Hypercontractivity of the semigroup of the fractional laplacian on the n-sphere

Abstract

For $1<p\le q$ we show that the Poisson semigroup $e^{-t\sqrt{-\Delta }}$ on the $n$-sphere is hypercontractive from $L^p$ to $L^q$ in dimensions $n\le 3$ if and only if $e^{-t\sqrt{n}}\le \sqrt{\frac{p-1}{q-1}}$. We also show that the equivalence fails in large dimensions.