Modified Logarithmic Sobolev Inequalities and Transportation Cost Inequalities in ℝ n

Abstract

In this paper, the modified logarithmic Sobolev inequalities and transportation cost inequalities for measures with density e − V in ℝn are established. It is proved by using Prekopa–Leindler inequalities following the idea of Bobkov–Ledoux, but a different type of condition is used which recovers Bakry–Emery criterion. As an application, we establish the modified logarithmic Sobolev and transportation cost inequalities for probability measures \(e^{-|x|^p} {\rm{d}}x/Z_p\) with p > 1 in ℝn, and give out explicit estimates for their constants.