Square Function Estimates in Spaces of Homogeneous Type and on Uniformly Rectifiable Euclidean Sets

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