scholarly journals CONCENTRATION INEQUALITIES FOR GIBBS MEASURES

2011 ◽  
Vol 14 (01) ◽  
pp. 79-104
Author(s):  
IOANNIS PAPAGEORGIOU
Keyword(s):  
Gibbs Measures ◽  
Type Inequality ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Infinite Dimensional ◽  
Unbounded Spin Systems ◽  
The One ◽  
Concentration Properties ◽  
Weaker Conditions ◽  
Unbounded Spin

We are interested in Sobolev type inequalities and their relationship with concentration properties on higher dimensions. We consider unbounded spin systems on the d-dimensional lattice with interactions that increase slower than a quadratic. At first we assume that the one-site measure satisfies a modified log-Sobolev inequality with a constant uniformly on the boundary conditions and we determine conditions so that the infinite-dimensional Gibbs measure satisfies a concentration as well as a Talagrand type inequality, similar to the ones obtained by Barthe and Roberto6 for the product measure. Then a modified log-Sobolev type concentration property is obtained under weaker conditions referring to the log-Sobolev inequalities for the boundary free measure.

scholarly journals Poincaré and Log–Sobolev Inequalities for Mixtures

Entropy ◽  
2019 ◽  
Vol 21 (1) ◽  
pp. 89 ◽  
Author(s):  
André Schlichting
Keyword(s):  
Blow Up ◽  
Functional Inequality ◽  
Multidimensional Case ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Mixture Ratio ◽  
Mixture Parameter ◽  
Sobolev Constants ◽  
The One ◽  
Poincaré Constant

This work studies mixtures of probability measures on R n and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the χ 2 -distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter, whereas the log–Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.

scholarly journals Evaluation of the One-Dimensional Lp Sobolev Type Inequality

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 296
Author(s):  
Kazuo Takemura ◽  
Yoshinori Kametaka
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Type Inequality ◽  
Sobolev Type ◽  
Sharp Constant ◽  
Order Ordinary Differential Equation ◽  
One Dimensional ◽  
The Green Function ◽  
The One ◽  
Sobolev Type Inequality

This study applies the extended L 2 Sobolev type inequality to the L p Sobolev type inequality using Hölder’s inequality. The sharp constant and best function of the L p Sobolev type inequality are found using a Green function for the nth order ordinary differential equation. The sharp constant is shown to be equal to the L p norm of the Green function and to the pth root of the value of the origin of the best function.

scholarly journals Entropy inequalities for unbounded spin systems

2002 ◽  
Vol 30 (4) ◽  
pp. 1959-1976 ◽  
Author(s):  
Paolo Dai Pra ◽  
Anna Maria Paganoni ◽  
Gustavo Posta
Keyword(s):  
Spin Systems ◽  
Unbounded Spin Systems ◽  
Entropy Inequalities ◽  
Unbounded Spin

scholarly journals New Sharp Gagliardo–Nirenberg–Sobolev Inequalities and an Improved Borell–Brascamp–Lieb Inequality

2018 ◽  
Vol 2020 (10) ◽  
pp. 3042-3083 ◽  
Author(s):  
François Bolley ◽  
Dario Cordero-Erausquin ◽  
Yasuhiro Fujita ◽  
Ivan Gentil ◽  
Arnaud Guillin
Keyword(s):  
Half Space ◽  
Minkowski Inequality ◽  
Point Of View ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Geometric Point ◽  
Functional Point

Abstract We propose a new Borell–Brascamp–Lieb inequality that leads to novel sharp Euclidean inequalities such as Gagliardo–Nirenberg–Sobolev inequalities in $ {\mathbb{R}}^n$ and in the half-space $ {\mathbb{R}}^n_+$. This gives a new bridge between the geometric point of view of the Brunn–Minkowski inequality and the functional point of view of the Sobolev-type inequalities. In this way we unify, simplify, and generalize results by S. Bobkov–M. Ledoux, M. del Pino–J. Dolbeault, and B. Nazaret.

Weighted Hardy–Littlewood–Sobolev inequalities on the upper half space

2016 ◽  
Vol 18 (05) ◽  
pp. 1550067 ◽  
Author(s):  
Jingbo Dou
Keyword(s):  
Half Space ◽  
Type Inequality ◽  
Boundary Term ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Biharmonic Operator ◽  
Trace Inequality ◽  
Hardy Type Inequality ◽  
Hardy Type

In this paper, we establish a weighted Hardy–Littlewood–Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. As an application, we can show a weighted Sobolev–Hardy trace inequality with [Formula: see text]-biharmonic operator.

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