{
  "abstract": "In the present paper, we consider the maximal operator \\[ \\mathcal M_Ωf(x) := \\sup_{r>0}\\frac{1}{r^n} \\int_{|y|<r} |f(x-y)| \\left| Ω\\!\\left(\\frac{y}{|y|}\\right) \\right|\\,dy. \\] A longstanding open conjecture raised by E.M.Stein asks whether the maximal operator $\\mathcal M_Ω$ is of weak type $(1,1)$ when $Ω$ is merely in $L^1(\\mathbb S^{n-1})$. We partially settle this problem by proving weak type $(1,1)$ bounds of $\\mathcal M_Ω$ with kernel $Ω\\in\\mathcal X(\\mathbb S^{n-1})$, yielding a significant improvement over the work of M.~Christ and Rubio de Francia. Here$\\mathcal X(\\mathbb S^{n-1})$ is the space related to the Hausdorff--Choquet angular space and $$ L\\log^+\\!L(\\mathbb S^{n-1}) \\subsetneq \\mathcal X(\\mathbb S^{n-1})\\subset L^1(\\mathbb S^{n-1}). $$ Prior to our work, the condition $Ω\\in L\\log^+\\!L(\\mathbb S^{n-1}) $ remained the best known size restriction guaranteeing the weak-type$ (1,1)$ bounds for all dimensions $n > 2$. To our knowledge, the present paper is the first to relax this threshold for $n>2$. Finally, for the general Hausdorff--Choquet scale $\\mathcal H\\mathcal C_α$, we show that $α=(n-1)/2$ is the sharp exponent for uniform weak type $(1,1)$ estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03785",
  "author_total": 2,
  "authors": [
    {
      "name": "Yanping Chen"
    },
    {
      "name": "Zhengyang Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9f3fw7rptg170.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03785",
  "fulltext_url": null,
  "id": "9f3fw7rptg170",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03785",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7208789851",
  "pdf_url": "https://arxiv.org/pdf/2609.03785",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9f3fw7rptg170.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2609.03785",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03785"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hausdorff-Choquet Angular Spaces and Weak-Type $(1,1)$ Bounds for Rough Maximal Operators",
  "title_text": "Hausdorff-Choquet Angular Spaces and Weak-Type (1,1) Bounds for Rough Maximal Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/9f3fw7rptg170",
  "venue": "arXiv",
  "versions": []
}