{
  "abstract": "We prove a weak-type $(1,1)$ estimate for the centered Hardy--Littlewood maximal function with respect to Euclidean balls with dimensional dependence $O(\\sqrt{n} \\log n)$. This improves the order of growth in the classical $O(n)$ estimate of Stein and Strömberg. The proof goes through a pointwise bound of the Hardy--Littlewood maximal operator by the heat maximal operator with $\\sqrt{n}$ loss. The key technical aspect of our result is an improvement of the weak-type bound for the heat maximal operator from $O(\\sqrt{n})$ to $O(\\log n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05377",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniel Spector"
    },
    {
      "name": "Cody B. Stockdale"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6w7qnpa06swmp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05377",
  "fulltext_url": null,
  "id": "6w7qnpa06swmp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05377",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7211883553",
  "pdf_url": "https://arxiv.org/pdf/2609.05377",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6w7qnpa06swmp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05377",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05377"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The weak-type (1,1) bound for the Hardy--Littlewood maximal function is $O(\\sqrt{n} \\log n)$",
  "title_text": "The weak-type (1,1) bound for the Hardy--Littlewood maximal function is O(√n log n)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/6w7qnpa06swmp",
  "venue": "arXiv",
  "versions": []
}