Rectifiable measures, square functions involving densities, and the Cauchy transform - Tolsa, Xavier (Memoirs of the American Mathematical Society, 2017)
"This monograph is devoted to the proof of two related results. The first one asserts that if is a Radon measure in satisfyingfor -a.e., then is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set with finite -dimensional Hausdorff measure is rectifiable if and only if Ĥ1x2E The second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform . Assuming that has linear growth, it is proved that is bounded in if and only iffor every square ."--Publisher website.