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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