{
  "abstract": "The Zygmund vector-field maximal function conjecture is a long-standing open problem in harmonic analysis. It is notoriously difficult in part because the vector-field maximal function is non-translation-invariant and lies beyond the scope of finite-type theory. Its resolution would yield a Lebesgue differentiation theorem along vector fields, i.e., pointwise recovery of functions from averages along vector fields. We establish an \\(L^2\\) boundedness criterion that substantially enlarges the class of admissible geometric degeneracies. More precisely, the power-type decay condition in Bourgain's theorem can be replaced by the much weaker logarithmic-polynomial decay \\((\\log\\tfrac{1}τ)^{-p}\\), \\(p>2\\). This condition no longer yields the power-type integrability underlying Bourgain's argument, yet \\(L^2\\) boundedness persists, permitting degeneracies far beyond any power-law regime. Our result also provides a unified geometric perspective on finite-type, intermediate-degenerate, and highly non-finite-type vector fields, with their behavior organized through the angular-variation quantity \\(w_x(t)\\). This perspective places several classical boundedness criteria within a common hierarchy. Motivated by Lacey and Li's work on the Stein vector-field Hilbert transform conjecture, we further introduce a non-centered rectangular maximal operator adapted to the variable geometry of vector fields and prove its weak-type \\((1,1)\\) boundedness. Together, these results provide new tools and a unified geometric framework for the study of two fundamental conjectural problems in non-translation-invariant analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.04394",
  "author_total": 1,
  "authors": [
    {
      "name": "Lingxiao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/am3y82hgexh6a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.04394",
  "fulltext_url": null,
  "id": "am3y82hgexh6a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.04394",
  "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": "W7160562517",
  "pdf_url": "https://arxiv.org/pdf/2605.04394",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/am3y82hgexh6a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2605.04394",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.04394"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform",
  "title_text": "A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/am3y82hgexh6a",
  "venue": "arXiv",
  "versions": []
}