On the Regularity of Characteristic Functions of Weakly Exterior Thick Domains

Let $E$ be a domain in $\mathbb R^d$. We investigate the regularity of the characteristic function $\mathcal X_E$ depending on the behavior of the $\delta $-neighborhoods of the boundary of $E$. The regularity is measured in terms of the Nikol'skii–Besov and Lizorkin–Triebel spaces.