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

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.