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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

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

On the Mixed Pólya–Szegö Principle

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Acta Mathematica Sinica,English Series 2021年第5期37卷 753-767页

作者： Niu Fa FANG Jia Zu ZHOU Chern Institute of Mathematics Nankai UniversityTianjin 300071P.R.China School of Mathematics and Statistics Southwest UniversityChongqing 400715P.R.China College of Science Wuhan University of Science and TechnologyWuhan 430081P.R.China

In this paper,the mixed Pólya-Szegöprinciple is *** the mixed Pólya-Szegöprinciple,the mixed Morrey-sobolev inequality and some new analytic inequalities are obtained.

关键词： Pólya–Szegöprinciple mixed Pólya–Szegöprinciple sobolev inequality spherically symmetric rearrangement

IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS

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Acta Mathematica Scientia 2011年第4期31卷 1583-1590页

作者：罗光洲 School of Mathematics and Statistics Hubei Normal University

Motivated by the idea of M. Ledoux who brings out the connection between sobolev embeddings and heat kernel bounds, we prove an analogous result for Kohn’s sub-Laplacian on the Heisenberg type groups. The main result... 详细信息

关键词： Heisenberg type group heat kernel sobolev inequality Galiardo-Nirenberg inequality

Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations

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Science China Mathematics 2024年第3期67卷 557-576页

作者： Jiaxiang Wang Bin Zhou Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China School of Mathematical Sciences Peking UniversityBeijing 100871China

In this paper, we study the relations between trace inequalities(sobolev and Moser-Trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a general nonnegative Borel measure. We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.

关键词： complex Monge-Ampère equations plurisubharmonic functions sobolev inequality Moser-Trudinger inequality

The Logarithmic sobolev and sobolev Inequalities Along the Ricci Flow

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Communications in Mathematics and Statistics 2015年第1期3卷 1-36页

作者： Rugang Ye Department of Mathematics University of CaliforniaSanta BarbaraUSA

Based on Perelman’s entropy monotonicity,uniform logarithmic sobolev inequalities along the Ricci flow are *** uniform sobolev inequalities along theRicci floware derived via harmonic analysis of the integral transform of the relevant heat *** inequalities are fundamental analytic properties of the Ricci *** are also extended to the volume-normalized Ricci flow and the Kähler-Ricci flow.

关键词： sobolev inequality Logarithmic sobolev inequality Ricci flow Heat operator

Convergence of the Generalized Kähler-Ricci Flow

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Communications in Mathematics and Statistics 2015年第2期3卷 239-261页

作者： Jiawei Liu Yue Wang School of Mathematical Sciences University of Science and Technology of ChinaHefeiPeople’s Republic of China Department of Mathematics China Jiliang UniversityHangzhouPeople’s Republic of China

In this paper,we consider the convergence of the generalized Kähler-Ricci flow with semi-positive twisted formθon Kähler manifold *** give detailed proofs of the uniform sobolev inequality and some uniform estimates for the metric potential and the generalized Ricci potential along the *** assuming that there exists a generalized Kähler-Einstein metric,if the twisting formθis strictly positive at a point or M admits no nontrivial Hamiltonian holomorphic vector field,we prove that the generalized Kähler-Ricci flow must converge in C^(∞)topology to a generalized Kähler-Einstein metric exponentially fast,where we get the exponential decay without using the Futaki invariant.

关键词： Complex Monge-Ampère equation Generalized Kähler-Einstein metric sobolev inequality Moser-Trudinger type inequality

L_(p)-Minkowski问题周期解的存在性

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哈尔滨商业大学学报（自然科学版） 2023年第4期39卷 437-440页

作者：何瑞瑞梁载涛安徽理工大学数学与大数据学院安徽淮南232001

L_(p)-Minkowski问题是凸几何分析L_(p)-Brunn-Minkowski理论的核心,其实质是分析给定测度是否为凸体的L_(p)表面积测度问题.这个问题可以简化为二阶微分方程的周期解的存在性,L_(p)-Minkowski问题中周期解的存在性问题如下u″+u=h(t)/... 详细信息

L_(p)-Minkowski问题是凸几何分析L_(p)-Brunn-Minkowski理论的核心,其实质是分析给定测度是否为凸体的L_(p)表面积测度问题.这个问题可以简化为二阶微分方程的周期解的存在性,L_(p)-Minkowski问题中周期解的存在性问题如下u″+u=h(t)/u^(ρ),其中h>0是连续的周期函数,常数ρ=1-p.利用了二阶微分方程周期解存在的充分条件,通过建立的一个方程的周期解的存在性判据,再利用sobolev inequality证明了这个二阶微分方程周期解的存在性,得到在一定条件下周期解存在,这个方法在一定程度上扩大p的取值范围.最后给出一个例子,验证文中所得到的主要结果的可行性.

关键词：周期解 sobolev inequality L_(p)-Minkowski问题微分方程周期函数不等式

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