  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.
