Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform
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130
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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 .Â
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Xavier Tolsa: ICREA, Barcelona, Spain and Universitat AutÃ˛noma de Barcelona, Spain
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