Mathematische Annalen

, Volume 361, Issue 3–4, pp 1055–1072 | Cite as

Multiscale analysis of 1-rectifiable measures: necessary conditions

Article

Abstract

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in \(\mathbb {R}^n\), \(n\ge 2\). To each locally finite Borel measure \(\mu \), we associate a function \(\widetilde{J}_2(\mu ,x)\) which uses a weighted sum to record how closely the mass of \(\mu \) is concentrated near a line in the triples of dyadic cubes containing \(x\). We show that \(\widetilde{J}_2(\mu ,\cdot )<\infty \ \mu \)-a.e. is a necessary condition for \(\mu \) to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze general 1-rectifiable measures, including measures which are singular with respect to 1-dimensional Hausdorff measure.

Mathematics Subject Classification

28A75 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA
  2. 2.Department of MathematicsStony Brook UniversityStony BrookUSA

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