{
  "abstract": "Let $n \\ge 3$ and fix $p>n$. We establish a sharp quantitative stability result in $W^{1,p}$ for Liouville's theorem for nonconstant, sense-preserving quasiregular mappings whose outer distortion $K$ is sufficiently close to one. We first reorganize Reshetnyak's classical local argument \\cite{R1976}, drawing systematically on modern techniques such as those developed in \\cite{FZ2022}. We then give two globalization schemes on bounded, connected domains. For bounded John domains, instead of Reshetnyak's original construction \\cite{R19762}, which glues Möbius transformations on adjacent Whitney cubes, we use Whitney chains and Bojarski's enlarged-cube estimate to obtain a quantitatively sharp Euclidean stability result. Finally, we extend this idea to general bounded, connected open subsets of $\\mathbb{R}^n$ via the Lorentzian representation of the Möbius group, yielding a compactified, weighted stability theorem whose underlying measure is absolutely continuous with respect to Lebesgue measure and has density bounded above by one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24041",
  "author_total": 1,
  "authors": [
    {
      "name": "Yi Ru-Ya Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cqxf8aqcc7rye.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24041",
  "fulltext_url": null,
  "id": "cqxf8aqcc7rye",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24041",
  "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",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7214025205",
  "pdf_url": "https://arxiv.org/pdf/2609.24041",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cqxf8aqcc7rye.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24041",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24041"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A sharp stability inequality of Liouville's theorem for quasiregular mappings on bounded domains",
  "title_text": "A sharp stability inequality of Liouville's theorem for quasiregular mappings on bounded domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/cqxf8aqcc7rye",
  "venue": "arXiv",
  "versions": []
}