Localized versions of function spaces and generic results

TL;DR

This paper introduces localized function spaces focusing on boundary approach properties, demonstrating generic non-extendability and nowhere differentiability within these spaces, which are modeled as Fréchet spaces.

Contribution

It generalizes classical function spaces by considering boundary approach restrictions and establishes generic non-extendability and differentiability results.

Findings

Generic non-extendability of functions in localized spaces

Generic nowhere differentiability on specified boundary parts

Spaces are structured as Fréchet spaces

Abstract

We consider generalizations of classical function spaces by requiring that a holomorphic in ${\Omega}$ function satisfies some property when we approach from ${\Omega}$, not the whole boundary, but only a part of it. These spaces endowed with their natural topology are Fr$\'{e}$chet spaces. We prove some generic non-extendability results in such spaces and generic nowhere differentiability on the corresponding part of ${\partial{\Omega}}$.