Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform

Download or read book Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform written by Xavier Tolsa and published by American Mathematical Soc.. This book was released on 2017-01-18 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 ifH^1x2EThe 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 .