Characterizing regularity of domains via Riesz transforms on their boundaries

TL;DR

This paper establishes a connection between the boundedness of Riesz transforms on boundary H"older spaces and the geometric regularity of domains, specifically characterizing Lyapunov domains via Riesz transform properties.

Contribution

It proves that Riesz transform boundedness on boundary H"older spaces characterizes Lyapunov domains and explores related conditions involving higher order transforms and special domain classes.

Findings

Boundedness of principal value Riesz transforms characterizes Lyapunov domains.

Higher order Riesz transforms are bounded on boundary H"older spaces for Lyapunov domains.

Results extend to VMO and Semmes-Kenig-Toro domains in limiting cases.

Abstract

Given a domain D in R^d with mild geometric measure theoretic assumptions on its boundary, we show that boundedness of the principal value Riesz tranforms (witn kernel of homogeneity -(d-1)) on H\"older spaces of order alpha on the boundary of D is equivalent to D being a Lyapunov domain of order alpha (i.e., the boundary of D is an hypersurface of class 1+alpha). Another equivalent condition involving Riesz transforms on D is discussed. We also prove that on Lyapunov domains of order alpha the higher order Riesz transforms associated with an odd polynomial are bounded on the H\"older space of order alpha on the boundary of D. Finally, a limiting case of the above results dealing with VMO and Semmes-Kenig-Toro domains is considered.