{
  "abstract": "We investigate the boundedness of the Hardy-Littlewood maximal operator $M$ on variable Lebesgue spaces $L^{p(\\cdot)}(Ω,d,μ)$ over spaces of homogeneous type, focusing on the endpoints $p_-=1$ and $p_+=\\infty$. At the lower endpoint, we prove that the boundedness of $M$ forces $p_->1$ for all exponents if and only if the space $(Ω,d,μ)$ is non-atomic. At the upper endpoint, we generalize the Muckenhoupt-Nekvinda sufficient condition of Adamadze, Diening, and Kopaliani. On bounded spaces of homogeneous type, whenever $p_->1$, this yields a boundedness criterion based on the Muckenhoupt condition alone. We also construct an unbounded space where the Muckenhoupt condition suffices for the maximal operator's boundedness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33792",
  "author_total": 1,
  "authors": [
    {
      "name": "Alina Shalukhina"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/759xrvpg3ndxb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33792",
  "fulltext_url": null,
  "id": "759xrvpg3ndxb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33792",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.33792",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/759xrvpg3ndxb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33792",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33792"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The endpoints $p_-=1$ and $p_+=\\infty$ for the maximal operator on $L^{p(\\cdot)}$ over spaces of homogeneous type",
  "title_text": "The endpoints p_-=1 and p_+=∞ for the maximal operator on L^p(·) over spaces of homogeneous type",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/759xrvpg3ndxb",
  "venue": "arXiv",
  "versions": []
}