Singularities of the Bergman kernel for certain weakly pseudoconvex domains

Details Singularities of the Bergman kernel for certain weakly pseudoconvex domains Singularities of the Bergman kernel for certain weakly pseudoconvex domains

1996-06-26

arxiv.org/abs/math/9606202v1

Mathematics

Complex Variables

Scientific paper

Consider the Bergman kernel $K^B(z)$ of the domain $\ellip = \{z \in \Comp^n ; \sum_{j=1}^n |z_j|^{2m_j}<1 \}$, where $m=(m_1,\ldots,m_n) \in \Natl^n$ and $m_n \neq 1$. Let $z^0 \in \partial \ellip$ be any weakly pseudoconvex point, $k \in \Natl$ the degenerate rank of the Levi form at $z^0$. An explicit formula for $K^B(z)$ modulo analytic functions is given in terms of the polar coordinates $(t_1, \ldots, t_k, r)$ around $z^0$. This formula provides detailed information about the singularities of $K^B(z)$, which improves the result of A. Bonami and N. Lohou\'e \cite{bol}. A similar result is established also for the Szeg\"o kernel $K^S(z)$ of $\ellip$.

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