Variétés Singulières et Extension des Fonctions Holomorphes

Abstract

Abstract. In this paper, we study a problem of extension of holomorphic functions given on a complex hypersurface with singularities on the boundary of a strictly pseudoconvex domain. §0. Introduction Les résultats principaux de cet article concernent un problème d'extension de fonction holomorphe, f , donnée sur une hypersurface complexe irréductible X définie au voisinage d'un domaine strictement pseudoconvexe D. Un théorème fondamental de Henkin [4] affirme que si X n'a pas de singularités sur ∂D alors f bornée admet une extension holomorphe bornée ; ce qui est faux si X a des singularités sur le bord de D (pour un contre-exemple, voir par exemple [3]). Ici, nous donnons une condition suffisante "raisonnable" sur f pour qu'une telle propriété d'extension H ∞ soit assurée pour certains modèles d'hypersurface complexe (voir définition 1-6-admettant notamment un paramétrage local). A notre connaissance, il n'existe qu'un résultat d'extension ([1]) dans le cas singulier mais en norme L 2 et la condition suffisante avancée est très forte : elle entraîne que f s'annule sur les singularités de X. Mais ce résultat est valable sans aucune restriction sur D pseudoconvexe. L'autre situation envisagée est le cas d'une hypersurface X de multiplicité inférieureà 2 sur ∂D (nous appelons très improprement de telles hypersurfaces de type "Morse"). A la différence, des hypersurfaces modèles cidessus, ce type d'hypersurface n'admet pas de paramétrage local, en général.Dans notre situation, la difficulté réside dans le fait que nous utilisons une extension de f donnée par l'action du courant résiduel associéà X, sur une certaine forme différentielle et malheureusement, il est bien connu qu'il

Extension with growth estimates of holomorphic functions defined on singular analytic spaces

“…In the last few years, many researches have been done on classical problems in complex analysis in the case of singular spaces; for example the ∂-Neumann operator has been studied in [33] by Ruppenthal, the Cauchy-Riemann equation in [6,17,21,31,32] by Andersson, Samuelsson, Diederich, Fornaess, Vassiliadou, Ruppenthal, ideals of holomorphic functions on analytic spaces in [5] by Andersson, Samuelsson and Sznajdman, problems of extensions and restrictions of holomorphic functions on analytic spaces in [18,20] by Diederich, Mazzilli and Duquenoy. In this article, we will be interested in problems of extension of holomorphic functions defined on an analytic space. Let D be a bounded pseudoconvex domain of C n with smooth boundary, let f be a holomorphic function in a neighborhood of D and let X = {z, f (z) = 0} be an analytic set such that D ∩ X = ∅.…”

“…In [3], Amar pointed out for the first time the importance of the current ∂ 1 f in the problem of extension. In [20] the extension is given by an operator constructed by Passare which uses the classical residue current ∂ 1 f (see [27]). However, as pointed out in [20], it is not so easy to handle the case of singularities of order greater than 2 and the classical currents do not give a good extension in this case.…”

“…In [20] the extension is given by an operator constructed by Passare which uses the classical residue current ∂ 1 f (see [27]). However, as pointed out in [20], it is not so easy to handle the case of singularities of order greater than 2 and the classical currents do not give a good extension in this case. To overcome this difficulty we have to adapt a construction due to the second author of new residue currents which will play the role of ∂ 1 f (see [24] and [25]).…”

“…It should be noticed that a condition using divided differences was already used in [20] but that only varieties with singularities of order 2 were considered there. Here we have no restriction on the order of the singularities, and our condition uses all the divided differences of degree at most the orders of the singularities.…”