Ivanisvili, P.; Nazarov, F. On Weissler’s conjecture on the Hamming cube. I. (English) Zbl 1487.42069 Int. Math. Res. Not. 2022, No. 9, 6991-7020 (2022). Summary: Let \(1\leq p\leq q<\infty\) and let \(w\in \mathbb{C}\). F. B. Weissler [J. Funct. Anal. 32, 102–121 (1979; Zbl 0433.47023)] 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 \[ |p-2-e^{2w}(q-2)|\leq p-|e^{2w}|q. \] It was proved in [W. Beckner, Ann. Math. (2) 102, 159–182 (1975; Zbl 0338.42017)], [A. Bonami, Ann. Inst. Fourier 20, No. 2, 335–402 (1970; Zbl 0195.42501)], and [Weissler, loc. cit.] 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. Cited in 2 Documents MSC: 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 42B35 Function spaces arising in harmonic analysis 47D03 Groups and semigroups of linear operators Keywords:Fourier-Walsh series; Hermite operator; hypercontractivity Citations:Zbl 0338.42017; Zbl 0195.42501; Zbl 0433.47023 PDFBibTeX XMLCite \textit{P. Ivanisvili} and \textit{F. Nazarov}, Int. Math. Res. Not. 2022, No. 9, 6991--7020 (2022; Zbl 1487.42069) Full Text: DOI arXiv