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Multiscale analysis of 1-rectifiable measures: necessary conditions

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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.

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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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Correspondence to Matthew Badger.

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M. Badger was partially supported by an NSF postdoctoral fellowship DMS 12-03497. R. Schul was partially supported by a fellowship from the Alfred P. Sloan Foundation as well as by NSF DMS 11-00008.

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Badger, M., Schul, R. Multiscale analysis of 1-rectifiable measures: necessary conditions. Math. Ann. 361, 1055–1072 (2015).

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