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ISSN 1088-6850(e) ISSN 0002-9947(p)

The Bergman kernel function of some Reinhardt domains

Author(s): Sheng Gong; Xuean Zheng
Journal: Trans. Amer. Math. Soc. 348 (1996), 1771-1803.
MSC (1991): Primary 32H10
MathSciNet review: 1329534
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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

$\begin{displaymath}D=\left\{ z\in\mathbf C^n\,|\,\|z\|_\alpha=\sum_{j=1}^n|z_j|^{2/\al_j}<1\right\} \end{displaymath}$

where $\al_j>0$ , $j=1,2,\dots, n$ ; 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

$\begin{displaymath}mF(z,\bar z) \le K(z,\bar z)\le MF(z,\bar z) \end{displaymath}$

holds for every $z\in D$ . Here

$\begin{displaymath}F(z,\bar z)=(-r(z))^{-n-1} \prod_{j=1}^n (-r(z)+|z_j|^{2/\al_j})^{1-\al_j} \end{displaymath}$

and $r(z)=\|z\|_\alpha-1$ is the defining function for $D$ . The constants $m$ and $M$ depend only on $\alpha=(\al_1,\dots, \al_n)$ and $n$ , not on $z$ .

References:

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