{
  "abstract": "Working at the level of a sectorial operator $A$ on a Banach space $X$ and an unbounded operator $\\partial$ defined on a subspace of $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\\|A^αx\\|_X \\lesssim \\|\\partial x\\|_Y$ for some $0 < α< 1$, combined with the condition $0 \\in ρ(A_0)$, where $A_0$ is the part of $A$ on the closure of the range of $A$, implies the Poincaré inequality $\\|x - P(x)\\|_X \\lesssim \\|\\partial x\\|_Y$, where $P$ is the Abel-ergodic projection onto the kernel of $A$. The condition $0 \\in ρ(A_0)$ is the natural abstract substitute for a spectral gap, and is sharp already in the Hilbertian case. We also obtain a companion divergence inequality. The arguments are remarkably short, yet the principle is genuinely unifying: it covers commutative and noncommutative situations on the same footing and can be used with arbitrary Banach spaces. As a consequence, we recover Poincaré inequalities associated with hypercontractive semigroups and extend their validity to semigroups satisfying only a reduced spectral gap and a reverse Riesz estimate. We then illustrate the flexibility of the method across a wide spectrum of geometries, ranging from Riemannian manifolds, Lie groups, metric measure spaces, spin manifolds to genuinely noncommutative settings such as group von Neumann algebras, semigroups of Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups and quantum tori, where we sometimes establish new inequalities and otherwise recover classical ones from a single principle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.08322",
  "author_total": 1,
  "authors": [
    {
      "name": "Cédric Arhancet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ebgghawzyfb9d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.08322",
  "fulltext_url": null,
  "id": "ebgghawzyfb9d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.08322",
  "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.DG"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7167938865",
  "pdf_url": "https://arxiv.org/pdf/2607.08322",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ebgghawzyfb9d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2607.08322",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.08322"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality",
  "title_text": "A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/ebgghawzyfb9d",
  "venue": "arXiv",
  "versions": []
}