{
  "abstract": "We prove that the positive ground state $u$ of the fractional Laplacian $(-Δ)^s$, $s\\in (0,1)$, on an arbitrary open set $Ω$ satisfies $-Δu>λ_s(Ω)^{1/s}u$ in $Ω$. In one dimension, this yields strong concavity and settles a conjecture of Bañuelos, Kulczycki, and Méndez-Hernández. We establish a hierarchy of pointwise inequalities comparing different powers of the Laplacian. In balls, these inequalities give a quantitative Hessian estimate for $\\log u$, and hence log-concavity in every dimension. By contrast, we show that log-concavity fails in general convex domains by constructing counterexamples in sufficiently thin ellipsoids. We also extend the superharmonicity principle to ground states of $ψ(-Δ)$ for complete Bernstein functions $ψ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.11686",
  "author_total": 4,
  "authors": [
    {
      "name": "Nicola De Nitti"
    },
    {
      "name": "Xavier Fernández-Real"
    },
    {
      "name": "Florian Grube"
    },
    {
      "name": "Harsh Prasad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bptghs518wryx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.11686",
  "fulltext_url": null,
  "id": "bptghs518wryx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.11686",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7202394015",
  "pdf_url": "https://arxiv.org/pdf/2608.11686",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bptghs518wryx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2608.11686",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.11686"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Superharmonicity of the fractional ground state",
  "title_text": "Superharmonicity of the fractional ground state",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/bptghs518wryx",
  "venue": "arXiv",
  "versions": []
}