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From universality to generic non-extendability and total unboundedness in spaces of holomorphic functions

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Abstract

It is well known that the notions of domain of holomorphy and weak domain of holomorphy are equivalent. If \(X({\varOmega })\) is a space of holomorphic functions we extend these notions to \(X({\varOmega })\)-domain of holomorphy and weak \(X({\varOmega })\)-domain of holomorphy. For several function spaces \(X(\varOmega )\), satisfying weak assumptions, we prove that the notions of \(X(\varOmega )\)-domain of holomorphy and weak \(X(\varOmega )\)-domain of holomorphy are equivalent and that in this case the set of non-extendable functions in \(X({\varOmega })\) is a dense \(G_\delta \)-subset of \(X({\varOmega })\). Similar results are obtained for the stronger notion of total unboundedness. Finally we provide examples of new spaces \(X({\varOmega })\), where all the above hold. Mainly they are localized versions of classical function spaces and combinations of them.

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Acknowledgements

The author would like to thank the anonymous referees for improving the presentation of the paper and the exposition.

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  1. Department of Mathematics, National and Kapodistrian University of Athens, Panepistemiopolis, 157 84, Athens, Greece

    Vassili Nestoridis

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  1. Vassili Nestoridis
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Correspondence to Vassili Nestoridis.

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Dedicated to Professor Stephen Gardiner on the occasion of his 60th birthday.

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Nestoridis, V. From universality to generic non-extendability and total unboundedness in spaces of holomorphic functions. Anal.Math.Phys. 9, 887–897 (2019). https://doi.org/10.1007/s13324-019-00315-9

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  • DOI: https://doi.org/10.1007/s13324-019-00315-9

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