{
  "abstract": "Let $\\mathcal{R}$ be the hyperfinite $\\mathrm{II}_1$ factor. Considering the partial sum operators $(\\mathcal{S}_n)_{n\\geq 1}$ of the noncommutative Vilenkin-Fourier series associated with an arbitrary admissible Vilenkin group, we prove that there exists a universal constant $c>0$ such that \\begin{equation*} \\sup_{n\\geq1}\\|\\mathcal{S}_n(f)\\|_{L_{1,\\infty}(\\mathcal{R})} \\leq c\\|f\\|_{L_1(\\mathcal{R})},\\quad f \\in L_1(\\mathcal{R}), \\end{equation*} and, for every $1<p<\\infty$, $$\\sup_{n\\geq1}\\|\\mathcal{S}_n(f)\\|_{L_p(\\mathcal{R})} \\leq c\\frac{p}{p-1}\\|f\\|_{L_p(\\mathcal{R})},\\quad f \\in L_p(\\mathcal{R}).$$ Besides the transference technique, the main novel ingredient is a modified version of noncommutative Calderón-Zygmund decomposition established in \\cite{CCP2022}. Consequently, we resolve the problem of weak type $(1,1)$ estimate communicated to the authors by Fedor Sukochev, and substantially improve the strong type (p,p) estimates obtained in \\cite{DFdePS2001} by achieving the optimal order $\\frac{p}{p-1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30779",
  "author_total": 2,
  "authors": [
    {
      "name": "Guixiang Hong"
    },
    {
      "name": "Tiantian Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11f3c9nkcdasy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30779",
  "fulltext_url": null,
  "id": "11f3c9nkcdasy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30779",
  "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.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.30779",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11f3c9nkcdasy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30779",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30779"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The weak type $(1,1)$ estimate of Dirichlet Means for unbounded noncommutative Vilenkin systems",
  "title_text": "The weak type (1,1) estimate of Dirichlet Means for unbounded noncommutative Vilenkin systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/11f3c9nkcdasy",
  "venue": "arXiv",
  "versions": []
}