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

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.