{
  "id": "1308.2373",
  "version": "v1",
  "published": "2013-08-11T08:11:39.000Z",
  "updated": "2013-08-11T08:11:39.000Z",
  "title": "Hardy and uncertainty inequalities on stratified Lie groups",
  "authors": [
    "Paolo Ciatti",
    "Michael G. Cowling",
    "Fulvio Ricci"
  ],
  "comment": "19 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove various Hardy-type and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_\\alpha: f \\mapsto |.|^{-\\alpha} L^{-\\alpha/2} f$, where $|.|$ is a homogeneous norm, $0 < \\alpha < Q/p$, and $L$ is the sub-Laplacian, are bounded on the Lebesgue space $L^p(G)$. As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg-Pauli-Weyl inequality, relating the $L^p$ norm of a function $f$ to the $L^q$ norm of $|.|^\\beta f$ and the $L^r$ norm of $L^{\\delta/2} f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-11T08:11:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B37",
      "43A80"
    ],
    "keywords": [
      "stratified lie group",
      "logarithmic uncertainty inequality",
      "lebesgue space",
      "general version",
      "heisenberg-pauli-weyl inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.2373C"
    }
  }
}