Best constants for Sobolev inequalities for higher order fractional derivatives

We obtain sharp constants for Sobolev inequalities for higher order fractional derivatives. As an application, we give a new proof of a theorem of W. Beckner concerning conformally invariant higher-order differential operators on the sphere.     

On Opial-type inequalities with higher order partial derivatives

In the present paper we establish some new Opial-type inequalities involving higher order partial derivatives. Our results in special cases yield some of the recent results on Opial’s inequality and provide new estimates on inequalities of this type. As an application, we prove the uniqueness of the solution of the initial value problem involving higher order partial differential equations.     

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     

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.     

Analytic inequalities,isoperimetric inequalities and logarithmic Sobolev inequalities

There is a simple equivalence between isoperimetric inequalities in Riemannian manifolds and certain analytic inequalities on the same manifold, more extensive than the familiar equivalence of the classical isoperimetric inequality in Euclidean space and the associated Sobolev inequality. By an isoperimetric inequality in this connection we mean any inequality involving the Riemannian volume and Riemannian surface measure of a subset α and its boundary, respectively. We exploit the equivalence to give log-Sobolev inequalities for Riemannian manifolds. Some applications to Schrödinger equations are also given.     

On logarithmic Sobolev inequalities for continuous time random walks on graphs

We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ ^d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals such as the graph distance in this setting. Received: 6 April 1998 / Revised version: 15 March 1999 / Published on line: 14 February 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].     

Critical dimensions and higher order Sobolev inequalities with remainder terms

Pucci and Serrin [21] conjecture that certain space dimensions behave 'critically' in a semilinear polyharmonic eigenvalue problem. Up to now only a considerably weakened version of this conjecture could be shown. We prove that exactly in these dimensions an embedding inequality for higher order Sobolev spaces on bounded domains with an optimal embedding constant may be improved by adding a 'linear' remainder term, thereby giving further evidence to the conjecture of Pucci and Serrin from a functional analytic point of view. Thanks to Brezis-Lieb [5] this result is already known for the space in dimension n=3; we extend it to the spaces (K>1) in the 'presumably' critical dimensions. Crucial tools are positivity results and a decomposition method with respect to dual cones. Received June 1999     

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.      

General logarithmic Sobolev inequalities and Orlicz imbeddings

We obtain, for a large class of measures μ, general inequalities of the form $\text{∫R}{}^{\mathrm{n}}\text{|u|}{}^{\mathrm{p}}\text{A(}\text{log}{}^1\text{|u|) dμ ? K(6u : W}{}^{\mathrm{m,p}}\text{(R}{}^{\mathrm{n}}\text{,dμ)6}{}^{\mathrm{p}}\text{+ 6 u 6}{}^{\mathrm{p}}\text{A(}\text{log}{}^1\text{6 u 6))}$, where $\text{6u6 = 6 u: L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{n}}\text{,dμ)6}{}^{\mathrm{p}}\text{,}\text{log}{}^1\text{t =}\text{max}\text{{1,}\text{log}\text{t}}$, and the function A depends in an appropriate way on μ. Our results extend similar results obtained by Rosen for the case p = 2, A(t) = t^s. We also investigate some implications of these inequalities for the imbedding of Sobolev spaces into Orlicz spaces.     

Clark formula and logarithmic Sobolev inequalities for Bernoulli measures

Using a Clark formula for the predictable representation of random variables in discrete time and adapting the method presented in [Electron. Commun. Probab. 2 (1997) 71–81] in the Brownian case, we obtain a proof of modified and L¹ logarithmic Sobolev inequalities for Bernoulli measures. We also prove a bound that improves these inequalities as well as the optimal constant inequality of [J. Funct. Anal. 156 (2) (1998) 347–365]. To cite this article: F. Gao, N. Privault, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Limit of fractional power Sobolev inequalities

We derive the Moser–Trudinger–Onofri inequalities on the 2-sphere and the 4-sphere as the limiting cases of the fractional power Sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by Thomas P. Branson.     

From concentration to logarithmic Sobolev and Poincaré inequalities

We give a new proof of the fact that Gaussian concentration implies the logarithmic Sobolev inequality when the curvature is bounded from below, and also that exponential concentration implies Poincaré inequality under null curvature condition. Our proof holds on non-smooth structures, such as length spaces, and provides a universal control of the constants. We also give a new proof of the equivalence between dimension free Gaussian concentration and Talagrand's transport inequality.     

Sharp Sobolev inequalities of second order

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and ₂ ² (M) be the Sobolev space consisting of functions in L²(M) whose derivatives up to the order two are also in L²(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H ₂ ² (M),

On refined Hardy-type inequalities with fractional integrals and fractional derivatives

The aim of this paper is to present new more general Hardy-type inequalities for different kinds of fractional integrals and fractional derivatives.     

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.