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

TL;DR

This paper characterizes Lyapunov domains using Riesz transforms and a generalized Plemelj-Privalov theorem, linking geometric regularity to singular integral operator behavior.

Contribution

It establishes new equivalences between domain regularity and the behavior of Riesz transforms and singular integrals on H"older spaces.

Findings

Modulus of continuity of the outward normal is equivalent to Riesz transform behavior.

Generalization of Plemelj-Privalov theorem to higher dimensions.

Boundedness of singular integral operators on generalized H"older spaces.

Abstract

We prove several characterizations of $\mathscr{C}^{1,\omega}$-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 H\"older spaces. This class includes the Cauchy-Clifford operator and the harmonic double layer operator, among others.