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arXiv · 2606.29694

Sharp hypercontractivity for free orthogonal quantum groups of Kac type

Haonan Zhang
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Abstract

We prove that the normalized heat semigroup $P_t=e^{-tL}$ on every free orthogonal quantum group $O_F^+$ of Kac type satisfies hypercontractivity with the optimal time. More precisely, for all $1<p\leq r<\infty$, \[ \|P_t:L_p(O_F^+)\to L_r(O_F^+)\|\leq1 \quad\Longleftrightarrow\quad t\geq\frac12\log\frac{r-1}{p-1}. \] Equivalently, the associated logarithmic Sobolev inequalities hold with sharp constants. For $F=I_N$, this determines the exact optimal time for $O_N^+$ and resolves a conjecture of Brannan, Vergnioux, and Youn \cite{BVY21}.

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Haonan Zhang. 2026-06-29. Sharp hypercontractivity for free orthogonal quantum groups of Kac type. https://arxiv.org/abs/2606.29694

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