On the Hartogs extension theorem for unbounded domains in n
Annales de l'Institut Fourier, Online first, 22 p.

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:
Online First:
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
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)
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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, Online first, 22 p.

[1] Baouendi, M. Salah; Ebenfelt, Peter; Rothschild, Linda Preiss Real submanifolds in complex space and their mappings, Princeton Mathematical Series, 47, Princeton University Press, 1999

[2] Boggess, Albert CR manifolds and the tangential Cauchy–Riemann complex, Studies in Advanced Mathematics, CRC Press, 1991 | Zbl

[3] Boggess, Albert; Dwilewicz, Roman; Słodkowski, Zbigniew Hartogs phenomenon on unbounded domains – conjectures and examples, Complex analysis and potential theory (CRM Proceedings & Lecture Notes), American Mathematical Society, 2012, pp. 117-134

[4] Boggess, Albert; Dwilewicz, Roman; Słodkowski, Zbigniew Hartogs extension for generalized tubes in n , J. Math. Anal. Appl., Volume 402 (2013) no. 2, pp. 574-578

[5] Boggess, Albert; Dwilewicz, Roman; Słodkowski, Zbigniew Hartogs-type extension for tube-like domains in 2 , Math. Ann., Volume 363 (2015) no. 1-2, pp. 35-60

[6] Boggess, Albert; Dwilewicz, Roman; Słodkowski, Zbigniew Hartogs-type extension for unbounded sets in 2 via construction of Riemann domains, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (2017) no. 1, pp. 93-111

[7] Chirka, Evgeni M. Analytic representation of CR functions, Math. USSR, Sb., Volume 27 (1975), pp. 526-553

[8] Colţoiu, Mihnea; Ruppenthal, Jean On Hartogs’ extension theorem on (n-1)-complete complex spaces, J. Reine Angew. Math., Volume 637 (2009), pp. 41-47

[9] Damiano, Alberto; Struppa, Daniele; Vajiac, Adrian; Vajiac, Mihaela Hartogs phenomena and antisyzygies for systems of differential equations, J. Geom. Anal., Volume 19 (2009) no. 2, pp. 288-300

[10] Docquier, Ferdinand; Grauert, Hans Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann., Volume 140 (1960), pp. 94-123

[11] Dwilewicz, Roman; Merker, Joël On the Hartogs–Bochner phenomenon for CR functions in P 2 (), Proc. Am. Math. Soc., Volume 130 (2002) no. 7, pp. 1975-1980

[12] Grauert, Hans; Remmert, Reinhold Konvexität in der komplexen Analysis, Comment. Math. Helv., Volume 31 (1956/57), pp. 152-183

[13] Guillemin, Victor; Pollack, Alan Differential topology, Prentice Hall, 1974

[14] Gunning, Robert C.; Rossi, Hugo Analytic functions of several complex variables, Prentice-Hall Series in Modern Analysis, Prentice Hall, 1965

[15] Harz, Tobias; Shcherbina, Nikolay; Tomassini, Giuseppe Wermer type sets and extension of CR functions, Indiana Univ. Math. J., Volume 61 (2012) no. 1, pp. 431-459

[16] Harz, Tobias; Shcherbina, Nikolay; Tomassini, Giuseppe On defining functions for unbounded pseudoconvex domains. I., Math. Z., Volume 286 (2017) no. 3-4, pp. 987-1002

[17] Hörmander, Lars The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften, 256, Springer, 1983

[18] Jarnicki, Marek; Pflug, Peter Extension of holomorphic functions, De Gruyter Expositions in Mathematics, 34, Walter de Gruyter, 2000

[19] Kerner, Hans Überlagerungen und Holomorphichüllen, Math. Ann., Volume 144 (1961), pp. 126-134

[20] Laurent-Thiébaut, Christine Holomorphic function theory in several variables. An introduction, Universitext, Springer, 2011

[21] Lewandowski, Arkadiusz A new Hartogs-type extension result for the cross-like objects, Kyushu J. Math., Volume 69 (2015) no. 1, pp. 77-94 | Zbl

[22] Lupacciolu, Guido Valeurs au bord de fonctions holomorphes dans des domaines non bornés de n , C. R. Math. Acad. Sci. Paris, Volume 304 (1987), pp. 67-69

[23] Merker, Joël; Porten, Egmont Characteristic foliations on maximally real submanifolds of n and removable singularities for CR functions, IMRP, Int. Math. Res. Pap., Volume 2006 (2006) no. 15, 72069, 131 pages

[24] Merker, Joël; Porten, Egmont Holomorphic extension of CR functions, envelopes of holomorphy, and removable singularities, IMRS, Int. Math. Res. Surv., Volume 2006 (2006) no. 1, 28925, 287 pages

[25] Merker, Joël; Porten, Egmont The Hartogs extension theorem on (n-1)-complete complex spaces, J. Reine Angew. Math., Volume 637 (2009), pp. 23-39

[26] Nacinovich, Mauro; Porten, Egmont 𝒞 -hypoellipticity and extension of CR functions, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 14 (2015) no. 3, pp. 677-703

[27] Ohsawa, Takeo Hartogs type extension theorems on some domains in Kähler manifolds, Ann. Pol. Math., Volume 106 (2012), pp. 243-254

[28] Øvrelid, Nils; Vassiliadou, Sophia Hartogs extension theorems on Stein spaces, J. Geom. Anal., Volume 20 (2010) no. 4, pp. 817-836

[29] Palamodov, Victor P. Hartogs phenomenon for systems of differential equations, J. Geom. Anal., Volume 24 (2014) no. 2, pp. 667-686

[30] Porten, Egmont On the Hartogs-phenomenon and extension of analytic hypersurfaces in non-separated Riemann domains, Complex Variables, Theory Appl., Volume 47 (2002) no. 4, pp. 325-332

[31] Porten, Egmont The Hartogs phenomenon on weakly pseudoconcave hypersurfaces, Math. Ann., Volume 354 (2012) no. 2, pp. 659-683

[32] Sarkis, Frédéric Hartogs–Bochner type theorem in projective space, Ark. Mat., Volume 41 (2003), pp. 151-163

[33] Stein, Karl Maximale holomorphe und meromorphe Abbildungen, Am. J. Math., Volume 85 (1963), pp. 298-315

[34] Sussmann, Hector J. Orbits of families of vector fields and integrability of distributions, Trans. Am. Math. Soc., Volume 180 (1973), pp. 171-188

[35] Trépreau, Jean-Marie Sur le prolongement holomorphe des fonctions CR définies sur une hypersurface réelle de classe C 2 dans n , Invent. Math., Volume 83 (1986), pp. 583-592

[36] Trèves, François Hypo-Analytic Structures. Local Theory, Princeton Mathematical Series, 40, Princeton University Press, 1992

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