On rectifiable measures in Carnot groups: existence of density

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.