一类量子Markov半群的超压缩性与对数Sobolev不等式

本文在有限von Neumann代数生成的非交换概率空间L~p(p≥1)框架下,证明了一类量子Markov半群的超压缩性等价于其对应的Dirichlet型满足对数Sobolev不等式.此结果包含前人的相关成果为特例.作为推论,细化了Biane的相关工作.     

Hypercontractivity of Solutions to Hamilton-Jacobi Equations

We show that solutions to some Hamilton-Jacobi Equations associated to the problem of optimal control of stochastic semilinear equations enjoy the hypercontractivity property.     

HAMILTON-JACOBI方程的超压缩性及相关不等式

本文主要研究了p≥2的对数-sobolev不等式与一般Hamilton-Jacobi方程解的超压缩性的相关性。同时我们也建立极小卷积不等式与p-阶费用传输不等式之间的关系。     

Nonlinear diffusions, hypercontractivity and the optimal L-Euclidean logarithmic Sobolev inequality

The equation u_t=Δ_p(u^1/(p−1)) for p>1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal L^p-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focus on the existence and the uniqueness of the solutions to the Cauchy problem and on the regularization properties (hypercontractivity and ultracontractivity) of the equation using the L^p-Euclidean logarithmic Sobolev inequality. A large deviation result based on a Hamilton-Jacobi equation and also related to the L^p-Euclidean logarithmic Sobolev inequality is then stated.     

Équations de Hamilton–Jacobi et inégalités entropiques généraliséesHamilton–Jacobi equations and inequalities for generalised entropies

We prove the equivalence between a general logarithmic Sobolev inequality and the hypercontractivity of a Hamilton–Jacobi equation. We also recover that this property imply a transportation inequality established by [5]. These results provide a natural generalization of the work performed in [3]. To cite this article: I. Gentil, F. Malrieu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 437–440.     

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.).     

An optimal logarithmic Sobolev inequality with Lipschitz constants

In this paper, we give an optimal logarithmic Sobolev inequality on Rn with Lipschitz constants. This inequality is a limit case of the Lp-logarithmic Sobolev inequality of Gentil (2003) [7] as p→∞. As a result of our inequality, we show that if a Lipschitz continuous function f on Rn fulfills some condition, then its Lipschitz constant can be expressed by using the entropy of f. We also show that a hypercontractivity of exponential type occurs in the heat equation on Rn. This is due to the Lipschitz regularizing effect of the heat equation.     

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.     

Dirichlet and Neumann problems to critical Emden–Fowler type equations

We describe recent results on attainability of sharp constants in the Sobolev inequality, the Sobolev–Poincaré inequality, the Hardy–Sobolev inequality and related inequalities. This gives us the solvability of boundary value problems to critical Emden–Fowler equations.      

Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup

We prove hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup on the entire algebra of bounded operators on a separable Hilbert space h. We exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain. Then we deduce a logarithmic Sobolev inequality for the semigroup and gain an elementary estimate of the best constant.     

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.     

A Harnack-type inequality for non-symmetric Markov semigroups

For a strong Feller and irreducible Markov semigroup on a locally compact Polish space, the Harnack-type inequality (1.1) holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive. Moreover, new sufficient conditions for this inequality to hold, as well as upper bound estimates of the underlying constant, are presented for diffusion semigroups on Riemannian manifolds.     

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.     

带有不同权函数的退化拟线性椭圆方程弱解的Holder正则性

我们研究了一类系数依赖于两不同权函数的退化拟线性椭圆方程．利用加权Sobolev不等式,加权Poincare不等式及Moser迭代技巧,得到了非负弱解的Harnack不等式,证明了弱解的H61der连续性．     

Intégalités de Sobolev et Temps d'Atteinte

We investigate the links between Sobolev and Nash inequalities, capacity and hitting times estimates and ultracontractive semigroups, in a non-symmetric setting.     

The Moser-Trudinger-Onofri Inequality

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks. Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods (in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality. In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally, a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.     

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.     

12-Laplacian problems with exponential nonlinearity

By exploiting a suitable Trudinger–Moser inequality for fractional Sobolev spaces, we obtain existence and multiplicity of solutions for a class of one-dimensional nonlocal equations with fractional diffusion and nonlinearity at exponential growth.     

Improved Gagliardo-Nirenberg inequalities on Heisenberg type groups

Motivated by the idea of M. Ledoux who brings out the connection between Sobolev embeddings and heat kernel bounds, we prove an analogous result for Kohn’s sub-Laplacian on the Heisenberg type groups. The main result includes features of an inequality of either Sobolev or Galiardo-Nirenberg type.     

Ultracontractivity for Brownian motion with Markov switching

Consider the transition density functions for Brownian motion with two-state Markov switching. The characteristic functions for transition density functions are presented. Then, we show that the semigroup-associated Brownian motion with Markov switching is ultracontractive. And an explicit time-dependent upper bound for heat kernels are presented. Moreover, we prove that the Dirichlet form associated Brownian motion with Markov switching satisfies the Nash inequality.