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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Bergman kernel function of
some Reinhardt domains


Authors: Sheng Gong and 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 $z$.


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Additional Information

Sheng Gong
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China; Department of Mathematics, University of California, San Diego, La Jolla, California 92093

Xuean Zheng
Affiliation: Department of Mathematics, Anhui University, Hefei, Anhui, 230039, People’s Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01526-7
PII: S 0002-9947(96)01526-7
Received by editor(s): October 13, 1994
Article copyright: © Copyright 1996 American Mathematical Society