Functional Inequalities for Gaussian and Log-Concave Probability Measures

We give three proofs of a functional inequality for the standard Gaussian measure originally due to William Beckner. The first uses the central limit theorem and a tensorial property of the inequality. The second uses the Ornstein-Uhlenbeck semigroup, and the third uses the heat semigroup. These latter two proofs yield a more general inequality than the one Beckner originally proved. We then generalize our new inequality to log-concave probability measures, study the relationship between this inequality and a generalized logarithmic Sobolev inequality, and prove several other inequalities for log-concave probability measures, including Brascamp and Lieb’s sharpened Poincaré inequality and Bobkov and Ledoux’s sharpened logarithmic Sobolev inequality of the same form. We discuss some of the potential applications of our work in economics.