Hypercontractivity of the semigroup of the fractional Laplacianon the n-sphere

摘要

For 1 < p <= q we show that the Poisson semigroup e(-t root-Delta) on the n-sphere is hypercontractive from L-p to L-q in dimensions n <= 3 if and only if e(-t root n) <= root p-1/q-1. We also show that the equivalence fails in large dimensions. (C) 2021 Elsevier Inc. All rights reserved.