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The Bergman and Szegő Kernels: a Direct Relationship

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Abstract

In this paper, we prove, for a fairly general class of domains, that the Bergman kernel of a domain is closely related to the normal derivative of the Szegő kernel. Such a result is useful in passing back and forth between estimates for the Bergman projection and estimates for the Szegő projection. We also make some remarks about comparability of the singularity of the Bergman kernel and the singularity of the Szegő kernel.

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Notes

  1. Of course, on a smoothly bounded pseudoconvex domain, this definition of Szegő kernel is equivalent to the fact that the it is the kernel of the projection operator from L2(∂Ω) to H2(∂Ω). And similarly for the Bergman kernel (see [4] and [7, Section 1.4]).

References

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  3. Boutet de Monvel, L, Sjöstrand, J: Sur la singularité des noyaux de Bergman et Szegő. Journées: Équations aux Dérivées Partielles de Rennes (1975), pp 123–164. Astérisque, No. 34–35, Soc. Math. France (1976)

  4. Catlin, D: Boundary behavior of holomorphic functions on pseudoconvex domains. J. Differ. Geom. 15, 605–625 (1980)

  5. Federer, H: Geometric Measure Theory. Springer, New York (1969)

  6. Fefferman, C: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 1–65 (1974)

  7. Krantz, SG: Function Theory of Several Complex Variables, 2nd ed. Am. Math. Soc., Providence, RI (2001)

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Authors and Affiliations

  1. Department of Mathematics, Washington University in St. Louis, St. Louis, MO, 63130, USA

    Steven G. Krantz

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  1. Steven G. Krantz
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Correspondence to Steven G. Krantz.

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Krantz, S.G. The Bergman and Szegő Kernels: a Direct Relationship. Acta Math Vietnam 46, 509–514 (2021). https://doi.org/10.1007/s40306-020-00407-w

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  • DOI: https://doi.org/10.1007/s40306-020-00407-w

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