Reinhardt domains. II (Q1913909)

Let \(K(z,\overline w)\) be the Bergman kernel function for a domain \[ D: =\left\{z \in\mathbb{C}^n: \;|z|_\alpha: =\sum^n_{j=1} |z_j |^{2/ \alpha_j}<1 \right\}, \] where \(0<\alpha_j\), \(j=1,2,\dots,n\). The main result is: There exist two positive constants \(m\) and \(M\), which depend on \(n\) and \(\alpha_j\), \(j=1,2, \dots, n\) only, such that \[ mF(z, \overline z) \leq K(z, \overline z) \leq MF(z, \overline z) \] holds for every \(z\in D\), where \[ F(z, \overline z) = \bigl(-r(z)\bigr)^{-n-1} \sum^n_{j=1} \bigl(-r(z) + z_j^{2/ \alpha_j} \bigr)^{1-\alpha_j} \] and \(r(z)= |z|_\alpha-1\) is the defining function for \(D\). [For part I see the second named author, Chin. Ann. Math., Ser. A8, 205-219 (1987; Zbl 0628.32003)].