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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A counterexample to the differentiability
of the Bergman kernel function


Author: So-Chin Chen
Journal: Proc. Amer. Math. Soc. 124 (1996), 1807-1810
MSC (1991): Primary 32H10
MathSciNet review: 1322916
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Abstract: In this paper we prove the following main result. Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf {C} ^n$ with $n\ge 2$. Suppose that there exists a complex variety sitting in the boundary $bD$; then we have

\begin{displaymath}K_{D}(z,w)\notin C^{\infty }(\overline {D}\times \overline{D}-\Delta (bD)). \end{displaymath}

In particular, the Bergman kernel function associated with the Diederich-Fornaess worm domain is not smooth up to the boundary in joint variables off the diagonal of the boundary.


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Additional Information

So-Chin Chen
Affiliation: Institute of Applied Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, Republic of China
Email: scchen@am.nthu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03290-X
PII: S 0002-9939(96)03290-X
Received by editor(s): December 1, 1994
Communicated by: Eric Bedford
Article copyright: © Copyright 1996 American Mathematical Society