{
  "abstract": "We prove that if a real-valued function $f\\in L^1$ on the complex unit circle has a gap of width at least $π$ in its essential range, then $\\exp(\\tilde f)$ is not integrable, where $\\tilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ satisfying $$ \\int_0^\\infty Y(x) e^{-x}\\, dx = \\infty. $$ We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, $Y(\\tilde f)$ fails to be integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Furthermore, we obtain corresponding results for complex-valued functions and a $Y$-integrability version of Zygmund's result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05348",
  "author_total": 2,
  "authors": [
    {
      "name": "David Norrbo"
    },
    {
      "name": "Jani Virtanen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f9h5rase9n0rz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05348",
  "fulltext_url": null,
  "id": "f9h5rase9n0rz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05348",
  "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.CV"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7211894745",
  "pdf_url": "https://arxiv.org/pdf/2609.05348",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f9h5rase9n0rz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.05348",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05348"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp exponential integrability of conjugate functions",
  "title_text": "Sharp exponential integrability of conjugate functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/f9h5rase9n0rz",
  "venue": "arXiv",
  "versions": []
}