Intertwinings, second-order Brascamp-Lieb inequalities and spectral estimates

TL;DR

This paper investigates how intertwinings between gradients and Markov diffusion operators can lead to second-order Brascamp-Lieb inequalities, providing new spectral bounds for eigenvalues of diffusion operators, especially in perturbed product measures.

Contribution

It extends previous inequalities using intertwinings, deriving new lower bounds on higher eigenvalues and applying these results to perturbed measures beyond classical methods.

Findings

Derived lower bounds on the (d+1)th eigenvalue based on spectral gap.

Extended second-order Brascamp-Lieb inequalities to broader classes of distributions.

Applied spectral estimates to perturbed product measures, surpassing classical approaches.

Abstract

We explore the consequences of the so-called intertwinings between gradients and Markov diffusion operators on $R^d$ in terms of second-order Brascamp-Lieb inequalities for log-concave distributions and beyond, extending our inequalities established in a previous paper. As a result, we derive some convenient lower bounds on the $(d+1)^{th}$ positive eigenvalue depending on the spectral gap of the dual Markov diffusion operator given by the intertwining. To see the relevance of our approach, we apply our spectral results in the case of perturbed product measures, freeing us from Helffer's classical method based on uniform spectral estimates for the one-dimensional conditional distributions.