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MULTISCALE ANALYSIS OF 1RECTIFIABLE MEASURES: NECESSARY CONDITIONS
"... Abstract. We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in Rn, n ≥ 2. To each locally finite Borel measure µ, we associate a function eJ2(µ, x) which uses a weighted sum to record how closely the mass of µ is concentrated near a ..."
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Abstract. We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in Rn, n ≥ 2. To each locally finite Borel measure µ, we associate a function eJ2(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 thateJ2(µ, ·) < ∞ µa.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 1rectifiable measures, including measures which are singular with respect to 1dimensional Hausdorff measure. 1.
TWO SUFFICIENT CONDITIONS FOR RECTIFIABLE MEASURES
"... Abstract. We identify two sufficient conditions for locally finite Borel measures on Rn to give full mass to a countable family of Lipschitz images of Rm. The first condition, extending a prior result of Pajot, is a sufficient test in terms of Lp affine approximability for a locally finite Borel mea ..."
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Abstract. We identify two sufficient conditions for locally finite Borel measures on Rn to give full mass to a countable family of Lipschitz images of Rm. The first condition, extending a prior result of Pajot, is a sufficient test in terms of Lp affine approximability for a locally finite Borel measure µ on Rn satisfying the global regularity hypothesis lim sup r↓0 µ(B(x, r))/rm < ∞ at µa.e. x ∈ Rn to be mrectifiable in the sense above. The second condition is an assumption on the growth rate of the 1density that ensures a locally finite Borel measure µ on Rn with lim r↓0 µ(B(x, r))/r = ∞ at µa.e. x ∈ Rn is 1rectifiable. 1.