Vol. 10, No. 6, 2017

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Bergman kernel and hyperconvexity index

Bo-Yong Chen

Vol. 10 (2017), No. 6, 1429–1454

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

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