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Extendibility of the Bergman kernel function. (English) Zbl 0626.32028
Complex analysis II, Proc. Spec. Year, College Park/Md. 1985-86, Lect. Notes Math. 1276, 33-41 (1987).
[For the entire collection see Zbl 0615.00005.]
Let \(\Omega\) be a bounded pseudoconvex domain in \({\mathbb{C}}^ n\) of finite type and let K(z,w) denote the Bergman kernel of \(\Omega\). By a result of N. Kerzman [Math. Ann. 195, 149-158 (1972; Zbl 0221.32012)], K(z,w) extends to be \(C^{\infty}(({\bar \Omega}\times {\bar \Omega})-\Delta),\) where \(\Delta =\{(z,z): z\in \partial \Omega \}\). In the paper, the author proves that if \(\Omega\) is a bounded strictly pseudoconvex domain with real analytic boundary, then the Bergman kernel K satisfies the following: if a and b are distinct points in \({\bar \Omega}\), then there exist balls \(B_ a\) and \(B_ b\) centered at a and b respectively, such that K(z,w) extends to be holomorphic in z and antiholomorphic in w on the set \(B_ a\times B_ b\).
Reviewer: M.Stoll

MSC:
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32T99 Pseudoconvex domains