

MSC:

32A36 Bergman spaces of functions in several complex variables
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References:

[1] Bekollé, D., Inégalité à poids pour le projecteur de Bergman dans la boule unité de \(C^n\), Studia Math., 71, 3, 305-323 (1981/82) · Zbl 0516.47016
[2] Blasco, O.; Pérez-Esteva, S., Averaging operators, Berezin transforms and atomic decomposition on Bergman-Herz spaces, Eur. J. Math., 1, 3, 560-581 (2015) · Zbl 1323.47039
[3] Chen, Z.; Ouyang, W., Maximal and area integral characterizations of Bergman spaces in the unit ball of \(C^n\), J. Funct. Spaces Appl., 1-13 (2013) · Zbl 1279.32004
[4] Zhao, R.; Zhu, K., Theory of Bergman spaces in the unit ball of \(C^n\), Mém. Soc. Math. Fr., 115 (2008) · Zbl 1176.32001
[5] Zhu, K., Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, vol. 226 (2005), Springer: Springer New York · Zbl 1067.32005
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