On Weissler's conjecture on the Hamming cube I

Abstract

Let $1\le p\le q<\infty$, and let $w\in \mathbb{C}$. Weissler conjectured that the Hermite operator $e^{w\Delta }$ is bounded as an operator from $L^p$ to $L^q$ on the Hamming cube $\{-1,1{\}}^n$ with the norm bound independent of $n$ if and only if \begin{align*} |p-2-e^{2w}(q-2)|\leq p-|e^{2w}|q. \end{align*} It was proved by Bonami (1970), Beckner (1975), and Weissler (1979) in all cases except $2<p\le q<3$ and $3/2<p\le q<2$, which stood open until now. The goal of this paper is to give a full proof of Weissler's conjecture in the case $p=q$. Several applications will be presented.

The final version accepted to IMRN