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arXiv · 2101.06209

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

Rupert L. Frank·Paata Ivanisvili
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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.

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Rupert L. Frank, Paata Ivanisvili. 2021-01-15. Hypercontractivity of the semigroup of the fractional laplacian on the n-sphere. https://arxiv.org/abs/2101.06209

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