Geodesic
On Poisson semigroup hypercontractivity for higher-dimensional spheres
Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 3, pp. 100-103 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this note we consider a variant of a question of Mueller and Weissler raised in 1982, thereby complementing a classical result of Beckner on Stein's conjecture and a recent result of Frank and Ivanisvili. More precisely, we show that, for $1$ and $n\geq1$, the Poisson semigroup $e^{-t\sqrt{-\Delta-(n-1)\mathbb{P}}}$ on the $n$-sphere is hypercontractive from $L^p$ to $L^q$ if and only if $e^{-t}\leq\sqrt{(p-1)/(q-1)}$; here $\Delta$ is the Laplace–Beltrami operator on the $n$-sphere and $\mathbb{P}$ is the projection operator onto spherical harmonics of degree $\geq1$.
Keywords: hypercontractivity, higher-dimensional sphere.
Mots-clés : Poisson semigroup 
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Yi C. Huang. On Poisson semigroup hypercontractivity for higher-dimensional spheres. Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 3, pp. 100-103. http://geodesic.mathdoc.fr/item/FAA_2022_56_3_a7/

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