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

TL;DR

This paper investigates the hypercontractivity properties of the Poisson semigroup associated with the fractional Laplacian on the n-sphere, establishing dimension-dependent conditions for dimensions up to three.

Contribution

It provides a precise characterization of hypercontractivity for the Poisson semigroup on the sphere in low dimensions and shows the failure of this equivalence in higher dimensions.

Findings

Hypercontractivity holds for n ≤ 3 under specific exponential conditions.

The equivalence between hypercontractivity and the exponential condition fails in large dimensions.

The results delineate the dimension-dependent behavior of the fractional Laplacian semigroup.

Abstract

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