Extension and restriction of holomorphic functions. (English) Zbl 0881.32005

Diederich, Klas; Mazzilli, Emmanuel

Ann. Inst. Fourier 47, No. 4, 1079-1099 (1997).

Summary: Strong pathologies with respect to growth properties can occur for the extension of holomorphic functions from submanifolds \(D'\) of pseudoconvex domains \(D\) to all of \(D\) even in quite simple situations; the spaces \(A^{p}(D'):={\mathcal O}(D')\cap L^{p}(D')\) are, in general, not at all preserved. Also the image of the Hilbert space \(A^{2}(D)\) under the restriction to \(D'\) can have a very strange structure.

MSC:

32D15	Continuation of analytic objects in several complex variables
32T99	Pseudoconvex domains
32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A10	Holomorphic functions of several complex variables
32A35	\(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables

Keywords:

extension of holomorphic functions; \(A^ p\)-spaces; weighted Bergman spaces; pseudoellipsoids

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