# On Weissler's conjecture on the Hamming cube I

## TL;DR

This paper proves Weissler's conjecture on the boundedness of the Hermite operator on the Hamming cube for the case p=q, resolving previously open cases and providing new applications.

## Contribution

It provides a complete proof of Weissler's conjecture for p=q on the Hamming cube, filling gaps in prior partial results.

## Key findings

- Confirmed Weissler's conjecture for p=q case.
- Established operator norm bounds independent of dimension n.
- Presented applications of the proven conjecture.

## Abstract

Let $1\leq p \leq q <\infty$, 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 $2<p\leq q <3$ and $3/2<p\leq q <2$, 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.