Multiscale analysis of 1-rectifiable measures: necessary conditions

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 $\hat{I}¼$, we associate a function ${\tilde{J}}_2(\hat{I}¼,x)$ which uses a weighted sum to record how closely the mass of $\hat{I}¼$ is concentrated on a line in the triples of dyadic cubes containing $x$. We show that ${\tilde{J}}_2(\hat{I}¼,x)<\infty$ $\hat{I}¼$-a.e. is a necessary condition for $\hat{I}¼$ 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 generic 1-rectifiable measures that are mutually singular with the 1-dimensional Hausdorff measure.

16 pages, v2: expanded introduction, improved Lemma 2.6 (now 2.7), corrected mistake in proof of Proposition 3.1