{
  "abstract": "Let $k\\geq 2$ be an integer, $T$ an infinite rooted $k$-ary tree, and $M$ the Hardy--Littlewood maximal operator on $T$. For any $p\\in(0,\\infty)$, we characterize the weight $w$ such that $M$ is bounded on $L^p(w)$. To this end, we introduce a two-layer Muckenhoupt weight class $\\mathscr A_p$ and prove that, for any $p\\in(\\frac{1}{2},\\infty)$, the boundedness of $M$ on $L^p(w)$, $w\\in\\mathscr A_p$, and the exponential decay boundedness of spherical averaging operators on $L^p(w)$ are mutually equivalent, and that, when $p\\in(0,\\frac{1}{2}]$, there exists no weight $w$ such that $M$ is bounded on $L^p(w)$. Moreover, for any $p\\in(\\frac{1}{2},\\infty)$, we establish the quantitative estimate, with the optimal exponent $\\frac{1}{p}$ of the weight constant, for the boundedness of $M$ on $L^p(w)$. For any $p\\in(1,\\infty)$, we also obtain two further equivalent characterizations of the boundedness of $M$ on $L^p(w)$, respectively, in terms of a global Sawyer-type testing condition and an estimate for the weighted product measure of distance incidence sets. As applications, for any $p\\in(\\frac{1}{2},\\infty)$, under the assumption that $M$ is bounded on $L^p(w)$, we establish the boundedness of exponentially decaying kernel operators on $L^p(w)$ and weighted Fefferman--Stein vector-valued inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12844",
  "author_total": 3,
  "authors": [
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Wen Yuan"
    },
    {
      "name": "Mingdong Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9zvqa75wddpsp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12844",
  "fulltext_url": null,
  "id": "9zvqa75wddpsp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12844",
  "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": "W7213020394",
  "pdf_url": "https://arxiv.org/pdf/2609.12844",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9zvqa75wddpsp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12844",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12844"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hardy--Littlewood Maximal Operator and Two-Layer Muckenhoupt Weights on Infinite Rooted $k$-Ary Trees",
  "title_text": "Hardy--Littlewood Maximal Operator and Two-Layer Muckenhoupt Weights on Infinite Rooted k-Ary Trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/9zvqa75wddpsp",
  "venue": "arXiv",
  "versions": []
}