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The Equivalence of Hypercontractivity and logarithmic sobolev inequality for q(-1≤q≤1)-Ornstein-Uhlenbeck Semigroup

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Chinese Annals of Mathematics,Series B 2020年第4期41卷 615-626页

作者： Lunchuan ZHANG School of Mathematics Renmin University of ChinaBeijing 100086China

In this paper the author proves the equivalence of hypercontractivity and logarithmic sobolev inequality for q-Ornstein-Uhlenbeck semigroup Ut(q)=Γq(e-tI)(-1≤q≤1),whereΓq is a q-Gaussian functor.

关键词： q-Ornstein-Uhlenbeck semigroup Hypercontractivity logarithmic sobolev inequality

The logarithmic sobolev inequality for a Submanifold in Manifolds with Nonnegative Sectional Curvature

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 487-496页

作者： Chengyang YI Yu ZHENG School of Mathematical Sciences East China Normal UniversityShanghai 200241China

The authors prove a sharp logarithmic sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifolds with nonnegative sectional curvature of arbitrary dimension and *** the Michael-Simon sobolev inequality,this inequality includes a term involving the mean *** extends a recent result of Brendle with Euclidean setting.

关键词： logarithmic sobolev inequality Nonnegative sectional curvature Submanifold

THE logarithmic sobolev inequality FOR A SUBMANIFOLD IN MANIFOLDS WITH ASYMPTOTICALLY NONNEGATIVE SECTIONAL CURVATURE

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Acta Mathematica Scientia 2024年第1期44卷 189-194页

作者：东瑜昕林和子陆琳根 School of Mathematical Sciences Fudan UniversityShanghai20043China School of Mathematics and Statistics&Laboratory of Analytical Mathematics and Applications(Ministry of Education)&FJKLMAA Fujian Normal UniversityFuzhou350108China

In this note,we prove a logarithmic sobolev inequality which holds for compact submanifolds without a boundary in manifolds with asymptotically nonnegative sectional *** the Michale-Simon sobolev inequality,this inequ... 详细信息

关键词： asymptotically nonnegative sectional curvature logarithmic sobolev inequality ABP method

The logarithmic sobolev inequality Along the Ricci Flow:The Caseλ0(g0)=0

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Communications in Mathematics and Statistics 2014年第3期2卷 363-368页

作者： Rugang Ye Department of Mathematics University of CaliforniaSanta BarbaraUSA

A uniform logarithmic sobolev inequality,a uniform sobolev inequality and a uniformκ-noncollapsing estimate along the Ricci flow are established in the situation that a certain smallest eigenvalue for the initial met... 详细信息

关键词： Uniform logarithmic sobolev inequality sobolev inequality Ricci flow Eigenvalue

On Supercontractivity for Markov Semigroups

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Acta Mathematica Sinica,English Series 2007年第5期23卷 905-914页

作者： Yong Hua MAO School of Mathematical Science Beijing Normal University Beijing 100875 P. R. China

In this paper, we study the supercontractivity for Maxkov semigroups and obtain some sufficient and necessary conditions; especially explicit formulae are obtained for birth-death process and diffusion on the line. Sufficient conditions and necessary conditions in terms of isoperimetric inequalities are also presented. Moreover, we prove that the supercontractivity is equivalent to the compact embedding of sobolev space into an Orlicz space.

关键词： Markov semigroup logarithmic sobolev inequality supercontractivity compact embedding

Sharp Interpolation Inequalities on the Sphere:New Methods and Consequences

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Chinese Annals of Mathematics,Series B 2013年第1期34卷 99-112页

作者： Jean DOLBEAULT Maria J. ESTEBAN Michal KOWALCZYK Michael LOSS Ceremade Université Paris DauphinePlace de Lattre de Tassigny Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS) Universidad de Chile Skiles Building Georgia Institute of Technology

This paper is devoted to various considerations on a family of sharp interpo- lation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincare, logarithmic sobolev and critical sobolev （Onofri in dimension two） inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.

关键词： sobolev inequality Interpolation Gagliardo-Nirenberg inequality logarithmic sobolev inequality Heat equation

Talagrand's T_2-Transportation inequality and Log-sobolev inequality for Dissipative SPDEs and Applications to Reaction-Diffusion Equations

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Chinese Annals of Mathematics,Series B 2006年第3期27卷 243-262页

作者： Liming WU Zhengliang ZHANG Laboratoire de Mathématiques Appliquées CNRS-UMR 6620 Université Blaise Pascal 63177 Aubière France School of Mathematics and Statistics Wuhan University Wuhan 430072 China. School of Mathematics and Statistics Wuhan University Wuhan 430072 China.

We establish Talagrand＇s T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type＇s approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic sobolev inequalities with sharp constants. Applications to Reaction- Diffusion equations are provided.

关键词： Stochastic partial differential equations (SPDEs) logarithmic sobolev inequality Talagrand's transportation inequality Poincaré inequality

Poincaréand logarithmic sobolev Inequalities for Nearly Radial Measures

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Acta Mathematica Sinica,English Series 2022年第8期38卷 1377-1398页

作者： Patrick CATTIAUX Arnaud GUILLIN Li Ming WU Institut de Mathématiques de Toulouse CNRS UMR 5219UniversitéPaul Sabatier118 Route de NarbonneF-31062 Toulouse Cedex 09 Laboratoire de Mathématiques Blaise Pascal CNRS UMR 6620UniversitéClermont-AuvergneAvenue des LandaisF-63177 Aubiere

Poincaréinequality has been studied by Bobkov for radial measures,but few are known about the logarithmic sobolev inequality in the radial *** try to fill this gap here using different methods:Bobkov's argument and super-Poincaréinequalities,direct approach via L_(1)-logarithmic sobolev *** also give various examples where the obtained bounds are quite *** bounds obtained by Lee–Vempala in the log-concave bounded case are refined for radial measures.

关键词： Radial measure log-concave measure Poincaréinequality logarithmic sobolev inequality super-Poincaréinequality

Variational Formulas of Poincare-Type Inequalities in Banach Spaces of Functions on the Line

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Acta Mathematica Sinica,English Series 2002年第3期18卷 417-436页

作者： Mu Fa CHEN Department of Mathematics. Beijing Normal. University. Beijing 100875. P. R. China Department of Mathematics Beijing Normal University Beijing P. R. China

Motivated from the study on logarithmic sobolev. Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic sobolev constant is carefully examined.

关键词： Variational formula Poincare inequality logarithmic sobolev inequality Orliez space