On Poisson semigroup hypercontractivity for higher-dimensional spheres

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 $1<p\leq q<\infty$ 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 $\geq1$.