A proof of the local $Tb$ Theorem for standard Calder\'{o}n-Zygmund operators

Abstract

We give a proof of a so-called "local $Tb$" Theorem for singular integrals whose kernels satisfy the standard Calder\'on-Zygmund conditions. The present theorem, which extends an earlier result of M. Christ \cite{Ch}, was proved in \cite{AHMTT} for "perfect dyadic" Calder\'on-Zygmund operators. The proof in \cite{AHMTT} essentially carries over to the case considered here, with some technical adjustments.