{
  "id": "2409.13233",
  "version": "v1",
  "published": "2024-09-20T05:39:54.000Z",
  "updated": "2024-09-20T05:39:54.000Z",
  "title": "$L^p$-boundedness of Riesz transforms on solvable extensions of Carnot groups",
  "authors": [
    "Alessio Martini",
    "Paweł Plewa"
  ],
  "comment": "27 pages",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "Let $G=N\\rtimes \\mathbb{R}$, where $N$ is a Carnot group and $\\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\\mathbb{R}$ can be lifted to $G$, and their sum is a left-invariant sub-Laplacian $\\Delta$ on $G$. We prove that the first-order Riesz transforms $X \\Delta^{-1/2}$ are bounded on $L^p(G)$ for all $p\\in(1,\\infty)$, where $X$ is any horizontal left-invariant vector field on $G$. This extends a previous result by Vallarino and the first-named author, who obtained the bound for $p\\in(1,2]$. The proof makes use of an operator-valued spectral multiplier theorem, recently proved by the authors, and hinges on estimates for products of modified Bessel functions and their derivatives.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-20T05:39:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E30",
      "42B20",
      "42B30"
    ],
    "keywords": [
      "carnot group",
      "solvable extensions",
      "boundedness",
      "horizontal left-invariant vector field",
      "left-invariant sub-laplacian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}