# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## The Bergman kernel function of some Reinhardt domainsHTML articles powered by AMS MathViewer

by Sheng Gong and Xuean Zheng
Trans. Amer. Math. Soc. 348 (1996), 1771-1803 Request permission

## 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 $z$.
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