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.     

On the equivalence of heat kernel estimates and logarithmic Sobolev inequalities for the Hodge Laplacian

In this paper we consider the Hodge Laplacian on differential k-forms over smooth open manifolds MN, not necessarily compact. We find sufficient conditions under which the existence of a family of logarithmic Sobolev inequalities for the Hodge Laplacian is equivalent to the ultracontractivity of its heat operator.We will also show how to obtain a logarithmic Sobolev inequality for the Hodge Laplacian when there exists one for the Laplacian on functions. In the particular case of Ricci curvature bounded below, we use the Gaussian type bound for the heat kernel of the Laplacian on functions in order to obtain a similar Gaussian type bound for the heat kernel of the Hodge Laplacian. This is done via logarithmic Sobolev inequalities and under the additional assumption that the volume of balls of radius one is uniformly bounded below.     

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.     

On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds

We study the second best constant problem for logarithmic Sobolev inequalities on complete Riemannian manifolds and investigate its relationship with optimal heat kernel bounds and the existence of extremal functions.     

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.     

Log-Sobolev inequality on non-convex Riemannian manifolds

Counterexamples are constructed to show that when the second fundamental form of the boundary is bounded below by a negative constant, any curvature lower bound is not enough to imply the log-Sobolev inequality. This indicates that in the study of functional inequalities on non-convex manifolds, the concavity of the boundary cannot be compensated by the positivity of the curvature. Next, when the boundary is merely concave on a bounded domain, a criterion on the log-Sobolev inequality known for convex manifolds is proved. Finally, when the concave part of the boundary is unbounded, a Sobolev inequality for a weighted volume measure is established, which implies an explicit sufficient condition for the log-Sobolev inequality to hold on non-convex manifolds.     

A new criterion for the logarithmic Sobolev inequality and two applications

We give a criterion for the logarithmic Sobolev inequality (LSI) on the product space X₁×?×XN. We have in mind an N-site lattice, unbounded continuous spin variables, and Glauber dynamics. The interactions are described by the Hamiltonian H of the Gibbs measure. The criterion for LSI is formulated in terms of the LSI constants of the single-site conditional measures and the size of the off-diagonal entries of the Hessian of H. It is optimal for Gaussians with positive covariance matrix. To illustrate, we give two applications: one with weak interactions and one with strong interactions and a decay of correlations condition.     

Extremal maps in Sobolev type inequalities: Some remarks

In this work we make some observations on the existence of extremal maps for sharp L²-Riemannian Sobolev type inequalities as Nash and logarithmic Sobolev ones. Among other results, we prove also that there exist smooth compact Riemannian manifolds with scalar curvature changing signal on which there exist extremal maps.     

Stochastic partial differential equations on manifolds,I

We present a general framework of treating SPDEs on manifolds by adapting the notion of well-weighted Sobolev spaces from [1]. Using this we extend the theory of SPDEs to the case of manifolds.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below

Using the coupling by parallel translation, along with Girsanov's theorem, a new version of a dimension-free Harnack inequality is established for diffusion semigroups on Riemannian manifolds with Ricci curvature bounded below by , where c>0 is a constant and ρo is the Riemannian distance function to a fixed point o on the manifold. As an application, in the symmetric case, a Li-Yau type heat kernel bound is presented for such semigroups.     

Logarithmic Sobolev inequalities and the spectrum of Sturm-Liouville operators

We continue our study of the logarithmic Sobolev inequality of Gross and its connections with the spectrum of Sturm-Liouville operators, confining our attention to a finite interval and the circle. The analysis begins with a certain variational problem, which turns out to be related to two other variational problems. One of these gives the connection with bounds from below for the spectrum of Sturm-Liouville operators, while the other furnishes examples of logarithmic Sobolev inequalities with probability measures. The “best constant” in the latter version turns out to be related to the mass gap of certain Sturm-Liouville operators. The extension of these results to higher-dimensional manifolds will appear in a subsequent paper.     

Harnack inequality on configuration spaces: The coupling approach and a unified treatment

In this paper, we establish the dimension-free Harnack inequality on configuration spaces by using the coupling argument. Furthermore, a unified treatment is also used to prove the equivalence between the Harnack inequality on configuration space and that on the corresponding base space under a very mild condition.     

Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution

Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincaré inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper bounds on spectral gaps. Received: 10 June 1996 / In revised form: 9 August 1996     

Differential calculus on path and loop spaces II. Irreducibility of Dirichlet forms on loop spaces

We prove the irreducibility of a Dirichlet form on the based loop space on a compact Riemannian manifold. The Dirichlet form is defined by the gradient operator due to Driver and Léandre. We also prove the uniqueness of the ground states of the Schrödinger operator for which the Dirichlet form satisfies the logarithmic Sobolev inequality. This is an extension of the corresponding results of Gross ([28], [29]) to the case of general compact Riemannian manifolds.     

Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential

In this paper, we study the Dirichlet problem for a class of infinitely degenerate nonlinear elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy's inequality, the existence and regularity of multiple nontrivial solutions have been proved.     

Convex Sobolev inequalities and spectral gap

This Note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux, and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities including in the limit case corresponding to the logarithmic Sobolev inequalities. To cite this article: J.-P. Bartier, J. Dolbeault, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

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     

Some functional inequalities on non-reversible Finsler manifolds

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by Ohta and Sturm. Following the approach of the \(\Gamma \)-calculus of Bakry et al (2014), we show the dimensional versions of the Poincaré–Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti and Mondino (2015) in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.     

Weighted Sobolev Inequalities and Ricci Flat Manifolds

In this paper, we prove a weighted Sobolev inequality and a Hardy inequality on manifolds with nonnegative Ricci curvature satisfying a reverse volume doubling condition. It enables us to obtain rigidity results for Ricci flat manifolds. Received: November 2006, Revision: April 2007, Accepted: April 2007     

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.