Analytic extension from non-pseudoconvex boundaries and $A\left(D\right)$-convexity

Laurent-Thiébaut, Christine; Porten, Egmon

Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 847-857.

Soit $D\subset \subset {ℂ}^n,n\ge 2$, un domaine à bord $C^2$ et $K\subset \partial D$ un compact tel que $\partial D\setminus K$ soit connexe. On étudie l’extension holomorphe des fonctions CR définies sur $\partial D\setminus K$ à des sous-ensembles de $D$. On dit que $K$ est CR-convexe si son enveloppe $A\left(D\right)$-convexe, $A\left(D\right)-\mathrm{hull}\left(K\right)$, vérifie $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ désigne l’espace des fonctions holomorphes sur $D$ et continues sur $\overline{D}$). Le théorème principal de cet article prouve l’extension holomorphe à $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, si $K$ est CR-convexe.

Let $D\subset \subset {ℂ}^n,n\ge 2$, be a domain with $C^2$-boundary and $K\subset \partial D$ be a compact set such that $\partial D\setminus K$ is connected. We study univalent analytic extension of CR-functions from $\partial D\setminus K$ to parts of $D$. Call $K$ CR-convex if its $A\left(D\right)$-convex hull, $A\left(D\right)-\mathrm{hull}\left(K\right)$, satisfies $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ denoting the space of functions, which are holomorphic on $D$ and continuous up to $\partial D$). The main theorem of the paper gives analytic extension to $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, if $K$ is CR- convex.

Classification : 32V25, 32D10, 32D20
Keywords: holomorphic hulls and holomorphic convexity, CR functions, removable singularities
Mot clés : enveloppes holomorphes et convexité holomorphe, CR fonctions, singularités éliminables

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