{
  "abstract": "In this paper we present partial progress towards two conjectures on the variation of maximal operators of convolution type $\\mathrm{M}_φ$, associated to non-negative and radial decreasing kernels $φ$ of integral $1$. In dimension $d=1$ we prove that $\\mathrm{var}(\\mathrm{M}_φ f) \\leq \\mathrm{var}(f)$ when $f$ is a characteristic function. In higher dimensions we prove that $\\mathrm{var}(\\mathrm{M}_φ f) \\leq C_d \\mathrm{var}(f)$ when $f$ is a radial characteristic function, with a constant $C_d$ depending only on the dimension $d$ but not on the kernel $φ$. We further show that $C_d \\sim \\sqrt{d}\\log d $ is the sharp dependence on $d$ for such an inequality. This last claim relies on a novel explicit asymptotic estimate for the Hardy-Littlewood maximal function of the unit ball in $\\mathbb R^d$, a result of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18544",
  "author_total": 2,
  "authors": [
    {
      "name": "Emanuel Carneiro"
    },
    {
      "name": "Julian Weigt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cqftfdqad74kc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18544",
  "fulltext_url": null,
  "id": "cqftfdqad74kc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18544",
  "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": "W7221781732",
  "pdf_url": "https://arxiv.org/pdf/2610.18544",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cqftfdqad74kc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18544",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18544"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the variation and asymptotics of maximal functions",
  "title_text": "On the variation and asymptotics of maximal functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/cqftfdqad74kc",
  "venue": "arXiv",
  "versions": []
}