Intertwinings, second-order Brascamp–Lieb inequalities and spectral estimates
We continue to explore the consequences of the so-called intertwinings between gradients and Markov diffusion operators on $\mathbb R ^d$ in terms of Brascamp–Lieb type inequalities for log-concave distributions and beyond, extending our inequalities established in a previous paper. First, we identify the extremal functions in the so-called generalized Brascamp–Lieb inequalities, an issue left open in our previous work. Moreover, we derive new generalized Brascamp–Lieb inequalities of second order from which some new lower bounds on the $(d+1)$th positive eigenvalue of the associated Markov diffusion operator are deduced. We apply our spectral results to perturbed product probability measures, freeing us from Helffer’s classical method based on uniform spectral estimates for the one-dimensional conditional distributions. In particular we exhibit new examples involving some non-classical nearest-neighbour interactions, for which our spectral estimates turn out to be dimension-free.