{
  "abstract": "We study Riesz and reverse Riesz inequalities for the fractional powers of the Laplacian $Δ$ on the unbounded Vicsek set. The space carries its Hausdorff measure $m$. The gradient $\\partial$ is a weak gradient defined on the skeleton, and it is measured with respect to a length measure $ν$ that is singular with respect to $m$. The relevant fractional order for the Riesz inequalities is not $1/2$ but $γ_{p}=\\frac{d_f+p-1}{pd_w}$, where $d_f=\\log_{3}5$ and $d_w=d_f+1$. For $p\\in[1,2]$, we prove the weak-type reverse Riesz inequality $\\|(-Δ)^{γ_p}f\\|_{L^{p,\\infty}(m)}\\lesssim\\|\\partial f\\|_{L^p(ν)}$. The corresponding strong estimate fails for $1\\le p<2$, whereas the weak estimate fails for $2<p<\\infty$, even after heat regularization. For the Riesz transform $\\mathcal{R}_p=\\partial(-Δ)^{-γ_p}$, we show that $\\mathcal{R}_1$ is not of weak type $(1,1)$ but that $\\mathcal{R}_p$ is bounded from $L^p(m)$ to $L^{p,\\infty}(ν)$ for $p\\in(1,2)$, and from $L^{p,1}(m)$ to $L^p(ν)$ for $p\\in(2,\\infty)$. However, if $p \\ge 1$, $\\mathcal{R}_p$ is bounded from $L^p(m)$ to $L^p(ν)$ only for $p=2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33858",
  "author_total": 3,
  "authors": [
    {
      "name": "Fabrice Baudoin"
    },
    {
      "name": "Aobo Chen"
    },
    {
      "name": "Li Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ejaenpdbhh3h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33858",
  "fulltext_url": null,
  "id": "4ejaenpdbhh3h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33858",
  "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.PR"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214750558",
  "pdf_url": "https://arxiv.org/pdf/2609.33858",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ejaenpdbhh3h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33858",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33858"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Riesz transforms on the Vicsek set",
  "title_text": "Riesz transforms on the Vicsek set",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4ejaenpdbhh3h",
  "venue": "arXiv",
  "versions": []
}