Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality

TL;DR

This paper characterizes uniform rectifiability using new coefficients related to Jones' beta numbers and proves boundedness of certain Calderon-Zygmund operators on uniformly rectifiable measures.

Contribution

It introduces new coefficients to characterize uniform rectifiability and establishes $L^2$ boundedness of odd kernel Calderon-Zygmund operators on such measures.

Findings

Uniform rectifiability characterized by new coefficients.

Boundedness of odd kernel Calderon-Zygmund operators on uniformly rectifiable measures.

Connection between geometric measure theory and singular integral operators.

Abstract

In this paper we study some questions in connection with uniform rectifiability and the $L^2$ boundedness of Calderon-Zygmund operators. We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the Jones' $\beta$ numbers. We also use these new coefficients to prove that n-dimensional Calderon-Zygmund operators with odd kernel of type $C^2$ are bounded in $L^2(\mu)$ if $\mu$ is an n-dimensional uniformly rectifiable measure.