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Mathematische Annalen

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

Multiscale analysis of 1-rectifiable measures: necessary conditions

  • Matthew Badger
  • Raanan Schul
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 

Notes

Acknowledgments

The authors would like to thank Marianna Csörnyei for insightful discussions about this project. The authors would also like to thank an anonymous referee for his or her careful reading of the paper. Part of this work was carried out while both authors visited the Institute for Pure and Applied Mathematics (IPAM), during the Spring 2013 long program on Interactions between Analysis and Geometry.

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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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