{
  "abstract": "We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t)_{t\\ge0}$ on a nonatomic $σ$-finite measure space $(X,\\mathcal F,μ)$ has maximal $L^p$ norm $p'=p/(p-1)$ for every $1<p<\\infty$ whenever $f\\mapsto T_{t_0}(\\mathbf 1_Ef)$ is compact on $L^2(μ)$ for some $t_0>0$ and some measurable set $E$ of positive measure. The lower bound follows by approximating finite dyadic conditional expectations at suitable semigroup times; the upper bound is Stein's maximal inequality in its sub-Markovian form. Important examples of such maximal operators we discuss include maximal functions of various semigroups (heat, Poisson, Ornstein--Uhlenbeck, Schrödinger, Dunkl, Laguerre, Jacobi) as well as the centered Hardy-Littlewood maximal operator on $\\mathbb{R}.$ In particular, the heat and Poisson maximal operators on every complete connected Riemannian manifold of positive dimension without boundary have exact $L^p$ norm $p'$, with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that $p'$ is the limit, as $d\\to \\infty,$ of the $L^p$ norms $B(p,d)$ of the centered Hardy-Littlewood maximal operators on $\\mathbb{R}^d.$ On the other hand, we show that the norm of this maximal operator is strictly larger than $p',$ in dimensions $d\\ge2.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10247",
  "author_total": 5,
  "authors": [
    {
      "name": "Valentina Ciccone"
    },
    {
      "name": "Mahdi Hormozi"
    },
    {
      "name": "José Madrid"
    },
    {
      "name": "Jakub Niksiński"
    },
    {
      "name": "Błażej Wróbel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dkzfv31eej2ac.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10247",
  "fulltext_url": null,
  "id": "dkzfv31eej2ac",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10247",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7220947739",
  "pdf_url": "https://arxiv.org/pdf/2610.10247",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dkzfv31eej2ac.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10247",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10247"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On $L^p$ norms of maximal operators",
  "title_text": "On L^p norms of maximal operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/dkzfv31eej2ac",
  "venue": "arXiv",
  "versions": []
}