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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