On Bergman type spaces of holomorphic functions and the density, in these spaces, of certain classes of singular functions

Abstract

We consider Bergman spaces and variations of them on domains in one or several complex variables. For certain domains , we show that the generic function in these spaces is totally unbounded in and hence non-extendable. We also show that generically these functions do not belong – not even locally – to Bergman spaces of higher order. Finally, in certain domains , we give examples of bounded non-extendable holomorphic functions – a generic result in the spaces of holomorphic functions in whose derivatives of order extend continuously to .

Keywords:

• Domain of holomorphy
• Bergman spaces
• Baire’s theorem
• generic property
• strictly pseudoconvex domains

AMS Subject Classifications:

• Primary: 30H20
• 32A36
• Secondary: 32D05
• 32T05
• 54E52

Acknowledgements

We would like to thank M. Fragoulopoulou and A. Siskakis for helpful communications.

Notes

No potential conflict of interest was reported by the authors.