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 \(\geq1\).