Characterizing regularity of domains via Riesz transforms on their boundaries

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.