{
  "id": "1804.04510",
  "version": "v1",
  "published": "2018-04-11T12:40:54.000Z",
  "updated": "2018-04-11T12:40:54.000Z",
  "title": "Riesz transforms on solvable extensions of stratified groups",
  "authors": [
    "Alessio Martini",
    "Maria Vallarino"
  ],
  "comment": "20 pages. arXiv admin note: text overlap with arXiv:1504.03862",
  "categories": [
    "math.FA",
    "math.AP",
    "math.CA",
    "math.DG"
  ],
  "abstract": "Let $G = N \\rtimes A$, where $N$ is a stratified group and $A = \\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous sub-Laplacians on $N$ and $A$ can be lifted to left-invariant operators on $G$ and their sum is a sub-Laplacian $\\Delta$ on $G$. Here we prove weak type $(1,1)$, $L^p$-boundedness for $p \\in (1,2]$ and $H^1 \\to L^1$ boundedness of the Riesz transforms $Y \\Delta^{-1/2}$ and $Y \\Delta^{-1} Z$, where $Y$ and $Z$ are any horizontal left-invariant vector fields on $G$, as well as the corresponding dual boundedness results. At the crux of the argument are large-time bounds for spatial derivatives of the heat kernel, which are new when $\\Delta$ is not elliptic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-11T12:40:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E30",
      "42B20",
      "42B30"
    ],
    "keywords": [
      "riesz transforms",
      "stratified group",
      "solvable extensions",
      "horizontal left-invariant vector fields",
      "corresponding dual boundedness results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}