The Bergman kernel function of some Reinhardt domains (II)

Abstract

The boundary behavior of the Bergman kernel function of a kind of Reinhardt domain is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points$Z,\bar Z$. Let Ω be the Reinhardt domain$\Omega = \left\{ {Z = (Z_1 ,Z_2 , \cdots ,Z_n ) \in \mathbb{C}^N ,Z_j \in \mathbb{C}^{N_j } ,j = 1,2. \cdots ,n,| \left\| Z \right\|_a = \sum\limits_{j = 1}^n {\left\| {Z_j } \right\|} _{a_j }^{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} }< 1} \right\}$ where$\alpha _j > 0, j - 1,2, \cdots ,n,N = N_1 + N_2 + \cdots + N_n ,||Z_j ||$ is the standard Euclidean norm in$\mathbb{C}^{N_j } ,j = 1,2, \cdots ,n$ j and let$K(Z,\overline W )$be the Bergman kernel function of Ω. Then there exist two positive constantsm andM, and a functionF such that$mF(Z,\overline Z ) \leqslant K(Z,\overline Z ) \leqslant MF(Z,\overline Z )$ holds for every Z∈Ω. Here$mF(Z,\overline Z ) = ( - r(Z))^{ - N - 1} \prod\limits_{j = 1}^n {( - r(Z) + ||Z_j ||_{a_j }^{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} } )^{(1 - a_j )N_j } }$ and r(Z)=||Z||_α-1 is the defining function of Ω. The constantsm and M depend only on α = (α₁,...,α_n) and N₁, N₂, N_n, not on Z. This result extends some previous known results.