Non-extendability of holomorphic functions with bounded or continuously extendable derivatives

Abstract

We consider the spaces $H_F^{\infty }(\Omega )$ and ${\mathcal{A}}_F(\Omega )$ containing all holomorphic functions $f$ on an open set $\Omega \subseteq \mathbb{C}$, such that all derivatives $f^{(l)}$, $l\in F\subseteq {\mathbb{N}}_0=\{0,1,...\}$, are bounded on $\Omega$, or continuously extendable on $\overline{\Omega }$, respectively. We endow these spaces with their natural topologies and they become Fréchet spaces. We prove that the set $S$ of non-extendable functions in each of these spaces is either void, or dense and $G_{\delta }$. We give examples where $S=\varnothing$ or not. Furthermore, we examine cases where $F$ can be replaced by $\tilde{F}=\{l\in {\mathbb{N}}_0:\min F⩽l⩽\sup F\}$, or ${\tilde{F}}_0=\{l\in {\mathbb{N}}_0:0⩽l⩽\sup F\}$ and the corresponding spaces stay unchanged.