{
  "abstract": "We investigate graph domains and graph curves from the viewpoints of $\\mathrm{BMOA}$, Zygmund regularity, and asymptotic geometry. First, we prove that every graph domain is a $\\mathrm{BMO}$ domain, which provides a proof of a statement of Astala and Zinsmeister. We then use a Weierstrass-type function to construct a Bishop-Jones quasicircle that is asymptotically conformal but not asymptotically smooth. This example yields the strict inclusions \\[ \\mathrm{VMOA}(\\mathbb H) \\subsetneq \\mathrm{BMOA}(\\mathbb H)\\cap B_0(\\mathbb H), \\qquad \\mathrm{SS}(\\mathbb R) \\subsetneq \\mathrm{SQS}(\\mathbb R)\\cap\\mathrm{S}(\\mathbb R). \\] Finally, by a localized lacunary construction, we produce a compactly supported locally absolutely continuous function $f$ with $f'\\in\\mathrm{BMO}(\\mathbb R)\\setminus\\mathrm{VMO}(\\mathbb R)$ whose graph is an asymptotically conformal chord-arc curve that is not asymptotically smooth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39567",
  "author_total": 2,
  "authors": [
    {
      "name": "Katsuhiko Matsuzaki"
    },
    {
      "name": "Fei Tao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dvkzf9k5bppnq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39567",
  "fulltext_url": null,
  "id": "dvkzf9k5bppnq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39567",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39567",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dvkzf9k5bppnq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39567",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39567"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "$\\mathrm{BMO}$ Domains and Quasicircles Associated with Zygmund Functions",
  "title_text": "BMO Domains and Quasicircles Associated with Zygmund Functions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/dvkzf9k5bppnq",
  "venue": "arXiv",
  "versions": []
}