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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

Received by editor(s): December 1, 1994
Communicated by: Eric Bedford
Article copyright: © Copyright 1996 American Mathematical Society

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