*Published Paper*

**Inserted:** 28 feb 2022

**Last Updated:** 12 apr 2022

**Journal:** Journal of Functional Analysis

**Year:** 2022

**Doi:** https://doi.org/10.1016/j.jfa.2022.109495

The present work consists of an elaboration of Sections 2 and 5 of the Preprint "On rectifiable measures in Carnot groups: structure theory". This is the second of two companion papers derived from "On rectifiable measures in Carnot groups: structure theory" that can be found as version 2 of the submission in https://arxiv.org/abs/2009.13941. The first companion paper is the version 3 of the submission in https://arxiv.org/abs/2009.13941.

**Abstract:**

In this paper we continue the study of the notion of $\mathscr{P}$-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. In this paper we prove a Marstrand--Mattila rectifiability criterion in arbitrary Carnot groups for $\mathscr{P}$-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 $A\subset \mathbb R$ to $\mathbb H^1$.

**Tags:**
GeoMeG