{
  "abstract": "Let \\(\\Gm=\\prod_{k\\ge0}\\mathbb Z_{m_k}\\) be a Vilenkin group that is not necessarily bounded, i.e., \\(\\sup_k m_k=\\infty\\). We prove that, for every UMD Banach space \\(X\\) and every \\(1<p<\\infty\\), the Vilenkin partial-sum operators are uniformly bounded on \\(L^p(\\Gm;X)\\), with a bound depending only on \\(p\\) and the UMD constant of \\(X\\), and not on \\(\\mathbf m\\). This resolves an open problem arising from the work of Clément et al.~\\cite{ClementDePagterSukochevWitvliet2000} and later recorded explicitly in the book of Hytönen et al.~\\cite[p.~362]{HNVWI}. The proof reduces the partial-sum estimate, via a Paley conjugation identity and a tangent-sequence decoupling argument, to a decoupling inequality for Fourier projections on finite cyclic groups, which appears to be new. The same approach also yields \\(\\mathcal R\\)-boundedness for the family of partial-sum operators associated with the finer block decomposition, thereby resolving another related problem communicated to us by Fedor Sukochev.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22714",
  "author_total": 2,
  "authors": [
    {
      "name": "Deyu Chen"
    },
    {
      "name": "Guixiang Hong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2fysgsty7b67x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22714",
  "fulltext_url": null,
  "id": "2fysgsty7b67x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22714",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7214023048",
  "pdf_url": "https://arxiv.org/pdf/2609.22714",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2fysgsty7b67x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22714",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22714"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Vector-valued partial sums on unbounded Vilenkin systems",
  "title_text": "Vector-valued partial sums on unbounded Vilenkin systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/2fysgsty7b67x",
  "venue": "arXiv",
  "versions": []
}