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

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.