Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets

Abstract

We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolationresults for $L^2$ square function estimates related to the analysis of integraloperators that act on Ahlfors-David regular sets of arbitrary codimension inambient quasi-metric spaces. The inductive scheme is a natural application ofthe local $T(b)$ theorem and it implies the stability of $L^2$ square functionestimates under the so-called big pieces functor. In particular, this analysisimplies $L^p$ and Hardy space square function estimates for integral operatorson uniformly rectifiable subsets of the Euclidean space.