Characterizations of Lyapunov domains in terms of Riesz transforms and the Plemelj-Privalov theorem

Abstract

We prove several characterizations of ${\mathcal{C}}^{1,\omega }$-domains (aka Lyapunov domains), where $\omega$ is a growth function satisfying natural assumptions. For example, given an Ahlfors regular domain $\Omega \subseteq {\mathbb{R}}^n$, we show that the modulus of continuity of the geometric measure theoretic outward unit normal $\nu$ to $\Omega$ is dominated by (a multiple of) $\omega$ if and only if the action of each Riesz transform $R_j$ associated with $\partial \Omega$ on the constant function $1$ has a modulus of continuity dominated by (a multiple of) $\omega$. The proof of this result requires that we establish a higher-dimensional generalization of the classical Plemelj-Privalov theorem, identifying a large class of singular integral operators that are bounded on generalized Hölder spaces. This class includes the Cauchy-Clifford operator and the harmonic double layer operator, among others.