{
  "abstract": "We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \\[ \\mathcal{C}_2f(x) := \\sup_λ \\left|\\operatorname{p.v.} \\int_{\\mathbb{R}}f(x-y)e^{2 πi λy^2} \\frac{\\mathrm{d} y }{y} \\right| \\] and its lacunary counterpart obtained by restricting $λ$ to $2^{\\mathbb{Z}}.$ We prove that if $Φ$ is a Young function satisfying $ Φ(t)=o\\bigl(t\\log_2t\\bigr)$ then $\\mathcal{C}_{2,\\mathsf{lac}}$, and therefore $\\mathcal{C}_2$ as well, does not satisfy a corresponding $Φ$-modular estimate. In particular, neither the lacunary nor the full quadratic operator is of weak type $(1,1)$. In the positive direction, we establish an $L\\log_1 L$ modular estimate for $\\mathcal{C}_2$ and a $L (\\log_2L)^2 \\log_4L$ estimate for $\\mathcal{C}_{2,\\mathsf{lac}}.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04101",
  "author_total": 3,
  "authors": [
    {
      "name": "Anastasios Fragkos"
    },
    {
      "name": "Ben Krause"
    },
    {
      "name": "Michael Lacey"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dfrytfbfs7zg6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04101",
  "fulltext_url": null,
  "id": "dfrytfbfs7zg6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04101",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7208705182",
  "pdf_url": "https://arxiv.org/pdf/2609.04101",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dfrytfbfs7zg6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.04101",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04101"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Endpoint Estimates for Stein's Purely Quadratic Carleson Operator",
  "title_text": "Endpoint Estimates for Stein's Purely Quadratic Carleson Operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/dfrytfbfs7zg6",
  "venue": "arXiv",
  "versions": []
}