$\Phi$-entropy inequalities and asymmetric covariance estimates for convex measures

Abstract

In this paper, we use the semi-group method and an adaptation of the $L^{2}$-method of Hörmander to establish some $\Phi$-entropy inequalities and asymmetric covariance estimates for the strictly convex measures in $\mathbb{R}^{n}$. These inequalities extends the ones for the strictly log-concave measures to more general setting of convex measures. The $\Phi$-entropy inequalities are turned out to be sharp in the special case of Cauchy measures. Finally, we show that the similar inequalities for log-concave measures can be obtained from our results in the limiting case.