{
  "abstract": "Let $N\\geq2$, let $F\\in GL_N(\\mathbb C)$, and let $O_F^+$ be the associated free orthogonal quantum group of Kac type, with tracial Haar state $h$. We prove that, for $1<p\\leq r<\\infty$, the normalized heat semigroup $P_t=e^{-tL}$ on $O_F^+$ \\cite{CFK14,BVY21} is hypercontractive with the optimal time \\[ \\|P_t:L_p(O_F^+)\\to L_r(O_F^+)\\|\\leq1 \\quad\\Longleftrightarrow\\quad t\\geq\\frac12\\log\\frac{r-1}{p-1}. \\] In particular, this solves a conjecture of Brannan, Vergnioux and Youn \\cite{BVY21}. To prove this hypercontractivity result, we establish the equivalent logarithmic Sobolev inequality \\[ \\operatorname{Ent}(|x|^2)\\leq2\\mathsf{E}(x,x), \\qquad x\\in D(\\mathsf{E}), \\] with sharp constant $2$, where $\\operatorname{Ent}(y)=h(y\\log y)-h(y)\\log h(y)$ is the relative entropy and $\\mathsf E(x,x)=\\|L^{1/2}x\\|_2^2$ is the Dirichlet form associated with $P_t$. The main ingredients are the cubic-majorant idea from the recent work of Frank--Ivanisvili \\cite{FI26} and Xie--Zhang \\cite{XZC26,XZ26}, and a centered third-moment estimate coming from the representation theory. We also prove the sharp Beckner inequalities \\[ \\frac{\\|x\\|_p^2-\\|x\\|_2^2}{p-2}\\leq\\mathsf E(x,x), \\qquad 1\\leq p\\leq3,\\quad p\\ne2, \\] whose limit as $p\\to2$ recovers the logarithmic Sobolev inequality above. Moreover, when $N=2$, the results can be sharpened by replacing the heat semigroup with $\\mathsf P_t=e^{-tΛ}$, where $Λ=\\sqrt{I+3L}-I$. The key step is to use the graded-twist realization of $SU_{-1}(2)$ to transfer Beckner's sharp $p=3$ inequality on $SU(2)$ \\cite{Beckner93}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.29694",
  "author_total": 1,
  "authors": [
    {
      "name": "Haonan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0zkt7bzc09js5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.29694",
  "fulltext_url": null,
  "id": "0zkt7bzc09js5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.29694",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7166697321",
  "pdf_url": "https://arxiv.org/pdf/2606.29694",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0zkt7bzc09js5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2606.29694",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.29694"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp hypercontractivity for free orthogonal quantum groups of Kac type",
  "title_text": "Sharp hypercontractivity for free orthogonal quantum groups of Kac type",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/0zkt7bzc09js5",
  "venue": "arXiv",
  "versions": []
}