On rectifiable measures in Carnot groups: existence of density

Abstract

In this paper we prove the one-dimensional Preiss' theorem in the first Heisenberg group H 1 . More precisely we show that a Radon measure φ on 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 mapsThe 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 P h -rectifiable, for h ∈ 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 P h -rectifiable measures. First, we compare 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 P h -rectifiable measure has almost everywhere positive and finite h-density whenever the tangents admit at least one complementary subgroup. Finally, we prove that a P h -rectifiabile measure with complemented tangents is supported on the union of intrinsically differentiable graphs.

Identifying 1-rectifiable measures in Carnot groups

“…to be the maximal cube containing x with 4 3 side Q ≤ r x and side Q ≤ 1. Then x ∈ Leaves(T x ), where…”

“…measures carried by C k,α submanifolds, see [5,29,35,60]. There is also much interest in understanding the rectifiability of sets and measures in non-Euclidean metric spaces; see [3,4,17,19,42,55,57] for a short sample.…”

On rectifiable measures in Carnot groups: representation

“…Remark: Some of the results presented in this paper use results proven in [4,. We recall the most important ones in the preliminary section of this work, see Section 2.…”

“…In [4] we started to study structure results for the class of P-rectifiable measures, proving a Marstrand-Mattila type rectifiability criterion in the co-normal case [4,Theorem 1.3]. The latter theorem directly leads to the proof of Preiss's theorem in the first Heisenberg group H 1 equipped with the Koranyi norm, see [4,Theorem 1.4].…”

“…In [4] we started to study structure results for the class of P-rectifiable measures, proving a Marstrand-Mattila type rectifiability criterion in the co-normal case [4,Theorem 1.3]. The latter theorem directly leads to the proof of Preiss's theorem in the first Heisenberg group H 1 equipped with the Koranyi norm, see [4,Theorem 1.4]. In this paper we complete the study of the notion of P-rectifiable measure, when the tangents are complemented, showing that the notion of P-rectifiability, which is infinitesimal in nature, is equivalent to the global one given in terms of intrinsic differentiable graphs, see Theorem 1.1.…”