Modified logarithmic Sobolev inequalities and transportation inequalities

We present a new class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type $\exp(-|x|^\al)$ or $\exp(-|x|^\al\log^\beta(2+|x|))$ ($\al\in]1,2[$ and $\be\in\dR$) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities.ou