{
  "abstract": "This paper establishes an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. In the Euclidean setting, under a mild boundedness condition, the product of the order and the norm raised to the power $p$ converges to a dimension-independent constant multiple of the corresponding $L^p$ norm. The proof relies on heat-kernel regularization, weak compactness, and an Abelian--Tauberian argument. The result is further extended to a general measure space equipped with a family of bounded and continuous operators that covering nonlinear and non-semigroup settings. We also present several applications in analysis and probability, including an absolute-continuity criterion for measures, a random-distribution regularity result, a construction of square-integrable local times for fractional Brownian motion, and limiting formulas for truncated maximal operators and martingales.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26002",
  "author_total": 1,
  "authors": [
    {
      "name": "Huaiqian Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8p4jm6d4g8gdt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26002",
  "fulltext_url": null,
  "id": "8p4jm6d4g8gdt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26002",
  "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.PR"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214121964",
  "pdf_url": "https://arxiv.org/pdf/2609.26002",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8p4jm6d4g8gdt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26002",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26002"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Asymptotic Formulas for Negative Sobolev Norms and Applications",
  "title_text": "Asymptotic Formulas for Negative Sobolev Norms and Applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/8p4jm6d4g8gdt",
  "venue": "arXiv",
  "versions": []
}