{
  "abstract": "Let $(X,d,ω)$ be an unbounded complete doubling metric measure space supporting a weak $(1,2)$-Poincaré inequality, and let $π$ be a positive Radon measure. For a ball $B=B(x,R)$, suppose that for some $δ>0$ and $C_{\\mathrm{gr}}>0$, \\[ π(B(y,r))\\le C_{\\mathrm{gr}} \\left(\\frac rR\\right)^δ\\frac{ω(B(y,r))}{r^2}, \\qquad y\\in B,\\quad 0<r\\le R. \\] Let $\\operatorname{cap}_{2,ω}(K;Ω)$ denote the relative variational $2$-capacity of $K$ in an open set $Ω$. We prove that, for every fixed $Λ_0>2$, there is a constant $C>0$ such that \\[ π(K)\\le C\\operatorname{cap}_{2,ω}(K;Λ_0B), \\qquad K\\subset B\\ \\text{compact}, \\] with the same outer ball for all compact $K$. No doubling or lower-Ahlfors regularity is imposed on $π$. This fixed-domain capacitary domination yields representative-independent $L^2(dπ)$ traces and, under a lower normalization $π(B)\\ge c_0ω(B)/R^2$ with $c_0>0$, the mixed $dω\\,dπ$ Poincaré estimate. Summation over bounded-overlap normalized covers gives two-sided global Fefferman--Phong inequalities and identifies the corresponding homogeneous energy spaces. In $\\mathbb R^d$ with a Muckenhoupt $A_2$ weight $w$, scale decay and controlled enlargement of $π$ generate the required critical-radius cover for every sufficiently large fixed threshold $A$; no additional lower reverse-doubling hypothesis on $w$ is needed. The resulting energy comparison gives equality of the measure-potential and critical-multiplier form domains, a smooth form core, a nonnegative self-adjoint realization, resolvent estimates, and local critical-multiplier bounds. The framework also covers reverse-Hölder function potentials, Carnot groups, and lower-dimensional singular measures, and provides a representative-aware capacity derivation of an earlier generalized Schrödinger energy estimate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.17315",
  "author_total": 1,
  "authors": [
    {
      "name": "Tan Duc Do"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/177sgh6w42y03.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.17315",
  "fulltext_url": null,
  "id": "177sgh6w42y03",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.17315",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7169826011",
  "pdf_url": "https://arxiv.org/pdf/2607.17315",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/177sgh6w42y03.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2607.17315",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.17315"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Capacitary domination and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces",
  "title_text": "Capacitary domination and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/177sgh6w42y03",
  "venue": "arXiv",
  "versions": []
}