{
  "abstract": "We develop a measure-theoretic framework for small-order limits of nonlocal energies with metric-valued maps. For globally $L^p$ maps, we consider interaction measures whose marginals have uniformly bounded $L^\\infty$ densities $a_α$ and $b_α$. If these densities converge weakly-* to $a_\\infty$ and $b_\\infty$, the normalized energies converge to the $L^p$ energy weighted by $a_\\infty+b_\\infty$ if and only if the interaction measures escape every bounded rectangle. For maps that are only locally $L^p$, we introduce target laws describing the distribution of the values sampled at infinity. Convergence of their $p$-distance profiles yields the limiting interaction energy for every $1\\leq p<\\infty$ and arbitrary Polish targets, with integrated $p$-Wasserstein convergence as a sufficient criterion. The arguments include the endpoint $p=1$ and require no linear structure on the target space. Applications include directional and anisotropic Maz'ya--Shaposhnikova formulas, a Bernoulli-law interpretation of fractional perimeters, and an Abelian principle for heat-semigroup energies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01958",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrea Pinamonti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2d9zernwvt2wd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01958",
  "fulltext_url": null,
  "id": "2d9zernwvt2wd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01958",
  "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.CA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7216324386",
  "pdf_url": "https://arxiv.org/pdf/2610.01958",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2d9zernwvt2wd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01958",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01958"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Decoupling and Tail Laws for Small-Order Metric-Valued Nonlocal Energies: A Structural View of the Maz'ya--Shaposhnikova Formula",
  "title_text": "Decoupling and Tail Laws for Small-Order Metric-Valued Nonlocal Energies: A Structural View of the Maz'ya--Shaposhnikova Formula",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/2d9zernwvt2wd",
  "venue": "arXiv",
  "versions": []
}