Variation for the Riesz transform and uniform rectifiability

TL;DR

This paper establishes that for certain measures in Euclidean space, the boundedness of the Riesz transform's r-variation characterizes uniform rectifiability, advancing understanding of geometric measure theory.

Contribution

It proves a new equivalence linking the boundedness of the Riesz transform's r-variation to uniform rectifiability of measures, partially resolving a longstanding open problem.

Findings

R-variation boundedness characterizes uniform rectifiability

Provides a partial solution to a problem posed by David and Semmes

Advances the understanding of the relationship between singular integrals and geometric measure theory

Abstract

For 0<n<d integers and r>2, we prove that an n-dimensional Ahlfors-David regular measure M in R^d is uniformly n-rectifiable if and only if the r-variation for the Riesz transform with respect to M is a bounded operator in L^2(M). This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L^2(M) boundedness of the Riesz transform to the uniform rectifiability of M.