Sharp hypercontractivity for free orthogonal quantum groups of Kac type

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 $1<p\le r<\infty$, \[ \|P_t:L_p(O_F^+)\to L_r(O_F^+)\|\leq1 \quad\Longleftrightarrow\quad t\geq\frac12\log\frac{r-1}{p-1}. \] 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}.

19 pages. AI acknowledgement added. Simpler proof of N=2 case added. More details added