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A holomorphic reproducing kernel for Kohn-Nirenberg domains in \({\mathbb C}^ 2\). (English) Zbl 0639.32002

Let \(\Omega\) be a truncated Kohn-Nirenberg domain in \({\mathbb{C}}^ 2:\) \(\Omega =\{w\in {\mathbb{C}}^ 2:\) \(Re w_ 2+P(w_ 1)<0\}\cap \{| w| <R\},\) where P is a real valued homogeneous polynomial which is strictly subharmonic for \(w\neq 0\). A Cauchy-Fantappiè kernel for \(\Omega\) is constructed and its properties are investigated. The construction is based on a modification of the Leray map for \(\Omega\) as constructed by the first author in Ann. Math., II. Ser. 123, 335-345 (1986; Zbl 0589.32038). It is shown that this kernel reproduces \(H(\Omega)\cap C(\Omega ^ -)\) and maps C(b\(\Omega)\) into H(\(\Omega)\), H denoting holomorphic functions and C denoting continuous functions.
Reviewer: J.E.Fornaess

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)

Citations:

Zbl 0589.32038
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