On rectifiable measures in Carnot groups: Marstrand-Mattila rectifiability criterion

In this paper we continue the study of the notion of $\mathcal{P}$-rectifiability in Carnot groups. We say that a Radon measure is ${\mathcal{P}}_h$-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. In this paper we prove a Marstrand–Mattila rectifiability criterion in arbitrary Carnot groups for $\mathcal{P}$-rectifiable measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if a priori the tangent planes at a point might not be the same at different scales, a posteriori the measure has a unique tangent almost everywhere. Since every horizontal subgroup of a Carnot group has a normal complement, our criterion applies in the particular case in which the tangents are one-dimensional horizontal subgroups. Hence, as an immediate consequence of our Marstrand–Mattila rectifiability criterion and a result of Chousionis–Magnani–Tyson, we obtain the one-dimensional Preiss’s theorem in the first Heisenberg group ${\mathbb{H}}^1$. More precisely, we show that a Radon measure $\varphi$ on ${\mathbb{H}}^1$ with positive and finite one-density with respect to the Koranyi distance is absolutely continuous with respect to the one-dimensional Hausdorff measure ${\mathcal{H}}^1$, and it is supported on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union of the images of Lipschitz maps from $A\subseteq \mathbb{R}$ to ${\mathbb{H}}^1$.