Reflectionless measures for Calder\'{o}n-Zygmund operators

TL;DR

This paper investigates reflectionless measures for Calderón-Zygmund operators, linking their properties to harmonic analysis and geometric measure theory, and provides a new proof of a theorem on the unboundedness of Riesz transforms.

Contribution

It introduces a new framework for understanding reflectionless measures and offers a novel proof of a key theorem regarding Riesz transforms in certain dimensions.

Findings

Reflectionless measures have specific properties related to Calderón-Zygmund operators.

The classification of these measures connects to well-known problems in harmonic analysis.

A new proof is provided for the unboundedness of Riesz transforms for certain measures.

Abstract

We study the properties of reflectionless measures for a Calder\'{o}n-Zygmund operator T. Roughly speaking, these are measures $\mu$ for which T(\mu) vanishes (in a weak sense) on the support of the measure. We describe the relationship between certain well-known problems in harmonic analysis and geometric measure theory and the classification of reflectionless measures. As an application of our theory, we give a new proof of a recent theorem of Eiderman, Nazarov, and Volberg, which states that in $\mathbb{R}^d$, the s-dimensional Riesz transform of a non-trivial $s$-dimensional measure is unbounded if $s\in (d-1,d)$.