2022/02/19 by Antonelli, Gioacchino, Merlo, Andrea
#22E25 #26A16 #28A75 #49Q15 #53C17 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2202.12741
In this paper we continue the study of the notion of \mathscrP-rectifiability in Carnot groups. We say that a Radon measure is \mathscrPh-rectifiable, for h∈\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 \mathscrP-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 H1. More precisely, we show that a Radon measure ϕ on \mathbb H1 with positive and finite one-density with respect to the Koranyi distance is absolutely continuous with respect to the one-dimensional Hausdorff measure H1, 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⊆ \mathbb R to \mathbb H1.