{
  "abstract": "We give a complete characterization of the kernels $Ω\\in \\Lup^1(\\Sp)$ for which the rough singular integral $\\T_Ω$ and the associated maximal operator $\\M_Ω$ satisfy pointwise bounds by $\\Int_1(|\\nabla \\cdot|)$ and $\\M_1(|\\nabla \\cdot|)$, respectively, where $\\Int_1$ is the Riesz potential and $\\M_1$ is the fractional maximal operator. These bounds hold if and only if $Ω$ belongs to the homogeneous dual Sobolev space $\\dot \\W^{-1,\\infty}(\\Sp)$. Equivalently, $Ω=\\text{div}_\\Sp \\mathbf{V}$ for some bounded vector field $\\mathbf{V}$ on the sphere, or $\\big|\\int_EΩ\\big|\\leq C\\,\\mathcal H^{d-2}(\\partial E)$ for every open set $E\\subseteq\\Sp$ with smooth boundary. On the Lebesgue scale, we show that for $d\\geq3$ the condition $Ω\\in\\Lup^{d-1,\\infty}(\\Sp)$, together with mean zero, suffices, and that the exponent $d-1$ is sharp. Finally, in dimension $d=2$, we use complex-analytic methods to prove a new pointwise bound in terms of the complex derivative $\\bar\\partial f$ that is valid for every mean-zero $Ω\\in\\Lup^1(\\mathbb S^1)$. As a consequence, we obtain new Sobolev bounds for rough singular integrals in the plane.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31888",
  "author_total": 2,
  "authors": [
    {
      "name": "Kabe Moen"
    },
    {
      "name": "Carlos Pérez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b5tkv9qann5h4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31888",
  "fulltext_url": null,
  "id": "b5tkv9qann5h4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31888",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7223660957",
  "pdf_url": "https://arxiv.org/pdf/2610.31888",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b5tkv9qann5h4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.31888",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31888"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Pointwise Riesz potential bounds for rough singular integrals",
  "title_text": "Pointwise Riesz potential bounds for rough singular integrals",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:08Z",
  "url": "https://research.draconitic.com/paper/b5tkv9qann5h4",
  "venue": "arXiv",
  "versions": []
}