Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 6 (2017), 1429-1454.

Bergman kernel and hyperconvexity index

Bo-Yong Chen

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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.

Article information

Anal. PDE, Volume 10, Number 6 (2017), 1429-1454.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.)
Secondary: 32U35: Pluricomplex Green functions

Bergman kernel hyperconvexity index


Chen, Bo-Yong. Bergman kernel and hyperconvexity index. Anal. PDE 10 (2017), no. 6, 1429--1454. doi:10.2140/apde.2017.10.1429.

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