Reflectionless measures for Calder\'{o}n-Zygmund operators II: Wolff potentials and rectifiability

TL;DR

This paper investigates the geometric implications of reflectionless measures for certain Calderón-Zygmund operators, showing that rigidity conditions lead to strong geometric constraints on measures where the operators are bounded.

Contribution

It establishes a connection between the rigidity of Calderón-Zygmund operators and geometric properties of measures, advancing understanding in harmonic analysis and geometric measure theory.

Findings

Rigidity of CZOs implies strong geometric conditions on measure supports.

Reduces problems in harmonic analysis to the study of reflectionless measures.

Provides new results linking operator properties to geometric measure theory.

Abstract

We continue our study of the reflectionless measures associated to an $s$-dimensional Calder\'{o}n-Zygmund operator (CZO) acting in $\mathbb{R}^d$ with $s\in (0,d)$. Here, our focus will be the study of CZOs that are rigid, in the sense that they have few reflectionless measures associated to them. Our goal is to prove that the rigidity properties of a CZO $T$ impose strong geometric conditions upon the support of any measure $\mu$ for which $T$ is a bounded operator in $L^2(\mu)$. In this way, we shall reduce certain well-known problems at the interface of harmonic analysis and geometric measure theory to a description of reflectionless measures of singular integral operators. What's more, we show that this approach yields promising new results.