On rectifiable measures in Carnot groups: existence of density

TL;DR

This paper introduces a new notion of rectifiability for measures in Carnot groups, compares it with existing notions, and establishes structural properties and density results for these measures.

Contribution

It defines $ ext{P}_h$-rectifiability in Carnot groups, compares it with prior notions, and proves key structural and density properties, including support on Lipschitz graphs.

Findings

$ ext{P}_h$-rectifiability is strictly weaker than existing notions.

Supports are covered by cone-like sets and Lipschitz graphs.

Measures have positive and finite $h$-density almost everywhere.

Abstract

In this paper we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is $\mathscr{P}_h$-rectifiable, for $h\in\mathbb N$, if it has positive $h$-lower density and finite $h$-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. First, we compare $\mathscr{P}_h$-rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups, and we prove that it is strictly weaker than them. Second, we prove several structure properties of $\mathscr{P}_h$-rectifiable measures. Namely, we prove that the support of a $\mathscr P_h$-rectifiabile measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of $\mathscr P_h$-rectifiabile measures with complemented tangents, we show that they are supported on the union of intrinsically Lipschitz and differentiable graphs. Such a covering property is used to prove the main result of this paper: we show that a $\mathscr{P}_h$-rectifiable measure has almost everywhere positive and finite $h$-density whenever the tangents admit at least one complementary subgroup.