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Hypercontractivity of the semigroup of the fractional laplacian on the n-sphere (2101.06209v1)
Published 15 Jan 2021 in math.CA and math.FA
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.