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Bergman Kernel Function for Hartogs Domains Over Bounded Homogeneous Domains

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Abstract

We obtain an explicit formula of the Bergman kernel for Hartogs domains over bounded homogeneous domains. In order to find a simple formula, we consider a Siegel domain biholomorphic to the bounded homogeneous domain and use its Bergman kernel obtained by Gindikin. The Bergman kernel of the Hartogs domain is expressed by two different forms and the main part of the Bergman kernel is a polynomial whose coefficients contain the Stirling number of the second kind. As an application of our formula, we investigate the Lu Qi-Keng problem for our Hartogs domains and give some important examples of Hartogs domains whose Bergman kernels are zero-free.

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Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Furo-Cho, Chikusa-Ku, Nagoya, 464-8602, Japan

    Hideyuki Ishi

  2. Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul, 130-701, Republic of Korea

    Jong-Do Park

  3. Institute of Mathematics, Academia Sinica 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan

    Atsushi Yamamori

Authors
  1. Hideyuki Ishi
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  2. Jong-Do Park
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  3. Atsushi Yamamori
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Correspondence to Jong-Do Park.

Additional information

The second author was supported by NRF-2010-0011841 from the National Research Foundation of Korea.

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Ishi, H., Park, JD. & Yamamori, A. Bergman Kernel Function for Hartogs Domains Over Bounded Homogeneous Domains. J Geom Anal 27, 1703–1736 (2017). https://doi.org/10.1007/s12220-016-9737-4

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  • DOI: https://doi.org/10.1007/s12220-016-9737-4

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