Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities

Modified logarithmic Sobolev inequalities and transportation inequalities

Direct and Reverse Gagliardo–nirenberg Inequalities from Logarithmic Sobolev Inequalities

We investigate the connection between the validity of certain logarithmic Sobolev inequality and the validity of suitable generalizations of Gagliardo–Nirenberg inequalities. A similar connection holds between reverse logarithmic Sobolev inequalities and a new class of reverse Gagliardo–Nirenberg inequalities, valid for a suitable class of functions. 0. Introduction The main concern of this pap...

Sobolev and Isoperimetric Inequalities with Monomial Weights

We consider the monomial weight |x1|1 · · · |xn|n in R, where Ai ≥ 0 is a real number for each i = 1, ..., n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by |x1|1 · · · |xn|ndx, and they contain the best or critical exponent (which depends on A1, ..., An). More i...

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...

Isoperimetry and Symmetrization for Logarithmic Sobolev Inequalities

Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.