Hostname: page-component-7bb8b95d7b-495rp Total loading time: 0 Render date: 2024-09-11T07:41:48.100Z Has data issue: false hasContentIssue false

Some estimates for the Bergman Kernel and Metric in Terms of Logarithmic Capacity

Published online by Cambridge University Press:  11 January 2016

Zbigniew Błocki*
Affiliation:
Jagiellonian University, Institute of Mathematics, Reymonta 4 30-059 Kraków, Poland, Zbigniew.Blocki@im.uj.edu.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a bounded domain Ω on the plane we show the inequality cΩ(z)22πKΩ(z), z ∈ Ω, where cΩ(z) is the logarithmic capacity of the complement ℂ\Ω with respect to z and KΩ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality cΩ(z)2 ≤ πKΩ(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.

Keywords

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

[B1] Berndtsson, B., Weighted estimates for in domains in ℂ, Duke Math. J., 66 (1992), 239255.Google Scholar
[B2] Berndtsson, B., The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman, Ann. Inst. Fourier, 46 (1996), 10831094.Google Scholar
[B3] Berndtsson, B., Personal communication, Beijing, August 2004.Google Scholar
[C] Chen, B. -Y., A remark on an extension theorem of Ohsawa, Chin. Ann. Math., Ser. A, 24 (2003), 129134 (in Chinese).Google Scholar
[Co] Conway, J. B., Functions of One Complex Variable. II, Springer-Verlag, 1995.Google Scholar
[O1] Ohsawa, T., Addendum to “On the Bergman kernel of hyperconvex domains”, Nagoya Math. J., 137 (1995), 145148.CrossRefGoogle Scholar
[O2] Ohsawa, T., On the extension of L2 holomorphic functions V - effects of generalization, Nagoya Math. J., 161 (2001), 121.Google Scholar
[OT] Ohsawa, T. and Takegoshi, K., On the extension of L2 holomorphic functions, Math. Z., 195 (1987), 197204.CrossRefGoogle Scholar
[S] Suita, N., Capacities and kernels on Riemann surfaces, Arch. Ration. Mech. Anal., 46 (1972), 212217.Google Scholar
[Z1] Zwonek, W., Regularity properties for the Azukawa metric, J. Math. Soc. Japan, 52 (2000), 899914.Google Scholar
[Z2] Zwonek, W., Wiener’s type criterion for Bergman exhaustiveness, Bull. Polish Acad. Sci. Math., 50 (2002), 297311.Google Scholar