Identifying 1-rectifiable measures in Carnot groups

TL;DR

This paper extends geometric measure theory tools to Carnot groups, providing tests to distinguish measures supported on rectifiable curves from singular parts, generalizing the Analyst's Traveling Salesman Theorem.

Contribution

It develops new criteria for identifying rectifiable measures in Carnot groups, extending classical theorems to a broader non-Euclidean setting.

Findings

Established tests for rectifiable measures in Carnot groups.

Extended the Analyst's Traveling Salesman Theorem to Carnot groups.

Constructed doubling measures charging rectifiable curves in metric spaces.

Abstract

We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in $\mathbb{R}^2$ (P. Jones, 1990), in $\mathbb{R}^n$ (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' $\beta$-numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in $\mathbb{R}^n$ that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space.