Domains of existence for finely holomorphic functions

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.

13 pages 1 figure. This new version has Bent Fuglede as coauthor. We extended the main result to include that regular fine domains are fine domains of existence and corrected many typo's and inaccuracies. In the third version a mistake at the end of the proof of Proposition 2.6 has been corrected