A supersolutions perspective on hypercontractivity
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A supersolutions perspective on hypercontractivity Yosuke Aoki1 · Jonathan Bennett2 · Neal Bez1 · Shuji Machihara1 · Kosuke Matsuura1 · Shobu Shiraki1 Received: 15 October 2019 / Accepted: 8 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Hypercontractivity · Markov semigroup · Supersolutions Mathematics Subject Classification 47D07
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]. * Neal Bez [email protected]‑u.ac.jp Yosuke Aoki [email protected]‑u.ac.jp Jonathan Bennett [email protected] Shuji Machihara [email protected]‑u.ac.jp Kosuke Matsuura [email protected]‑u.ac.jp Shobu Shiraki [email protected]‑u.ac.jp 1
Department of Mathematics, Saitama University, Saitama 338‑8570, Japan
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School of Mathematics, University of Birmingham, Birmingham B15 2TT, England
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1 Introduction The semigroup method has been extremely fruitful in generating and furnishing a greater depth of understanding of a wide variety of fundamental inequalities across mathematics. The method rests on an astute choice of a monotone time-dependent functional which generates the desired inequality by comparing the functional at two distinct times. Typically, the functional involves the time evolution of the input functions under appropriately chosen diffusion flows. For greater perspective on the pervasiveness of the method in geometric analysis and harmonic analysis, the reader may consult, for example, [1, 2] and [3], as well as [4] for a very recent application of the technique in the multilinear theory of the restriction and Kakeya problems. Recently in [5], an abstract framework for generating monotone quantities for nonnegative solutions of certain diffusion equations was exposed. Underpinning this framework is the observation that families of supersolutions to certain diffusion equations obey a variety of algebraic closure properties. Its potency stems from the fact that supersolutions satisfy many more closure properties than families of bona fide solutions and this makes it possible to sequentially combine several algebraic closure properties to generate a wide variety of monotone quantities (we refer t
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