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

TL;DR

This paper investigates the extendability of holomorphic functions with bounded or extendable derivatives, showing that non-extendable functions are either nonexistent or form a dense, large subset in certain function spaces.

Contribution

It establishes conditions under which the set of non-extendable functions in these spaces is either empty or dense and $G_\delta$, and explores the effects of modifying derivative index sets.

Findings

The set of non-extendable functions is either void or dense and $G_\delta$.

Examples demonstrate cases where this set is empty or non-empty.

The structure of the derivative index set influences the properties of the function spaces.

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 Fr\'echet 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 $\widetilde{F}=\{ l\in \mathbb{N}_0:\min F \leqslant l \leqslant \sup F\}$, or $\widetilde{F}_0= \{ l\in \mathbb{N}_0:0\leqslant l \leqslant \sup F\}$ and the corresponding spaces stay unchanged.