The Bergman kernel function of some Reinhardt domains. II (Q1585593)

The authors estimate the size of the Bergman kernel function on the diagonal for a general class of Reinhardt domains. For \(1\leq j \leq n\), suppose \(\alpha_j\) is a positive real number, \(N_j\) is a positive integer, \(Z_j\) is the variable in \(\mathbb{C}^{N_j}\), and \(N=N_1+\dots+N_n\). For \(Z=(Z_1,\dots,Z_n)\in \mathbb{C}^N\), set \(r(Z)=-1+ \sum_{j=1}^n \|Z_j\|^{2/\alpha_j}\), and let \(\Omega\) be the Reinhardt domain in \(\mathbb{C}^N\) defined by \(r(Z)<0\). The main result states that \(K_\Omega(Z)\), the Bergman kernel function of \(\Omega\) on the diagonal, is bounded above and below by positive constants (depending only on the \(\alpha_j\) and the \(N_j\)) times the quantity \[ (-r(Z))^{-N-1} \prod_{j=1}^n (-r(Z) + \|Z_j\|^{2/\alpha_j})^ {(1-\alpha_j)N_j}. \] This generalizes the authors' previous result [Trans. Am. Math. Soc. 348, No. 5, 1771-1803 (1996; Zbl 0865.32017)] that covered the case when \(N_j=1\) for every~\(j\).