On the regularity of characteristic functions

Let $E \subset \mathbb{R}$ be a domain such that $0 < |E| < \infty$. In my talk I will discuss the regularity of the associated characteristic function in the framework of Besov and Lizorkin–Triebel spaces. The main goal will be a description of if and only if — conditions given in terms of the volume of the $\delta$ neighborhoods of the boundary of $E$ for a certain class of domains.