Singularities of the Bergman Kernel for Certain Weakly Pseudoconvex Domains

摘要

Consider the Bergman kernel K~B(z) of the domain ε_m = {z ∈ C~n; ∑_(j = 1)~n |z_j|~(2m_j) < 1}, where m = {m_1, …, m_n} ∈ N~n and m_n ≠ 1. Let z~0 ∈ partial derivε_m be any weakly pseudoconvex point, k ∈ N 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, …, 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. Lohoue [4]. A similar result is established also for the Szego kernel K~S(z) of ε_m.