Abstract
We obtain new explicit formulas for the Bergman kernel function on two families of Hartogs domains. To do so, we first compute the Bergman kernels on the slices of these Hartogs domains with some coordinates fixed, evaluate these kernel functions at certain points off the diagonal, and then apply a first order differential operator to them. We find, for example, explicit formulas for the kernel function on
and on
We use our formulas to determine the boundary behavior of the kernel function of these domains on the diagonal.
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Acknowledgments
These results are part of the author’s PhD thesis at the University of Illinois at Urbana–Champaign. The author acknowledges his thesis advisor Professor John D’Angelo for his patience, encouragement, and valuable advice. The author acknowledges Professor Jeff McNeal for discussions about the boundary behavior of the Bergman kernel. The author also thanks Luke Edholm for helpful conversations. The author thanks the two referees who provided constructive criticisms. This paper is supported by NSF Grant DMS 13-61001 of D’Angelo.
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Huo, Z. The Bergman Kernel on Some Hartogs Domains. J Geom Anal 27, 271–299 (2017). https://doi.org/10.1007/s12220-016-9681-3
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DOI: https://doi.org/10.1007/s12220-016-9681-3