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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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The Bergman kernel function of some Reinhardt domains
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by Sheng Gong and Xuean Zheng PDF
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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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
  • Received by editor(s): October 13, 1994
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 1771-1803
  • MSC (1991): Primary 32H10
  • DOI: https://doi.org/10.1090/S0002-9947-96-01526-7
  • MathSciNet review: 1329534