Multiscale Analysis of 1-rectifiable Measures II: Characterizations

  title={Multiscale Analysis of 1-rectifiable Measures II: Characterizations},
  author={Matthew Badger and Raanan Schul},
  journal={Analysis and Geometry in Metric Spaces},
  pages={1 - 39}
Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the… 

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