Inequalities for the Gaussian measure and an application to Wiener space

This paper deals with a generalization of a result due to Brascamp and Lieb which states that in the space of probabilities with log-concave density with respect to a Gaussian measure on $\text{R}{}^{\mathrm{n}}$, this Gaussian measure is the one which has strongest moments. We show that this theorem remains true if we replace x^α by a general convex function. Then, we deduce a correlation inequality for convex functions quite better than the one already known. Finally, we prove a result concerning stochastic analysis on Wiener space through the notion of approximate limit.