{
  "abstract": "We obtain sharp two-sided estimates for the relative capacity of fat and thin annuli in the fractional Sobolev spaces $W^s_{p,q}$, where $s\\in(0,1)$, $p\\in(0,\\infty)$, $q\\in(0,\\infty]$, in terms of an explicit volume growth quantity. The underlying space is a general metric measure space equipped with a doubling and reverse doubling measure. In particular, whenever the capacity is finite, the estimates are independent of the parameter $q$, and in the Ahlfors regular case with $p=q$ they recover Costea's logarithmic law at $sp=Q$. For $p<q$, we also introduce a sharp condition on the measure that is necessary for the existence of admissible functions in the space $W^s_{p,q}$. For thin annuli $(B_r,B_{r+\\varepsilon r})$ under an $η$-annular decay property, we uncover a purely nonlocal phenomenon: for $sp <η$, the capacity remains bounded as $\\varepsilon\\to 0$, because the limiting competitor $χ_{B_r}$ already has finite energy, while if $sp\\geqη$ it blows up as $ \\varepsilon\\to0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24752",
  "author_total": 1,
  "authors": [
    {
      "name": "Angha Agarwal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d90pn6qtpaz9c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24752",
  "fulltext_url": null,
  "id": "d90pn6qtpaz9c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24752",
  "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.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7222996015",
  "pdf_url": "https://arxiv.org/pdf/2610.24752",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d90pn6qtpaz9c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.24752",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24752"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Capacity Estimates and Volume Growth for Fractional Sobolev Spaces",
  "title_text": "Sharp Capacity Estimates and Volume Growth for Fractional Sobolev Spaces",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/d90pn6qtpaz9c",
  "venue": "arXiv",
  "versions": []
}