Rectifiable measures, square functions involving densities, and the Cauchy transform

© 2016 American Mathematical Society. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. This monograph is devoted to the proof of two related results. The first one asserts that if μ is a Radon measure in ℝd satisfying (Equation presented) for μ-a.e. x ∈ Rd, then μ is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set E ⊂ ℝd with finite 1-dimensional Hausdorff measure H1 is rectifiable if and only if (Equation presented) The second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform Cμf(z) = ∫ 1/z-ξf(ξ)dμ(ξ). Assuming that μ has linear growth, it is proved that Cμ is bounded in L2(μ) if and only if (Equation presented) for every square Q ⊂ C.