On Sharp Sobolev Embedding and The Logarithmic Sobolev Inequality

The purpose of this note is to give a short proof of the Grosslogarithmic Sobolev inequality using the asymptotics of thesharp L² Sobolev constant and the product structure of Euclideanspace. Let FL^r(Rⁿ) for some positive r with ||F||_r=1. 1991 MathematicsSubject Classification 58G35.     

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...     

Weighted Logarithmic Sobolev Inequalities for Sub-Gaussian Measures

In this short paper, We establish the weighted Logarithmic Sobolev inequalities for sub-Gaussian measure of high dimension with explicit constants via phase transition and the well-known Bakry-émery criterion.     

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.     

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.     

Uniform Lieb-Thirring Inequality for the Three-Dimensional Pauli Operator with a Strong Non-Homogeneous Magnetic Field

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin in a magnetic field and an external potential. A new Lieb- Thirring type inequality on the sum of the negative eigenvalues is presented. The main feature compared to earlier results is that in the large field regime the present estimate grows with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain an optimal upper bound on the pointwise density of zero energy eigenfunctions of the Dirac operator. The main technical tools are: (i) a new localization scheme for the square of the resolvent of a general class of second order elliptic operators; (ii) a geometric construction of a Dirac operator with a constant magnetic field that approximates the original Dirac operator in a tubular neighborhood of a fixed field line. The errors may depend on the regularity of the magnetic field but they are uniform in the field strength. Communicated by Gian Michele GrafSubmitted 31/08/03, accepted 28/01/04     

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.     

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.     

Logarithmic Sobolev inequalities for some nonlinear PDE's

The aim of this paper is to study the behavior of solutions of some nonlinear partial differential equations of Mac Kean–Vlasov type. The main tools used are, on one hand, the logarithmic Sobolev inequality and its connections with the concentration of measure and the transportation inequality with quadratic cost; on the other hand, the propagation of chaos for particle systems in mean field interaction.     

Logarithmic Sobolev Inequalities for Pinned Loop Groups

LetGbe a connected compact type Lie group equipped with anAd_G-invariant inner product on the Lie algebra ofG. Given this data there is a well known left invariant “H¹-Riemannian structure” on =(G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”ν_T(g₀, ·) associated to the Laplace–Beltrami operator on (G). HereT>0,g₀∈(G), andν_T(g₀,·) is a certain probability measure on (G). For fixedg₀∈(G) andT>0, we use the measureν_T(g₀,·) and the Riemannian structure on (G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.     

Heisenberg's Inequality and Logarithmic Heisenberg's Inequality for Ambiguity Function

In this article we discuss the relation between Heisenberg's inequality and logarithmic Heisenberg's (entropy) inequality for ambiguity function. After building up a Heisenberg's inequality, we obtain a connection of variance with entropy by variational method. Using classical Taylor's expansion, we prove that the equality in Heisenberg's inequality holds if and only if the entropy of 2k - 1 order is equal to (2k - 1}!.     

扩散过程轨道空间上的对数Sobolev不等式(英语)

设M是连通Riemann流形,Z是M上C′类向量场,L=(△ Z),本文使用Kendall的耦合分析,给出了参考测度为L-扩散过程在t时刻分布的对数Sobolev常数的估计,并由此建立了轨道空间上的对数Sobolev不等式。此外,本文还给出了流形上的对数Sobolev常数的一个上界估计,所获结果,是对文[1],[2]和[3]的相应结果的推广。     

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.     

Logarithmic Sobolev Inequalities of Diffusions for the L^2 Metric

Under the Bakry–Emery's -minoration condition, we establish the logarithmic Sobolev inequality for the Brownian motion with drift in the metric instead of the usual Cameron–Martin metric. The involved constant is sharp and does not explode for large time. This inequality with respect to the -metric provides us the gaussian concentration inequalities for the large time behavior of the diffusion. An erratum to this article can be found at     

关于广义对数平均的一个不等式的加细

In this article,we show that the generalized logarithmic mean is strictly Schurconvex function for p＞2 and strictly Schur-concave function for P＜2 on R2+.And then we give a refinement of an inequality for the generalized logarithmic mean inequality using a simple majoricotion relation of the vector.