Holomorphic extendability in $\mathbf C^n$ as a rare phenomenon

TL;DR

This paper investigates the rarity of holomorphic extendability phenomena in complex functions on subsets of ^n, demonstrating that such extendability is topologically uncommon in relevant function spaces for dimensions n  2 and higher.

Contribution

It generalizes previous results by showing holomorphic extendability is a rare phenomenon in higher dimensions within various function spaces.

Findings

Holomorphic extendability is topologically rare in ^n for n  2.

Generalization of earlier results to higher dimensions.

Includes analysis of one-sided extendability phenomena.

Abstract

We consider various notions of holomorphic extendability of complex valued functions defined on subsets of $\mathbf C^n$, including one-sided extendability. We show that in the relevant function spaces, these phenomena of holomorphic extendability are rare in the topological sense, generalizing several results of the article "One sided extendability and $p$-continuous analytic capacities" by E. Bolkas, V. Nestoridis, C. Panagiotis and M. Papadimitrakis, in dimensions $n\ge 2$.