One sided extendability and p-continuous analytic capacities

TL;DR

This paper investigates the rarity of one-sided extendability and real analyticity in function spaces using complex analysis and Baire's Theorem, and introduces p-continuous analytic capacities to characterize singularity removability.

Contribution

It introduces p-continuous analytic capacities for compact sets and uses them to characterize the removability of singularities in function spaces.

Findings

One-sided extendability and real analyticity are rare phenomena.

p-continuous capacities effectively characterize singularity removability.

The methods combine complex analysis with Baire's Theorem.

Abstract

Using complex methods combined with Baire's Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to introduce the p-continuous analytic capacity and variants of it, $p \in \{ 0, 1, 2, \cdots \} \cup \{ \infty \}$, for compact or closed sets in $\mathbb{C}$. We use these capacities in order to characterize the removability of singularities of functions in the spaces $A^p$.