Calculus of Variations and Geometric Measure Theory

G. Antonelli - A. Merlo

On rectifiable measures in Carnot groups: existence of density

created by merlo on 30 Sep 2020
modified by antonelli on 11 Jul 2022


Accepted Paper

Inserted: 30 sep 2020
Last Updated: 11 jul 2022

Journal: The Journal of Geometric Analysis
Year: 2020

ArXiv: 2009.13941 PDF

The present work consists of an elaboration of Sections 2, 3, 4, and 6 of the Preprint "On rectifiable measures in Carnot groups: structure theory". This is the first of two companion papers derived from "On rectifiable measures in Carnot groups: structure theory" that can be found as version 2 of the submission in The second companion paper is in


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 $\mathscr{P}_h$-rectifiable measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of $\mathscr{P}_h$-rectifiable 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.

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