{
  "abstract": "It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space $X$ and $1<p<\\infty$, the UMD$_p$ property for $X$ is equivalent to boundedness of the Hilbert transform on $L^p(\\R;X)$, and that the UMD constant $β_{p,X}$ and the Hilbert transform constant $\\hbar_{p,X}$ are related by the quadratic bounds \\begin{equation*} \\hbar_{p,X}\\lesssim(β_{p,X})^2, \\qquad β_{p,X}\\lesssim(\\hbar_{p,X})^2. \\end{equation*} In this paper we present examples showing that both bounds are sharp. More precisely, we construct explicit $2^n$-dimensional Banach spaces for which the Hilbert transform constant grows like $n$ and the UMD constant like $\\sqrt n$, and a second family with the reverse behaviour.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10444",
  "author_total": 2,
  "authors": [
    {
      "name": "Emiel Lorist"
    },
    {
      "name": "Jan van Neerven"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/79ecyzs4jstre.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10444",
  "fulltext_url": null,
  "id": "79ecyzs4jstre",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10444",
  "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": "W7212176837",
  "pdf_url": "https://arxiv.org/pdf/2609.10444",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/79ecyzs4jstre.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:16Z",
      "native_id": "2609.10444",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10444"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The bounds $\\hbar_{p,X}\\lesssim(β_{p,X})^2$ and $β_{p,X}\\lesssim(\\hbar_{p,X})^2$ are sharp",
  "title_text": "The bounds ℏ_p,X≲(β_p,X)^2 and β_p,X≲(ℏ_p,X)^2 are sharp",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:16Z",
  "url": "https://research.draconitic.com/paper/79ecyzs4jstre",
  "venue": "arXiv",
  "versions": []
}