On rectifiable measures in Carnot groups: structure theory

In this paper we prove the one-dimensional Preiss theorem in the first Heisenberg group $mathbb H^1$. More precisely we show that a Radon measure $phi$ on $mathbb H^1$ with positive and finite one-density with respect to the Koranyi distance 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 $Asubseteq mathbb Rtomathbb H^1$. The previous theorem is a consequence of a Marstrand-Mattila type rectifiability criterion, which we prove in arbitrary Carnot groups for measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if we a priori ask that the tangent planes at a point might rotate at different scales, a posteriori the measure has a unique tangent almost everywhere. Since every horizontal subgroup has a normal complement, our criterion applies in the particular case of one-dimensional horizontal subgroups. These results are the outcome of a detailed study of a new notion of rectifiability: we say that a Radon measure on a Carnot group is $mathscr{P}_h$-rectifiable, for $hinmathbb N$, if it has positive $h$-lower density and finite $h$-upper density almost everywhere, and, at almost every point, it admits as tangent measures only (multiple of) the Haar measure of a homogeneous subgroup of Hausdorff dimension $h$. We also prove several structure properties of $mathscr{P}_h$-rectifiable measures. 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 realize that it is strictly weaker than them. Furthermore, 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.