Multiscale Analysis of 1-rectifiable Measures: Necessary Conditions
نویسندگان
چکیده
We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in R, n ≥ 2. To each locally finite Borel measure μ, we associate a function J̃2(μ, x) which uses a weighted sum to record how closely the mass of μ is concentrated near a line in the triples of dyadic cubes containing x. We show that J̃2(μ, ·) < ∞ μ-a.e. is a necessary condition for μ 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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