Reflectionless measures for Calder\'on-Zygmund Operators I: Basic Theory

TL;DR

This paper develops the foundational theory of reflectionless measures for Calderón-Zygmund operators, exploring their properties and connections to key problems in harmonic analysis and geometric measure theory.

Contribution

It introduces the basic theory of reflectionless measures for Calderón-Zygmund operators and links their properties to classical problems in harmonic analysis and geometric measure theory.

Findings

Reflectionless measures have constant operator action on their support.

The paper establishes foundational properties of these measures.

Connections to classical harmonic analysis problems are identified.

Abstract

We study the properties of reflectionless measures for an $s$-dimensional Calder\'on-Zygmund operator $T$ acting in $\mathbb{R}^d$, where $s\in (0,d)$. Roughly speaking, these are the measures $\mu$ for which $T(\mu)$ is constant on the support of the measure. In this series of papers, we develop the basic theory of reflectionless measures, and describe the relationship between the description of reflectionless measures and certain well-known problems in harmonic analysis and geometric measure theory.