{
  "abstract": "In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping $f\\colon\\mathbb{R}^n\\to\\mathbb{R}^m$ lies in the local Sobolev space $W^{1,p}_\\text{loc}(\\mathbb{R}^n\\colon\\mathbb{R}^m)$ if and only if $f$ is a $(p,1-n/p)$-compactly Hölder mapping, where $p>n$. In fact, we prove the following more general result: whenever $X$ is a complete $Q$-Ahlfors regular metric space supporting a $p$-Poincaré inequality with $p>Q$, and $V$ is any Banach space, the compactly Hölder class $CH^{p,1-Q/p}(X\\colon V)$ consists precisely of the continuous representatives of the local Hajłasz and Newtonian Sobolev spaces $M^{1,p}_\\text{loc} (X \\colon V)$ and $N^{1,p}_\\text{loc}(X\\colon V)$. We also establish quantitative comparisons between the corresponding seminorms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24520",
  "author_total": 2,
  "authors": [
    {
      "name": "Ryan Alvarado"
    },
    {
      "name": "Efstathios Konstantinos Chrontsios Garitsis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/faxsf7ma3wc3x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24520",
  "fulltext_url": null,
  "id": "faxsf7ma3wc3x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24520",
  "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.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214056993",
  "pdf_url": "https://arxiv.org/pdf/2609.24520",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/faxsf7ma3wc3x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24520",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24520"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A purely metric characterization of supercritical Sobolev spaces",
  "title_text": "A purely metric characterization of supercritical Sobolev spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/faxsf7ma3wc3x",
  "venue": "arXiv",
  "versions": []
}