Abstract
The purpose of this article is to expose an algebraic closure property of supersolutions to certain diffusion equations. This closure property quickly gives rise to a monotone quantity which generates a hypercontractivity inequality. Our abstract argument applies to a general Markov semigroup whose generator is a diffusion and satisfies a curvature condition.
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Notes
In fact, if the diffusion property holds, then the curvature condition is equivalent to (8).
The identity \(\varGamma (\psi (u))= \psi '(u)^2\varGamma (u)\) holds for smooth \(\psi \) as a result of the diffusion property (6) and thus (13) holds in a formal sense by taking \(\psi (u) = u^{1/p}\). In the case of the Ornstein–Uhlenbeck semigroup, (13) may be rigorously verified by direct calculations.
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We would like to thank the anonymous referee for their helpful comments which led to numerous improvements in the paper.
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The first, fifth and sixth authors were supported by JSPS Grant-in-Aid for Young Scientists A [Grant Number 16H05995], the second author was partially supported by ERC Grant 307617, the third author was supported by JSPS Grant-in-Aid for Young Scientists A [Grant Number 16H05995] and JSPS Grant-in-Aid for Scientific Research B [Grant Number 19H01796], and the fourth author was supported by JSPS Grant-in-Aid for Scientific Research C [Grant Number 16K05191].
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Aoki, Y., Bennett, J., Bez, N. et al. A supersolutions perspective on hypercontractivity. Annali di Matematica 199, 2105–2116 (2020). https://doi.org/10.1007/s10231-020-00958-7
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DOI: https://doi.org/10.1007/s10231-020-00958-7