Regularity of Characteristic Functions in Besov-Type and Triebel--Lizorkin-Type Spaces

TL;DR

This paper investigates the precise regularity conditions for characteristic functions in Besov-type and Triebel--Lizorkin-type spaces, focusing on domains with complex boundaries like snowflakes and spirals.

Contribution

It provides necessary and sufficient conditions for characteristic functions to belong to these function spaces, considering boundary neighborhood measures.

Findings

Characterizes regularity of characteristic functions in complex domains

Establishes optimal conditions for membership in Besov and Triebel--Lizorkin spaces

Analyzes domains with fractal boundaries like snowflakes and spirals

Abstract

In this article, the authors determine the optimal regularity of characteristic functions in Besov-type and Triebel--Lizorkin-type spaces under restrictions on the measure of the $\delta$-neighborhoods of the boundary. In particular, the necessary and sufficient conditions for the membership in these spaces of characteristic functions of the snowflake domain and also some spiral type domains are obtained.