# Some remarks on weighted logarithmic Sobolev inequality

@article{Cattiaux2010SomeRO, title={Some remarks on weighted logarithmic Sobolev inequality}, author={Patrick Cattiaux and Arnaud Guillin and Liming Wu}, journal={arXiv: Probability}, year={2010} }

We give here a simple proof of weighted logarithmic Sobolev inequality, for example for Cauchy type measures, with optimal weight, sharpening results of Bobkov-Ledoux. Some consequences are also discussed.

## 22 Citations

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## References

SHOWING 1-10 OF 41 REFERENCES

### Modified logarithmic Sobolev inequalities in null curvature

- Mathematics
- 2005

We present a new logarithmic Sobolev inequality adapted to a log-concave measure on $\dR$ between the exponential and the Gaussian measure.

### Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities

- Mathematics
- 1999

Abstract We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a…

### Weak logarithmic Sobolev inequalities and entropic convergence

- Mathematics
- 2005

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…

### A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality

- Mathematics
- 2008

We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand’s transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient…

### Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution

- Mathematics
- 1997

Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every…

### A Pathwise Approach of Some Classical Inequalities

- Mathematics
- 2004

The aim of this pedagogical paper is to show how some renowned inequalities may be obtained via a simple argument: entropy projection from the path space onto finite-dimensional coordinates spaces.…

### A simple proof of the Poincaré inequality for a large class of probability measures

- Mathematics
- 2008

Abstract. We give a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on…

### Mass Transport and Variants of the Logarithmic Sobolev Inequality

- Mathematics
- 2007

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…

### METRIC HOPF-LAX FORMULA WITH SEMICONTINUOUS DATA

- Mathematics
- 2007

In this paper we study a metric Hopf-Lax formula
looking in particular at the Carnot-Caratheodory case. We
generalize many properties of the classical euclidean Hopf-Lax
formula and we use it in…