{
  "abstract": "For a class of regular strongly local $p$-energy forms on complete doubling metric measure spaces, $1<p<\\infty$, we derive a macroscopic Poincaré inequality from a Poincaré inequality and a capacity upper bound at a doubling scale $Ψ$. The macroscopic inequality compares $L^p$-oscillations at any two scales and yields a uniform two-scale comparison $J_s\\lesssim J_r$, $0<r<s$, for the Korevaar-Schoen functionals, strengthening their usual weak monotonicity property. Conversely, under a density assumption on compactly supported functions, the macroscopic inequality yields a regular strongly local Korevaar-Schoen $p$-energy satisfying a $Ψ$-Poincaré inequality. We also show that the macroscopic inequality forces lower $p$-scaling of $Ψ$ on scales not exceeding $\\mathrm{diam} X$; together with doubling, this gives two-sided power bounds. Finally, we establish a weak-type characterization of the energy domain of Brezis-Van Schaftingen-Yung type.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29789",
  "author_total": 3,
  "authors": [
    {
      "name": "Fabrice Baudoin"
    },
    {
      "name": "Andrea Pinamonti"
    },
    {
      "name": "Meng Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fasn5emfpz5wj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29789",
  "fulltext_url": null,
  "id": "fasn5emfpz5wj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29789",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223227962",
  "pdf_url": "https://arxiv.org/pdf/2610.29789",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fasn5emfpz5wj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29789",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29789"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Macroscopic Poincaré Inequalities and Weak-Type Characterizations of $p$-Energy Forms",
  "title_text": "Macroscopic Poincaré Inequalities and Weak-Type Characterizations of p-Energy Forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/fasn5emfpz5wj",
  "venue": "arXiv",
  "versions": []
}