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.     

Modified Logarithmic Sobolev Inequalities in Discrete Settings

Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare’ (spectral gap) inequality. We show that, in contrast with the spectral gap, for bounded degree expander graphs, various log-Sobolev constants go to zero with the size of the graph. We also derive a hypercontractivity formulation equivalent to our main modified log-Sobolev inequality. Along the way we survey various recent results that have been obtained in this topic by other researchers.      

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     

Logarithmic Sobolev Inequalities,Matrix Models and Free Entropy

We give two applications of logarithmic Sobolev inequalities to matrix models and free probability. We also provide a new characterization of semi-circular systems through a Poincare-type inequality.     

生灭过程与一维扩散过程的对数sobolev不等式

本文运用加权的Hardy不等式的方法给出了生灭过程与一维扩散过程满足对数Sobolev不等式的显式判别准则。     

Modified Logarithmic Sobolev Inequalities on $\mathbb{R}$

We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Götze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo.     

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 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.

     

Logarithmic Sobolev inequality revisited

We provide a new characterization of the logarithmic Sobolev inequality.     

对称扩散过程的对数Sobolev不等式、指数可积性和熵的指数衰减性(英)

本文讨论了对称扩散过程的指数可积性，熵的指数衰减性与Sobolev不等式之间的等价关系．并给出了对称扩散过程的对数Sobolev不等式成立的一个充分条件．     

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.     

一组对数不等式的应用

利用辅助函数的单调性可证对数不等式 x 1＋ x ≤ ln（1＋ x）≤1＋ x （x ≥0）。通过实例介绍这组对x数不等式在证明不等式、求函数最大（小）值等方面的应用。     

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.     

Logarithmic Version of Interpolation Inequalities for Derivatives

A version of interpolation inequalities for derivatives in logarithmicOrlicz spaces is obtained where the first gradient of u is estimatedin terms of u and its second gradient. One of the Orlicz functionsconsidered is supposed to be ^p. The motivation, examples andapplications are discussed.     

New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow

In this note, we prove an ?‐regularity theorem for the Ricci flow. Let (Mⁿ,g(t)) with t ? [?T,0] be a Ricci flow, and let H_x0(y,s) be the conjugate heat kernel centered at some point (x₀,0) in the final time slice. By substituting H_x0(?,s) into Perelman's W‐functional, we obtain a monotone quantity W_x0(s) that we refer to as the pointed entropy. This satisfies W_x0(s) ≤ 0, and W_x0(s) = 0 if and only if (Mⁿ,g(t)) is isometric to the trivial flow on Rⁿ. Then our main theorem asserts the following: There exists ? > 0, depending only on T and on lower scalar curvature and μ‐entropy bounds for the initial slice (Mⁿ,g(?T)) such that W_x0(s) ≥ ?? implies |Rm| ≤ r^?2 on P_{? r}(x₀,0), where r² ≡ |s| and P_ρ(x,t) ≡ B_ρ(x,t) × (t?ρ²,t] is our notation for parabolic balls. The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s‐average of W_x(s). To accomplish this, we require a new log‐Sobolev inequality. Perelman's work implies that the metric measure spaces (Mⁿ,g(t),dvol_g(t)) satisfy a log‐Sobolev; we show that this is also true for the heat kernel weighted spaces (Mⁿ,g(t),H_x0(?,t)dvol_g(t)). Our log‐Sobolev constants for these weighted spaces are in fact universal and sharp. The weighted log‐Sobolev has other consequences as well, including certain average Gaussian upper bounds on the conjugate heat kernel. © 2014 Wiley Periodicals, Inc.     

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.     

Extremals for Logarithmic Hardy-Littlewood-Sobelov Inequalities on Compact Manifolds

Let M be a closed, connected surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g Γ, denote the area element by dV and the Laplace–Beltrami operator by Δ _g . We define the Robin mass m(x) at the point x M to be the value of the Green’s function G(x, y) at y = x after the logarithmic singularity has been subtracted off. The regularized trace of Δ _g ⁻¹ is then defined by trace Δ⁻¹ = ∫ _M m dV. (This essentially agrees with the zeta functional regularization and is thus a spectral invariant.) Let be the Laplace–Beltrami operator on the round sphere of volume V. We show that if there exists g Γ with trace Δ _g ⁻¹ < trace then the minimum of trace Δ⁻¹ over Γ is attained by a metric in Γ for which the Robin mass is constant. Otherwise, the minimum of trace Δ⁻¹ over Γ is equal to trace . In fact we prove these results in the general setting where M is an n-dimensional closed, connected manifold and the Laplace–Beltrami operator is replaced by any non-negative elliptic operator A of degree n which is conformally covariant in the sense that for the metric g we have . In this case the role of is assumed by the Paneitz or GJMS operator on the round n-sphere of volume V. Explicitly these results are logarithmic HLS inequalities for (M, g). By duality we obtain analogs of the Onofri–Beckner theorem. Received: February 2006, Accepted: March 2006