## Singularities of the Bergman Kernel for Certain Weakly Pseudoconvex Domains

J. Math. Sci. Univ. Tokyo
Vol. 5 (1998), No. 1, Page 99--117.

Kamimoto, Joe
Singularities of the Bergman Kernel for Certain Weakly Pseudoconvex Domains
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. LohouÃ© \cite{bol}. A similar result is established also for the SzegÃ¶ kernel $K^S(z)$ of $\ellip$.