Sharp hypercontractivity for free orthogonal quantum groups of Kac type

Abstract

Let $N\ge 2$, let $F\in G{L}_N(\mathbb{C})$, and let $O_F^{+}$ be the associated free orthogonal quantum group of Kac type, with tracial Haar state $h$. We prove that, for $1<p\le r<\infty$, 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 $\mathrm{Ent}(y)=h(y\log y)-h(y)\log h(y)$ is the relative entropy and $\mathsf{E}(x,x)=\Vert L^{1/2}x{\Vert }_2^2$ 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 $p\to 2$ recovers the logarithmic Sobolev inequality above. Moreover, when $N=2$, the results can be sharpened by replacing the heat semigroup with ${\mathsf{P}}_t=e^{-t\Lambda }$, where $\Lambda =\sqrt{I+3L}-I$. The key step is to use the graded-twist realization of $S{U}_{-1}(2)$ to transfer Beckner's sharp $p=3$ inequality on $SU(2)$ \cite{Beckner93}.

arXiv comment27 pages. Beckner inequalities added. Minor revision and reorganization