A $Tb$ type theorem for suppressed kernels

Abstract

In this article, a non-homogeneous $Tb$ type theorem for arbitrary dimensional Calderón-Zygmund singular integral operators is proved. This is an extension of an analogous non-homogeneous $Tb$ theorem for the Cauchy transform, in the planar setting, due to Nazarov, Treil and Volberg. The novelties of the present work are the change of dimension and the fact that the operators to which the theorem applies are not necessarily antisymmetric. The techniques used in the proof include, among others, suppressed kernels, decompositions in $L^2(μ)$, where $μ$ is a Radon measure in $\mathbb{R}^d$, and a probabilistic argument resulting from taking averages of the operators involved.