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null (Ed.)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 in [ 1], [ 2], and [ 17] 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.more » « less
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null (Ed.)A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier’s inequality on the discrete cube.more » « less
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