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Extension de fonctions holomorphes et intégrales singulières. (Extension of holomorphic functions and singular integrals). (French) Zbl 0587.32023
Let \(\Omega\) be a pseudo-convex domain in \({\mathbb{C}}^ n\) and X an analytic subvariety of \(\Omega\), \(C^{\infty}\) up to \(b\Omega\) and transversal to \(b\Omega\). If f is a holomorphic function on X satisfying a certain growth condition, it is natural to ask whether it extends to an analytic function on \(\Omega\) satisfying the same type of growth condition. For strictly pseudo-convex domains \(\Omega\) compare the author’s paper [Bull. Sci. Math., II. Ser. 107, 25-48 (1983; Zbl 0543.32007)], Frank Beatrous jun. [Math. Z. 191, 91-116 (1986)]; and G. M. Henkin [Math. USSR, Izv. 6(1972), 536-563 (1973; Zbl 0255.32008)].
In the note under review, the author considers weakly pseudoconvex domains which still satisfy some additional geometric conditions, and subvarieties X which have complex dimension 1. This last assumption enables the author to use Cauchy integral and Hilbert transform techniques. The main result is: Under the above assumptions, each function f in \(H^ p(X)\) (1\(\leq p\leq \infty)\) or in \(A^ k(X)\) (k\(\in N\cup \{\infty \})\) has a continuation to \(\Omega\) in \(H^ p(\Omega)\) or \(A^ k(\Omega)\) respectively. Here, \(H^ p\) denotes the Hardy classes, and \(A^ k\) the space of functions, analytic and \(C^ k\) up to the boundary.
Reviewer: E.Straube

MSC:
32D15 Continuation of analytic objects in several complex variables
32T99 Pseudoconvex domains
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
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