{
  "abstract": "We give a detailed, direct proof of a dimension-free weak-type $(1,1)$ estimate for the first-order Gaussian vector Riesz transform, using the fractional obstacle method of Ouyang, Spector, and Stockdale. For $dγ_n=π^{-n/2}e^{-|x|^2}\\,dx$, let $L=-\\tfrac12Δ+x\\cdot\\nabla$, $D=2^{-1/2}\\nabla$, and $R_γ=DL^{-1/2}Π_0$, where $Π_0$ removes the mean. We prove that $R_γ$ has weak-type constant at most $2$ on real-valued inputs, independently of $n$. The main construction is the unique unshifted decomposition $F=g+L^{1/2}u$, for nonnegative $F\\in L^2(γ_n)$ and $a>\\int F\\,dγ_n$, with $u\\geq0$, $\\min\\{F,a\\}\\leq g\\leq a$, preservation of mass, and $g=a$ on $\\{u>0\\}$. We retain the details of coercivity, the Lewy--Stampacchia estimate, the upgrade from the form domain to the operator domain, and locality. We also identify the $L^1$ extension with the principal-value transform and obtain the constant $8/π$ for complex-valued inputs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26478",
  "author_total": 2,
  "authors": [
    {
      "name": "Cristian E. Gutiérrez"
    },
    {
      "name": "Juan Manfredi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3w68pnv6zt1x7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26478",
  "fulltext_url": null,
  "id": "3w68pnv6zt1x7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26478",
  "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.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7222997174",
  "pdf_url": "https://arxiv.org/pdf/2610.26478",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3w68pnv6zt1x7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26478",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26478"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A direct obstacle proof of a dimension-free weak-type $(1,1)$ bound for the Gaussian Riesz transforms",
  "title_text": "A direct obstacle proof of a dimension-free weak-type (1,1) bound for the Gaussian Riesz transforms",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/3w68pnv6zt1x7",
  "venue": "arXiv",
  "versions": []
}