The Usual Square Function on Weakly Flat Sets

We study the usual square function estimate associated with the Cauchy single-layer kernel in the plane, without assuming Ahlfors-David regularity. We prove that a finite Radon measure with positive and finite upper density is rectifiable if it satisfies the usual square function estimate and a weak flatness condition. We also prove that, under the same finiteness and density hypotheses, the weak flatness condition follows when the support is contained in a locally two-sided NTA curve. As a corollary, rectifiability follows when the support is contained in a quasicircle.