{
  "abstract": "For the fractional heat equation $\\partial_t u + (-Δ)^{β/2}u=0$ on $\\mathbb{R}^d$ with $β\\in(0,2)$, Weber and Zacher introduced the optimal constant $C_{\\rm LY}(β,d)$ in the Li--Yau type estimate $$(-Δ)^{β/2}\\log u(t,\\cdot)\\leq \\frac {C_{\\rm LY}(β,d)}{t},\\quad t>0.$$ They asked whether $C_{\\rm LY}(β,d)$ converges to the classical Li--Yau constant $d/2$ as $β\\uparrow2$. In this note we prove that the answer is affirmative. More precisely, for every fixed $d\\geq1$ there exists a constant $K_d>0$ such that $$\\frac{d}β\\leq C_{\\rm LY}(β,d)\\leq\\frac{d}β+K_d(2-β)\\log\\frac{e}{2-β},\\quad 1\\leqβ<2.$$ Consequently, $\\lim_{β\\uparrow2}C_{\\rm LY}(β,d)=d/2$. The proof has two main ingredients. First, the Bochner subordination formula, together with a concavity argument, shows that the supremum in Weber--Zacher's formula for $C_{\\rm LY}(β,d)$ is attained at the origin. Second, the factor $2-β$ is derived from a lower bound for the fractional heat kernel profile in the large-distance regime, where the Lévy--Itô decomposition plays an important role.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34700",
  "author_total": 1,
  "authors": [
    {
      "name": "Huaiqian Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3tbrxh2a9h8ky.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34700",
  "fulltext_url": null,
  "id": "3tbrxh2a9h8ky",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34700",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.34700",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3tbrxh2a9h8ky.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34700",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34700"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The limit of the optimal Li--Yau constant for the fractional heat equation",
  "title_text": "The limit of the optimal Li--Yau constant for the fractional heat equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/3tbrxh2a9h8ky",
  "venue": "arXiv",
  "versions": []
}