{
  "abstract": "We prove the Zygmund conjecture in all parameters: the maximal operator associated with products of cubes with side lengths $(s_1,\\ldots,s_{m-1},ϕ(s_1,\\ldots,s_{m-1}))$, where $ϕ$ is any positive coordinatewise nondecreasing function, satisfies the weak $L(\\log L)^{m-2}$ estimate for every $m\\geq3$. The estimate is new for $m>3$. Our approach is to prove Rey's recent conjecture on the exponential integrability of the overlap function $h_{\\mathcal G}=\\sum_{I\\in\\mathcal G}\\mathbf{1}_I$, with exponent $1/(m-1)$, for sparse incomparable families $\\mathcal G$ of $m$-parameter dyadic rectangles, $m\\geq2$. The Zygmund conjecture, even in the continuous form above, follows by strengthening a reduction given by Rey.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23895",
  "author_total": 1,
  "authors": [
    {
      "name": "Henri Martikainen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7n4dyhx23xwva.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23895",
  "fulltext_url": null,
  "id": "7n4dyhx23xwva",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23895",
  "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": "W7214036516",
  "pdf_url": "https://arxiv.org/pdf/2609.23895",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7n4dyhx23xwva.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.23895",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23895"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Zygmund conjecture and Rey's exponential integrability conjecture",
  "title_text": "The Zygmund conjecture and Rey's exponential integrability conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/7n4dyhx23xwva",
  "venue": "arXiv",
  "versions": []
}