A supersolutions perspective on hypercontractivity
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...
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| Published in | Annali di matematica pura ed applicata Vol. 199; no. 5; pp. 2105 - 2116 |
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| Main Authors | , , , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
2020
Springer Nature B.V |
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| Online Access | Get full text |
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| 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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| AbstractList | 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. |
| Author | Bennett, Jonathan Shiraki, Shobu Matsuura, Kosuke Machihara, Shuji Bez, Neal Aoki, Yosuke |
| Author_xml | – sequence: 1 givenname: Yosuke surname: Aoki fullname: Aoki, Yosuke organization: Department of Mathematics, Saitama University – sequence: 2 givenname: Jonathan surname: Bennett fullname: Bennett, Jonathan organization: School of Mathematics, University of Birmingham – sequence: 3 givenname: Neal surname: Bez fullname: Bez, Neal email: nealbez@mail.saitama-u.ac.jp organization: Department of Mathematics, Saitama University – sequence: 4 givenname: Shuji surname: Machihara fullname: Machihara, Shuji organization: Department of Mathematics, Saitama University – sequence: 5 givenname: Kosuke surname: Matsuura fullname: Matsuura, Kosuke organization: Department of Mathematics, Saitama University – sequence: 6 givenname: Shobu surname: Shiraki fullname: Shiraki, Shobu organization: Department of Mathematics, Saitama University |
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| Copyright | Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020. |
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| DOI | 10.1007/s10231-020-00958-7 |
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| Keywords | Hypercontractivity Supersolutions Markov semigroup 47D07 |
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| References | BakryDGentilILedouxMAnalysis and Geometry of Markov Diffusion Operators2014BerlinSpringer10.1007/978-3-319-00227-9 Barthe, F.: The Brunn–Minkowski theorem and related geometric and functional inequalities. In: International Congress of Mathematicians. Eur. Math. Soc., Zürich, vol. 2, pp. 1529–1546 (2006) BakryDÉmeryMDiffusions hypercontractives, Séminaire de Probabilités XIX1985SpringerLecture Notes in Math177206 Ledoux, M.: Remarks on Gaussian noise stability, Brascamp–Lieb and Slepian inequalities. In: Geometric aspects of functional analysis, Lecture Notes in Math. vol. 2116, pp. 309–333. Springer, Berlin (2014) Ledoux, M.: Heat flows, geometric and functional inequalities. In: Proceedings of the International Congress of Mathematicians, vol. 4, pp. 117–135. Seoul (2014) NelsonEThe free Markov fieldJ. Funct. Anal.19731221122710.1016/0022-1236(73)90025-6 GrossLLogarithmic Sobolev inequalitiesAm. J. Math.1975971061108342024910.2307/2373688 BennettJCarberyAChristMTaoTThe Brascamp–Lieb inequalities: finiteness, structure and extremalsGeom. Funct. Anal.20071713431415237749310.1007/s00039-007-0619-6 Hu, Y.: A unified approach to several inequalities for Gaussian and diffusion measures, Séminaire de Probabilités XXXIV. Lecture Notes in Math. vol. 1729, pp. 329–335. Springer, Berlin (2000) Davies, E.B., Gross, L., Simon, B.: Hypercontractivity: a bibliographic review. In: Ideas and Methods in Quantum and Statistical Physics (Oslo, 1988), pp. 370–389 (1992) BonamiAÉtude des coefficients Fourier des fonctiones de Lp(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p(G)$$\end{document}Ann. Inst. Fourier19702033540210.5802/aif.357 BorellCPositivity improving operators and hypercontractivityMath. Z.198218022523466169910.1007/BF01318906 BakryDBolleyFGentilIDimension dependent hypercontractivity for Gaussian kernelsProbab. Theory Relat. Fields2012154845874300056410.1007/s00440-011-0387-y Bennett, J.: Aspects of multilinear harmonic analysis related to transversality, Harmonic analysis and partial differential equations, Contemp. Math., Amer. Math. Soc., vol. 612 pp. 1–28 Providence, RI (2014) HuYAnalysis on Gaussian Spaces2017SingaporeWorld Scientific Publishing Co.1386.60005 LedouxMThe geometry of Markov diffusion generatorsAnn. Fac. Sci. Toulouse Math.20009305366181380410.5802/afst.962 Gross, L.: Hypercontractivity, logarithmic Sobolev inequalities, and applications: a survey of surveys. In: Diffusion, quantum theory, and radically elementary mathematics, Math. Notes vol. 47, pp. 45–73. Princeton University Press, Princeton (2006) Tao, T.: Sharp bounds for multilinear curved Kakeya, restriction and oscillatory integral estimates away from the endpoint, arXiv:1907.11342 BakryDTransformations de Riesz pour les semigroupes symétriques, Séminaire de Probabilités XIX1985SpringerLecture Notes in Math145174 CarlenEALiebEHLossMA sharp analog of Young’s inequality on SN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^N$$\end{document} and related entropy inequalitiesJ. Geom. Anal.200414487520207716210.1007/BF02922101 MatkowskiJThe converse of the Hölder inequality and its generalizationsStudia Math.1994109171182126977410.4064/sm-109-2-171-182 BennettJBezNGenerating monotone quantities for the heat equationJ. Reine Angew. Math.20197563763402644810.1515/crelle-2017-0025 BakryDÉtude des transformations de Riesz dans les variétés Riemanniennes à courbure de Ricci minorée, Séminaire de Probabilités XXI1987SpringerLecture Notes in Math137172 BennettJBezNClosure properties of solutions to heat inequalitiesJ. Geom. Anal.200919584600249656710.1007/s12220-009-9070-2 |
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| Title | A supersolutions perspective on hypercontractivity |
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