Logarithmic Sobolev inequalities and the growth of $L^p$ norms

Modified logarithmic Sobolev inequalities and transportation inequalities

SHARP AFFINE Lp SOBOLEV INEQUALITIES

In this paper we prove a sharp affine Lp Sobolev inequality for functions on R. The new inequality is significantly stronger than (and directly implies) the classical sharp Lp Sobolev inequality of Aubin [A2] and Talenti [T], even though it uses only the vector space structure and standard Lebesgue measure on R. For the new inequality, no inner product, norm, or conformal structure is needed at...

Logarithmic Sobolev Inequalities and the Information Theory

In this paper we present an overview on logarithmic Sobolev inequalities. These inequalities have become a subject of intense research activity during the past years, from analysis and geometry in finite and infinite dimension, to probability and statistical mechanics, and of course many others developments and applications are expected. We have divided this paper into three parts. The first pa...

Concentration of measure and logarithmic Sobolev inequalities

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Weak logarithmic Sobolev inequalities and entropic convergence

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 inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for d...