The sharp quantitative Sobolev inequality for functions of bounded variation

The classical Sobolev embedding theorem of the space of functions of bounded variation BV(Rn) into Ln^′(Rn) is proved in a sharp quantitative form.     

A quantitative Sobolev inequality in BV

A quantitative version of the standard Sobolev inequality, with sharp constant, for functions u in W1,1(Rn) (or BV(Rn)) is established in terms of a distance of u from the manifold of all multiples of characteristic functions of balls. Inequalities involving non-Euclidean norms of the gradient are discussed as well.     

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.     

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.

     

Mass Transport and Variants of the Logarithmic Sobolev Inequality

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting. The research of A.V. Kolesnikov was supported by RFBR 07-01-00536, DFG Grant 436 RUS 113/343/0 and GFEN 06-01-39003.     

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.     

Pointwise symmetrization inequalities for Sobolev functions and applications

We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations.     

The best constant of Sobolev inequality on a bounded interval

The best constants Cm,j of Sobolev embedding of Hm(0,a) into Cj[0,a](0?j?m−1) are obtained. Especially, when a=∞, these constants can be represented in a closed form.     

Lipschitz constants and logarithmic Sobolev inequality

We revisit two results of  [3]; they are a logarithmic Sobolev inequality on Rⁿ${\mathbb{R}}^n$ with Lipschitz constants and an expression of Lipschitz constants as the limit of a functional by the entropy. We have two goals in this paper. The first goal is to clarify when the strict inequality holds in this inequality. The second goal is to investigate the asymptotic behavior of this functional by the Abelian and Tauberian theorems of Laplace transforms.     

A Sobolev inequality with reciprocal weights

We give a Sobolev inequality with the weight K(x)$K\left(x\right)$ belonging to the class _A2∩_Gn$A_2\cap G_n$ for the function |u|^t${\left|u\right|}^t$ and the weight K(x)⁻¹$K{\left(x\right)}^{-1}$ for |∇u|²${\left|\nabla u\right|}^2$. The constant in the relevant inequality is seen to depend on the _Gn$G_n$ and _A2$A_2$ constants of the weight.     

Logarithmic Sobolev inequality for symmetric forms

Some estimates of logarithmic Sobolev constant for general symmetric forms are obtained in terms of new Cheeger’s constants. The estimates can be sharp in some sense.     

A proof of Hessian Sobolev inequality

In this paper, taking the Hessian Sobolev inequality (0<p≤k) (X.-J. Wang, 1994 [2]) as the starting point, we give a proof of the Hessian Sobolev inequality when k<p≤k^∗, where k^∗ is the critical Sobolev embedding index of k-Hessian type. We also prove that k^∗ is optimal by one-dimensional Hardy’s inequality.     

Bounds on the deficit in the logarithmic Sobolev inequality

The deficit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and information-theoretic distances.     

A sharp exponential inequality for Lorentz-Sobolev spaces on bounded domains

This paper generalizes an inequality of Moser from the case that is in the Lebesgue space to certain subspaces, namely the Lorentz spaces , where . The conclusion is that is integrable, where . This is a higher degree of integrability than in the Moser inequality when . A formula for is given and it is also shown that no larger value of works.

     

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.     

Logarithmic Sobolev inequality and strong ergodicity for birth-death processes

We give two examples to show that the strong ergodicity and the logarithmic Sobolev inequality are incomparable for ergodic birth-death processes.     

Asymptotic behaviour of sharp Sobolev constants in thin domains

In this paper we study the asymptotic behaviour of the constants in Sobolev inequalities in thin domains with respect to the thickness of the domain ε. We prove that the sharp Sobolev constants in thin domains converge to the sharp Sobolev constant on the lower-dimensional domain, as ε tends to zero.     

The general optimal L-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations

We prove a general optimal L^p-Euclidean logarithmic Sobolev inequality by using Prékopa-Leindler inequality and a special Hamilton-Jacobi equation. In particular we generalize the inequality proved by Del Pino and Dolbeault in (J. Funt. Anal.).