{
  "id": "2104.06684",
  "version": "v1",
  "published": "2021-04-14T08:10:37.000Z",
  "updated": "2021-04-14T08:10:37.000Z",
  "title": "Loomis-Whitney inequalities in Heisenberg groups",
  "authors": [
    "Katrin Fässler",
    "Andrea Pinamonti"
  ],
  "comment": "27 pages. This paper supersedes arXiv:2003.05862v1; Loomis-Whitney inequalities are obtained in higher-dimensional Heisenberg groups by a simpler approach. Some preliminaries and Section 4 draw heavily from arXiv:2003.05862v1, where they were stated for the first Heisenberg group",
  "categories": [
    "math.CA"
  ],
  "abstract": "This note concerns Loomis-Whitney inequalities in Heisenberg groups $\\mathbb{H}^n$: $$|K| \\lesssim \\prod_{j=1}^{2n}|\\pi_j(K)|^{\\frac{n+1}{n(2n+1)}}, \\qquad K \\subset \\mathbb{H}^n.$$ Here $\\pi_{j}$, $j=1,\\ldots,2n$, are the vertical Heisenberg projections to the hyperplanes $\\{x_j=0\\}$, respectively, and $|\\cdot|$ refers to a natural Haar measure on either $\\mathbb{H}^n$, or one of the hyperplanes. The Loomis-Whitney inequality in the first Heisenberg group $\\mathbb{H}^1$ is a direct consequence of known $L^p$ improving properties of the standard Radon transform in $\\mathbb{R}^2$. In this note, we show how the Loomis-Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in $\\mathbb{H}^1$. The same approach, combined with multilinear interpolation, also yields the following strong type bound: $$\\int_{\\mathbb{H}^n} \\prod_{j=1}^{2n} f_j(\\pi_j(p))\\;dp\\lesssim \\prod_{j=1}^{2n} \\|f_j\\|_{\\frac{n(2n+1)}{n+1}}$$ for all nonnegative measurable functions $f_1,\\ldots,f_{2n}$ on $\\mathbb{R}^{2n}$. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis-Whitney inequalities in $\\mathbb{H}^n$, we mention the following sharper version of the classical geometric Sobolev inequality in $\\mathbb{H}^n$: $$\\|u\\|_{\\frac{2n+2}{2n+1}} \\lesssim \\prod_{j=1}^{2n}\\|X_ju\\|^{\\frac{1}{2n}}, \\qquad u \\in BV(\\mathbb{H}^n),$$ where $X_j$, $j=1,\\ldots,2n$, are the standard horizontal vector fields in $\\mathbb{H}^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-14T08:10:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "52C99",
      "46E35",
      "35R03"
    ],
    "keywords": [
      "loomis-whitney inequality",
      "standard horizontal vector fields",
      "note concerns loomis-whitney inequalities",
      "higher dimensional heisenberg groups",
      "classical geometric sobolev inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}