Asymmetric Covariance Estimates of Brascamp-Lieb Type and Related Inequalities for Log-concave Measures

An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^2$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto \cite{OM} proved a variant of this with the two $L^2$ norms replaced by $L^1$ and $L^{\mathrm{\infty }}$ norms, but only for ${\mathbb{R}}^1$. We prove a generalization of both by extending these inequalities to $L^p$ and $L^q$ norms and on ${\mathbb{R}}^n$, for any $n\ge 1$. We also prove an inequality for integrals of divided differences of functions in terms of integrals of their gradients.