  We establish square function estimates for integral operators on uniformly
rectifiable sets by proving a local $T(b)$ theorem and applying it to show that
such estimates are stable under the so-called big pieces functor. More
generally, we consider integral operators associated with Ahlfors-David regular
sets of arbitrary codimension in ambient quasi-metric spaces. The local $T(b)$
theorem is then used to establish an inductive scheme in which square function
estimates on so-called big pieces of an Ahlfors-David regular set are proved to
be sufficient for square function estimates to hold on the entire set.
Extrapolation results for $L^p$ and Hardy space versions of these estimates are
also established. Moreover, we prove square function estimates for integral
operators associated with variable coefficient kernels, including the Schwartz
kernels of pseudodifferential operators acting between vector bundles on
subdomains with uniformly rectifiable boundaries on manifolds.
