On the Hartogs extension theorem for unbounded domains in n
Annales de l'Institut Fourier, Volume 72 (2022) no. 3, pp. 1185-1206.

Let Ω n , n2, be a domain with smooth connected boundary. If Ω is relatively compact, the Hartogs–Bochner theorem ensures that every CR distribution on Ω has a holomorphic extension to Ω. For unbounded domains this extension property may fail, for example if Ω contains a complex hypersurface. The main result in this paper tells that the extension property holds if and only if the envelope of holomorphy of n Ω ¯ is n . It seems that it is the first result in the literature which gives a geometric characterization of unbounded domains in n for which the Hartogs phenomenon holds.

Comparing this to earlier work by the first two authors and Z. Słodkowski, one observes that the extension problem changes in character if one restricts to CR functions of higher regularity.

Soit Ω n , n2, un domaine à bord lisse et connexe. Si Ω est relativement compact, le théorème de Hartogs et Bochner assure que toute distribution CR définie sur Ω admet une extension holomorphe à Ω. Cela ne se généralise pas forcément aux domaines non-bornés, par exemple si Ω contient une hypersurface complexe. Le résultat principal de l’article dit que la propriété d’extension holomorphe a lieu si et seulement si l’enveloppe d’holomorphie de n Ω ¯ est n . Il apparaît que cela est le premier résultat connu à donner une caracterisation géometrique des domaines non-bornés pour lesquels le phénomène de Hartogs est valide.

En se référant à des résultats antérieurs des deux premiers auteurs et Z. Słodkowski, on constate que le problème d’extension change de nature si on se restreint à des fonctions CR plus régulières.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3514
Classification: 32D10, 32D2, 32V15, 32V25, 32Q28, 35F45
Keywords: Hartogs–Bochner extension theorem, CR functions, unbounded domains in Stein manifolds, envelopes of holomorphy
Mot clés : Théorème d’extension de Hartogs et Bochner, fonctions CR, sous-domaines non-bornés de variétés de Stein, enveloppes d’holomorphie

Boggess, Al 1; Dwilewicz, Roman 2, 3; Porten, Egmont 4

1 School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287 (USA)
2 Department of Mathematics Missouri University of Science and Technology Rolla, MO 65409 (USA)
3 Faculty of Mathematics Cardinal Stefan Wyszyński University Wóycickiego 1/3, 01-938 Warsaw (Poland)
4 Department of Mathematics and Educational Science Mid-Sweden University 85170 Sundsvall (Sweden)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Boggess, Al; Dwilewicz, Roman; Porten, Egmont. On the Hartogs extension theorem for unbounded domains in $\protect \mathbb{C}^n$. Annales de l'Institut Fourier, Volume 72 (2022) no. 3, pp. 1185-1206. doi : 10.5802/aif.3514. https://aif.centre-mersenne.org/articles/10.5802/aif.3514/

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