{
  "id": "2202.12741",
  "version": "v1",
  "published": "2022-02-19T17:13:50.000Z",
  "updated": "2022-02-19T17:13:50.000Z",
  "title": "On rectifiable measures in Carnot groups: Marstrand-Mattila rectifiability criterion",
  "authors": [
    "Gioacchino Antonelli",
    "Andrea Merlo"
  ],
  "comment": "This is the second of two companion papers derived from arXiv:2009.13941v2. The present work consists of an elaboration of Sections 2 and 5 of the Preprint 2009.13941v2, while the first of the two (that will appear as 2009.13941v3) is an elaboration of Sections 2, 3, 4, and 6 of 2009.13941v2",
  "categories": [
    "math.MG"
  ],
  "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 $\\phi$ 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 $\\mathbb H^1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-19T17:13:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C17",
      "22E25",
      "28A75",
      "49Q15",
      "26A16"
    ],
    "keywords": [
      "marstrand-mattila rectifiability criterion",
      "rectifiable measures",
      "radon measure",
      "tangent planes",
      "one-dimensional hausdorff measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}