Logarithmic Sobolev trace inequality

A logarithmic Sobolev trace inequality is derived. Bounds on the best constant for this inequality from above and below are investigated using the sharp Sobolev inequality and the sharp logarithmic Sobolev inequality.

     

Poincaré and Logarithmic Sobolev Inequalities for Nearly Radial Measures

Acta Mathematica Sinica, English Series - Poincaré inequality has been studied by Bobkov for radial measures, but few are known about the logarithmic Sobolev inequality in the radial case. We...     

Asymptotic Estimates for Gagliardo–Nirenberg Embedding Constants

Sharp asymptotic information is determined for the Gagliardo–Nirenberg embedding constants in high dimension. This analysis is motivated by the earlier observation that the logarithmic Sobolev inequality controls the Nash inequality. Moreover, one sees here that Hardy's inequality can be interpreted as the asymptotic limit of the logarithmic Sobolev inequality.     

A Generalization of the Logarithmic Gross–Sobolev Inequality

Doklady Mathematics - A sharp integral inequality is proved that is used to derive a Sobolev interpolation inequality. A generalization of the logarithmic Sobolev inequality is proposed based on...     

A new modified logarithmic Sobolev inequality for Poisson point processes and several applications

By means of the martingale representation, we establish a new modified logarithmic Sobolev inequality, which covers the previous modified logarithmic Sobolev inequalities of Bobkov-Ledoux and the L ¹-logarithmic Sobolev inequality obtained in our previous work. From it we derive several sharp deviation inequalities of Talagrand's type, by following the powerful Herbst method developed recently by Ledoux and al. Moreover this new modified logarithmic Sobolev inequality is transported on the discontinuous path space with respect to the law of a Lévy process. Received: 16 June 1999 / Revised version: 13 March 2000 / Published online: 12 October 2000     

A general two-scale criteria for logarithmic Sobolev inequalities

We present a general criteria to prove that a probability measure satisfies a logarithmic Sobolev inequality, knowing that some of its marginals and associated conditional laws satisfy a logarithmic Sobolev inequality. This is a generalization of a result by N. Grunewald et al. [N. Grunewald, F. Otto, C. Villani, M.G. Westdickenberg, A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit, Ann. Inst. H. Poincaré Probab. Statist., in press].     

Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator

We establish, by simple semigroup arguments, a Lévy-Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian, isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature.Oblatum 19-VI-1995     

Modified logarithmic Sobolev inequalities and transportation inequalities

We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type exp (−|x|^α) or exp (−|x|^α log ^β(2+|x|)) (α ∈]1,2[ and β ∈ ℝ) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities. Send offprint requests to: Ivan Gentil     

Rearrangement and the weighted logarithmic Sobolev inequality

Here we consider some weighted logarithmic Sobolev inequalities which can be used in the theory of singular Riemannian manifolds. We give the necessary and sufficient conditions such that the 1-dimension weighted logarithmic Sobolev inequality is true and obtain a new estimate on the entropy.     

The logarithmic Sobolev inequality for the Wasserstein diffusion

We prove that the Dirichlet form associated with the Wasserstein diffusion on the set of all probability measures on the unit interval, introduced in von Renesse and Sturm (Entropic measure and Wasserstein diffusion. Ann Probab, 2008) satisfies a logarithmic Sobolev inequality. This implies hypercontractivity of the associated transition semigroup. We also study functional inequalities for related diffusion processes.     

The Logarithmic Sobolev Inequality for a Submanifold in Manifolds with Nonnegative Sectional Curvature

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 codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of Brendle with Euclidean setting.     

Sobolev type inequalities for general symmetric forms

A general version of the Sobolev type inequality, including both the classical Sobolev inequality and the logarithmic Sobolev one, is studied for general symmetric forms by using isoperimetric constants. Some necessary and sufficient conditions are presented as results. The main results are illustrated by two examples of birth-death processes.

     

Improved interpolation inequalities,relative entropy and fast diffusion equations

We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolev?s inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.     

Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator

 We establish, by simple semigroup arguments, a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature. Oblatum 19-VI-1995     

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.      

A Sobolev Interpolation Inequality and a Gross-Sobolev Logarithmic Inequality

Mathematical Notes - A sharp integral inequality is proved and used to obtain a Sobolev interpolation inequality. Further, a new proof of a Gross-Sobolev logarithmic inequality is constructed on...     

Ultracontractive bounds on Hamilton-Jacobi solutions

Following the equivalence between logarithmic Sobolev inequality, hypercontractivity of the heat semigroup showed by Gross and hypercontractivity of Hamilton-Jacobi equations, we prove, like the Varopoulos theorem, the equivalence between Euclidean-type Sobolev inequality and an ultracontractive control of the Hamilton-Jacobi equations. We obtain also ultracontractive estimations under general Sobolev inequality which imply in the particular case of a probability measure, transportation inequalities.     

A sharp Sobolev trace inequality for the fractional-order derivatives

We establish a sharp Sobolev trace inequality for the fractional-order derivatives. As a close connection with this best estimate, we show a fractional-order logarithmic Sobolev trace inequality with the asymptotically optimal constant, but also sharpen the Poincaré embedding for the conformal invariant energy and BMO spaces.     

General Sobolev type inequalities for symmetric forms

A general Sobolev type inequality is introduced and studied for general symmetric forms by defining a new type of Cheeger's isoperimetric constant. Finally, concentration of measure for the Lp type logarithmic Sobolev inequality is presented.     

Time-dependent Sobolev inequality along the Ricci flow

In this article, we get a time-dependent Sobolev inequality along the Ricci flow in a more general situation than those in Zhang (A uniform Sobolev inequality under Ricci flow. Int Math Res Not IMRN 2007, no 17, Art ID rnm056, 17 pp), Ye (The logarithmic Sobolev inequality along the Ricci flow. arXiv:0707.2424v2) and Hsu (Uniform Sobolev inequalities for manifolds evolving by Ricci flow. arXiv:0708.0893v1) which also generalizes the results of them. As an application of the time-dependent Sobolev inequality, we get a growth of the ratio of non-collapsing along immortal solutions of Ricci flow.