The Bergman kernel function of some Reinhardt domains

Abstract:

The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points $(z, \bar {z})$. Let $D$ be the Reinhardt domain \[ D = \left \{ z \in \mathbf {C}^n | \|z\|_\alpha =\sum _{j=1}^n|z_j|^{2/\alpha _j}<1 \right \} \] where $\alpha _j>0$, $j=1,2,\dots , n$; and let $K(z,\bar w)$ be the Bergman kernel function of $D$. Then there exist two positive constants $m$ and $M$ and a function $F$ such that \[ mF(z, \bar {z}) \le K(z, \bar {z})\le MF(z, \bar {z}) \] holds for every $z\in D$. Here \[ F(z, \bar {z})=(-r(z))^{-n-1} \prod _{j=1}^n (-r(z)+|z_j|^{2/\alpha _j})^{1-\alpha _j} \] and $r(z)=\|z\|_\alpha -1$ is the defining function for $D$. The constants $m$ and $M$ depend only on $\alpha =(\alpha _1,\dots , \alpha _n)$ and $n$, not on $z$.