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On Weissler’s Conjecture on the Hamming Cube I
International Mathematics Research Notices ( IF 0.9 ) Pub Date : 2020-12-26 , DOI: 10.1093/imrn/rnaa363
P Ivanisvili 1 , F Nazarov 2
Affiliation  

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

中文翻译:

关于威斯勒关于汉明立方体的猜想 I

令 $1\leq p \leq q <\infty$,并令 $w \in \mathbb{C}$。Weissler 推测 Hermite 算子 $e^{w\Delta}$ 在汉明立方体 $\{-1,1\}^ 上作为从 $L^{p}$ 到 $L^{q}$ 的算子有界{n}$ 具有独立于 $n$ 的范数边界当且仅当 \begin{align*} |p-2-e^{2w}(q-2)|\leq p-|e^{2w}| q. \end{align*} 在所有情况下,Bonami (1970)、Beckner (1975) 和 Weissler (1979) 都证明了这一点,除了 $2
更新日期:2020-12-26
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