{
  "id": "2003.05862",
  "version": "v1",
  "published": "2020-03-12T15:54:55.000Z",
  "updated": "2020-03-12T15:54:55.000Z",
  "title": "Planar incidences and geometric inequalities in the Heisenberg group",
  "authors": [
    "Katrin Fässler",
    "Tuomas Orponen",
    "Andrea Pinamonti"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove that if $P,\\mathcal{L}$ are finite sets of $\\delta$-separated points and lines in $\\mathbb{R}^{2}$, the number of $\\delta$-incidences between $P$ and $\\mathcal{L}$ is no larger than a constant times $$|P|^{2/3}|\\mathcal{L}|^{2/3} \\cdot \\delta^{-1/3}.$$ We apply the bound to obtain the following variant of the Loomis-Whitney inequality in the Heisenberg group: $$ |K| \\lesssim |\\pi_{x}(K)|^{2/3} \\cdot |\\pi_{y}(K)|^{2/3}, \\qquad K \\subset \\mathbb{H}. $$ Here $\\pi_{x}$ and $\\pi_{y}$ are the vertical projections to the $xt$- and $yt$-planes, respectively, and $|\\cdot|$ refers to natural Haar measure on either $\\mathbb{H}$, or one of the planes. Finally, as a corollary of the Loomis-Whitney inequality, we deduce that $$ \\|f\\|_{4/3} \\lesssim \\sqrt{\\|Xf\\| \\|Yf\\| }, \\qquad f \\in BV(\\mathbb{H}), $$ where $X,Y$ are the standard horizontal vector fields in $\\mathbb{H}$. This is a sharper version of the classical geometric Sobolev inequality $\\|f\\|_{4/3} \\lesssim \\|\\nabla_{\\mathbb{H}}f\\|$ for $f \\in BV(\\mathbb{H})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-12T15:54:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "52C99",
      "46E35",
      "35R03"
    ],
    "keywords": [
      "heisenberg group",
      "planar incidences",
      "geometric inequalities",
      "loomis-whitney inequality",
      "standard horizontal vector fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}