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

$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets