On the Hartogs extension theorem for unbounded domains in $\protect \mathbb{C}^n$

Boggess, Al; Dwilewicz, Roman; Porten, Egmont

doi:10.5802/aif.3514

On the Hartogs extension theorem for unbounded domains in

ℂ^{n}

[Sur le théorème d’extension de Hartogs pour des domaines non bornés de

ℂ^{n}

]

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)

Annales de l'Institut Fourier, Tome 72 (2022) no. 3, pp. 1185-1206.

Résumé
Abstract

Soit $Ω \subset ℂ^{n}$ , $n \geq 2$ , 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 $\partial Ω$ 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 $ℂ^{n} ∖ \bar{Ω}$ est $ℂ^{n}$ . 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.

Let $Ω \subset ℂ^{n}$ , $n \geq 2$ , be a domain with smooth connected boundary. If $Ω$ is relatively compact, the Hartogs–Bochner theorem ensures that every CR distribution on $\partial Ω$ 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 $ℂ^{n} ∖ \bar{Ω}$ is $ℂ^{n}$ . It seems that it is the first result in the literature which gives a geometric characterization of unbounded domains in $ℂ^{n}$ 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.

Reçu le : 2017-12-14
Révisé le : 2019-02-17
Accepté le : 2019-04-04
Publié le : 2022-09-12

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

Affiliations des auteurs :

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Annales de l'Institut Fourier},
     pages = {1185--1206},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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, Tome 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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