Multiscale analysis of 1-rectifiable measures II: characterizations

TL;DR

This paper provides a comprehensive characterization of 1-rectifiable and purely 1-unrectifiable Radon measures in Euclidean spaces using geometric square functions and density conditions, advancing the understanding of measure rectifiability.

Contribution

It introduces new characterizations of 1-rectifiable measures without assuming a relationship with Hausdorff measure, and develops an $L^2$ variant of Jones' traveling salesman theorem.

Findings

Characterization of 1-rectifiable measures via density and geometric square functions.

Development of an $L^2$ version of Jones' traveling salesman construction.

Identification of purely 1-unrectifiable measures with measure zero on finite length curves.

Abstract

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an $L^2$ gauge the extent to which $\mu$ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between $\mu$ and 1-dimensional Hausdorff measure. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an $L^2$ variant of P. Jones' traveling salesman construction, which is of independent interest.