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

TL;DR

This paper investigates the properties of Bergman spaces of holomorphic functions, demonstrating that generically functions are unbounded, non-extendable, and that certain derivative spaces contain mostly non-extendable functions.

Contribution

It provides new results on the generic behavior of functions in Bergman spaces and their derivatives, especially regarding unboundedness and non-extendability.

Findings

Generic functions in these spaces are totally unbounded.

Most functions are not extendable, even locally.

Derivative spaces contain predominantly non-extendable functions.

Abstract

We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic function is not - even locally - in Bergman spaces of higher order. Finally, in certain domains we consider the space of holomorphic functions whose derivatives up to some order extend continuously to the closure of the domain endowed with its natural topology. Generically, every function in this space is proven to be non - extendable, although bounded.