# Domains of existence for finely holomorphic functions

## TL;DR

This paper characterizes certain fine domains in the complex plane as domains of existence for finely holomorphic functions, highlighting conditions under which they are or are not such domains.

## Contribution

It establishes criteria for when fine domains are domains of existence for finely holomorphic functions, including Euclidean $F_\sigma$ and $G_\delta$ properties and regularity.

## Key findings

- Euclidean $F_\sigma$ and $G_\delta$ fine domains are domains of existence.
- Regular fine domains are also domains of existence.
- Certain fine domains with polar set complements are not domains of existence.

## Abstract

We show that fine domains in $\mathbf{C}$ with the property that they are Euclidean $F_\sigma$ and $G_\delta$, are in fact fine domains of existence for finely holomorphic functions. Moreover \emph{regular} fine domains are also fine domains of existence. Next we show that fine domains such as $\mathbf{C}\setminus \mathbf{Q}$ or $\mathbf{C}\setminus (\mathbf{Q}\times i\mathbf{Q})$, more specifically fine domains $V$ with the properties that their complement contains a non-empty polar set $E$ that is of the first Baire category in its Euclidean closure $K$ and that $(K\setminus E)\subset V$, are NOT fine domains of existence.