Reflectionless measures and the Mattila-Melnikov-Verdera uniform rectifiability theorem

TL;DR

This paper presents a new proof of a theorem linking the uniform rectifiability of certain measures to the boundedness of their associated Cauchy transform, advancing understanding in geometric measure theory.

Contribution

It provides a novel proof of the Mattila-Melnikov-Verdera theorem, connecting reflectionless measures with uniform rectifiability.

Findings

Established a new proof of the Mattila-Melnikov-Verdera theorem.

Linked reflectionless measures to uniform rectifiability.

Enhanced theoretical understanding of measure rectifiability.

Abstract

A new proof is given of the Mattila-Melnikov-Verdera theorem on the uniform rectifiability of an Ahlfors-David regular measure whose associated Cauchy transform operator is bounded.