Multiscale analysis of 1-rectifiable measures: necessary conditions

TL;DR

This paper establishes a necessary condition for a measure in Euclidean space to be supported on rectifiable curves, using a new multiscale function that captures the measure's concentration on lines, confirming a long-standing conjecture.

Contribution

It introduces a novel multiscale function $ ilde J_2$ to characterize 1-rectifiability without density assumptions, confirming a conjecture by Peter Jones from 2000.

Findings

$ ilde J_2( u, x) < ty$ $ u$-a.e. for measures supported on rectifiable curves

The result applies to measures singular with respect to Hausdorff measure

Confirms a necessary condition for rectifiability in a broad setting

Abstract

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\Bbb{R}^n$, $n\geq 2$. 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 on a line in the triples of dyadic cubes containing $x$. We show that $\widetilde J_2(\mu, x) < \infty$ $\mu$-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 generic 1-rectifiable measures that are mutually singular with the 1-dimensional Hausdorff measure.