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2017 Bergman kernel and hyperconvexity index
Bo-Yong Chen
Anal. PDE 10(6): 1429-1454 (2017). DOI: 10.2140/apde.2017.10.1429

Abstract

Let Ω n be a bounded domain with the hyperconvexity index α(Ω) > 0. Let ϱ be the relative extremal function of a fixed closed ball in Ω, and set μ := |ϱ|(1 + |log|ϱ||)1 and ν := |ϱ|(1 + |log|ϱ||)n. We obtain the following estimates for the Bergman kernel. (1) For every 0 < α < α(Ω) and 2 p < 2 + 2α(Ω)(2n α(Ω)), there exists a constant C > 0 such that Ω|KΩ( ,w)KΩ (w)|p C|μ(w)|(p2)nα for all w Ω. (2) For every 0 < r < 1, there exists a constant C > 0 such that |KΩ(z,w)|2(KΩ(z)KΩ(w)) C(min{ν(z)μ(w),ν(w)μ(z)})r for all z,w Ω. Various applications of these estimates are given.

Citation

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Bo-Yong Chen. "Bergman kernel and hyperconvexity index." Anal. PDE 10 (6) 1429 - 1454, 2017. https://doi.org/10.2140/apde.2017.10.1429

Information

Received: 11 November 2016; Revised: 27 February 2017; Accepted: 24 April 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1368.32003
MathSciNet: MR3678493
Digital Object Identifier: 10.2140/apde.2017.10.1429

Subjects:
Primary: 32A25
Secondary: 32U35

Keywords: Bergman kernel , hyperconvexity index

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.10 • No. 6 • 2017
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