Vol. 10, No. 6, 2017

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Bergman kernel and hyperconvexity index

Bo-Yong Chen

Vol. 10 (2017), No. 6, 1429–1454
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.

Dedicated to Professor John Erik Fornaess on the occasion of his 70th birthday

Keywords
Bergman kernel, hyperconvexity index
Mathematical Subject Classification 2010
Primary: 32A25
Secondary: 32U35
Milestones
Received: 11 November 2016
Revised: 27 February 2017
Accepted: 24 April 2017
Published: 14 July 2017
Authors
Bo-Yong Chen
School of Mathematical Sciences
Fudan University
220 Handan Road
Shanghai 200433
China