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Annales mathématiques du Québec

, Volume 42, Issue 1, pp 101–105 | Cite as

Domains of holomorphy

  • V. NestoridisEmail author
Article

Abstract

We give a simple proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire’s Category Theorey and Montel’s Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of functions \(X=X({\varOmega })\subset H({\varOmega })\). We show that the set of non-extendable functions not only contains a \(G_{\delta }\)-dense subset of \(X({\varOmega })\), but it is itself a \(G_{\delta }\)-dense set. We give an example of a domain in \(\mathbb {C}\) which is a \(H({\varOmega })\)-domain of holomorphy but not a \(A({\varOmega })\)-domain of holomorphy.

Keywords

Extendability Domain of holomorphy Weak domain of holomorphy Baire’s theorem Generic property Montel’s theorem Analytic continuation 

Résumé

Nous donnons une preuve simple que les notions de Domaine d’Holomorphie et Domaine Faible d’Holomorphie sont équivalentes. Cette preuve est fondée sur une combinaison du Théorème de Baire et du Théorème de Montel. Nous obtenons aussi des généralisations en demandant que les fonctions non-prolongeables appartiennent à une classe particulière de fonctions \(X = X(\Omega ) \subset H(\Omega )\). Nous montrons que l’ensemble des fonctions non-prolongeables non-seulement contient un sousensemble \(G_{\delta }\) demse de \(X(\Omega )\), mais est lui-même un ensemble \(G_{\delta }\). Nous donnons un exemple d’un domaine de \(\mathbb {C}\) qui est un \(H(\Omega )\)-domaine d’holomorphie mais pas un \(A(\Omega )\)-domaine d’holomorphie.

Mathematics Subject Classification

Primary 32T05 Secondary 30D45 

Notes

Acknowledgements

I would like to thank R. Aron, B. Braga, S. Charpentier, P. M. Gauthier, T. Hatziafratis, M. Maestro, P. Pflug and A. Siskakis for helpful communications.

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Copyright information

© Fondation Carl-Herz and Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece

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