{
  "abstract": "We study heat semigroups on bounded Lipschitz domains $Ω\\subset \\mathbb{R}^{d}$ with dimension $d>2$ under non-local Robin boundary conditions. The boundary operator $B \\in \\mathcal{L}( \\mathrm{L}^{2}(\\partialΩ))$ is allowed to destroy the positivity of the semigroup. We assume that there exists a positive operator $C \\in \\mathcal{L}(\\mathrm{L}^{2}(\\partialΩ))$ such that $$ |Bu| \\leq C|u| \\quad\\text{for every }u\\in \\mathrm{L}^{2}(\\partialΩ), \\qquad C{\\bf 1}\\in \\mathrm{L}^{\\infty}(\\partialΩ). $$ Under this assumption, we prove that the semigroup generated by the corresponding uniformly elliptic operator is ultracontractive. More precisely, its norm from $\\mathrm{L}^{2}(Ω)$ to $\\mathrm{L}^{\\infty}(Ω)$ has short-time order $t^{-d/4}$. The main step is the construction of a positive eigenfunction $ϕ$ of a dominating elliptic operator such that $$ 0 \\ < \\ δ\\ \\leq \\ ϕ\\ \\leq \\ M $$ almost everywhere in $Ω$. The upper bound is obtained by power truncations and an iteration of Sobolev exponents. The lower bound follows by comparison with the Neumann semigroup. The two-sided estimate gives an $\\mathrm{L}^{\\infty}$-bound for the comparison semigroup. Nash's inequality and duality then yield ultracontractivity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30763",
  "author_total": 1,
  "authors": [
    {
      "name": "Christoph Schwerdt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b9066rje7r9xz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30763",
  "fulltext_url": null,
  "id": "b9066rje7r9xz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30763",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204963970",
  "pdf_url": "https://arxiv.org/pdf/2608.30763",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b9066rje7r9xz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30763",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30763"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Ultracontractivity of heat semigroups with non-local Robin boundary conditions via two-sided bounds for a positive eigenfunction",
  "title_text": "Ultracontractivity of heat semigroups with non-local Robin boundary conditions via two-sided bounds for a positive eigenfunction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/b9066rje7r9xz",
  "venue": "arXiv",
  "versions": []
}