Reflectionless measures for Calderón-Zygmund Operators I: Basic Theory

Abstract

We study the properties of reflectionless measures for an $s$-dimensional Calderón-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.

41 pages. This paper is the first of a three part series that expands (and will replace) the previous submission arXiv:1309.6661

References in corpus (1)

• On the uniform rectifiability of AD regular measures with bounded Riesz transform operator: the case of codimension 1