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2007 Some estimates for the Bergman kernel and metric in terms of logarithmic capacity
Zbigniew Błocki
Nagoya Math. J. 185: 143-150 (2007).

Abstract

For a bounded domain $\Omega$ on the plane we show the inequality $c_{\Omega}(z)^{2} \leq 2\pi K_{\Omega}(z)$, $z \in \Omega$, where $c_{\Omega}(z)$ is the logarithmic capacity of the complement $\mathbb{C} \setminus \Omega$ with respect to $z$ and $K_{\Omega}$ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality $c_{\Omega}(z)^{2} \leq \pi K_{\Omega}(z)$ conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.

Citation

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Zbigniew Błocki. "Some estimates for the Bergman kernel and metric in terms of logarithmic capacity." Nagoya Math. J. 185 143 - 150, 2007.

Information

Published: 2007
First available in Project Euclid: 23 March 2007

zbMATH: 1127.30006
MathSciNet: MR2301462

Subjects:
Primary: 30C40 , 31A35

Rights: Copyright © 2007 Editorial Board, Nagoya Mathematical Journal

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Vol.185 • 2007
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