Sharp hypercontractivity for free group von Neumann algebras

Abstract

In this paper, we settle the problem of optimal hypercontractivity for free group von Neumann algebras. Namely, for $n\ge 2$ and the free group ${\mathbb{F}}_n$ on $n$ generators, we prove that for any $1<p\le q<\infty$, the Poisson semigroup $P_t$ associated with the word-length function satisfies $$\Vert P_t:L_p(\widehat{{\mathbb{F}}_n})\to L_q(\widehat{{\mathbb{F}}_n})\Vert \le 1\text{ if and only if }t\ge \frac{1}{2}\log \frac{q-1}{p-1}.$$ The main idea is to apply a refined cubic majorant estimate from a recent work of Frank and Ivanisvili \cite{FrankIvanisvili2026} to the equivalent logarithmic Sobolev inequality, and use the Haagerup-type cancellation estimate \cite{Haagerup1979}. Similar ideas and techniques extend to free products \[ G=\left(*_{α\in A}\mathbb Z\right)*\left(*_{β\in B}\mathbb Z_2\right) \] and the free Gaussian von Neumann algebras. In the former setting, partial sharp estimates were previously obtained by Junge--Palazuelos--Parcet--Perrin--Ricard \cite{JungePalazuelosParcetPerrinRicard2015}; in the latter, our approach recovers Biane's free hypercontractivity theorem \cite{Biane1997}.

12 pages. Minor revision. More discussion added. Acknowlegement (including GPT) added