{
  "abstract": "We study the Riesz transform $\\mathcal{R}=\\partial(-Δ)^{-\\frac{1}{2}}$ on uniform local trees, metric measure spaces that are locally real trees and whose canonical Dirichlet form is built from weak derivatives along the skeleton. The reference measure $m$ may be singular with respect to the length measure $ν$, so boundedness of $\\mathcal R$ is understood from $L^{p}(m)$ to $L^{p}(ν)$. In contrast with fractal-like manifolds and cable systems, the diffusion is sub-Gaussian at small scales and Gaussian at large scales. Under uniform volume growth, two-sided heat kernel estimates and a pointwise gradient estimate for the heat kernel, we prove that a local Dini condition on the scale function implies boundedness of $\\mathcal{R}$ on $L^{p}$ for every $p\\in[2,\\infty)$, and hence the reverse Riesz inequality for every $p\\in(1,2]$. Conversely, boundedness of $\\mathcal{R}$ for some $p<2$, or a reverse Riesz inequality for some $p>2$, forces the space to be one-dimensional at small scales. We show that a reverse Hölder inequality for harmonic functions yields the gradient estimate, and verify all hypotheses for spaces carrying a geometric group action whose generators have bounded displacement. As an application, for the alternating Vicsek fractafold in $\\mathbb Z^{d}$ we determine the exact ranges of $p$ for which the Riesz and reverse Riesz inequalities hold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34331",
  "author_total": 3,
  "authors": [
    {
      "name": "Fabrice Baudoin"
    },
    {
      "name": "Aobo Chen"
    },
    {
      "name": "Li Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b7dg8bwwkttmr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34331",
  "fulltext_url": null,
  "id": "b7dg8bwwkttmr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34331",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214736106",
  "pdf_url": "https://arxiv.org/pdf/2609.34331",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b7dg8bwwkttmr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34331",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34331"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Riesz transform on eventually Gaussian local trees",
  "title_text": "Riesz transform on eventually Gaussian local trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/b7dg8bwwkttmr",
  "venue": "arXiv",
  "versions": []
}