Bergman kernel and hyperconvexity index

Chen, Bo-Yong

doi:10.2140/apde.2017.10.1429

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Bergman kernel and hyperconvexity index

Bo-Yong Chen

Vol. 10 (2017), No. 6, 1429–1454

DOI: 10.2140/apde.2017.10.1429

Abstract

Let $Ω \subset ℂ^{n}$ 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 $2 \leq p < 2 + 2 α (Ω) ∕ (2 n - α (Ω))$ , there exists a constant $C > 0$ such that $\int_{Ω} | K_{Ω} (\cdot, w) ∕ \sqrt{K_{Ω} (w)} |^{p} \leq C | μ (w) |^{- (p - 2) n ∕ α}$ for all $w \in Ω$ . (2) For every $0 < r < 1$ , there exists a constant $C > 0$ such that $| K_{Ω} (z, w) |^{2} ∕ (K_{Ω} (z) K_{Ω} (w)) \leq C {(min {ν (z) ∕ μ (w), ν (w) ∕ μ (z)})}^{r}$ for all $z, w \in Ω$ . 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

Vol. 10, No. 6, 2017