{
  "id": "2210.07148",
  "version": "v1",
  "published": "2022-10-13T16:31:37.000Z",
  "updated": "2022-10-13T16:31:37.000Z",
  "title": "Boundedness of a Riesz transform on the homogeneous tree",
  "authors": [
    "Federico Santagati",
    "Maria Vallarino"
  ],
  "comment": "11 pages. arXiv admin note: text overlap with arXiv:2107.06620",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\mathbb T_{q+1}$ denote the homogeneous tree of degree $q+1$ endowed with the standard graph distance $d$ and the canonical flow $\\mu$. The metric measure space $(\\mathbb T_{q+1},d,\\mu)$ is of exponential growth. In this note, we prove that the first order Riesz transform associated with a flow Laplacian on $\\mathbb T_{q+1}$ is bounded on $L^p(\\mu)$, for $p\\in (1,2]$ and of weak type $(1,1)$. This result was proved in a previous paper by Hebisch and Steger: we give a different proof which might pave the way to further generalizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-13T16:31:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C21",
      "42B20",
      "43A99"
    ],
    "keywords": [
      "homogeneous tree",
      "boundedness",
      "first order riesz transform",
      "metric measure space",
      "standard graph distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}