A note on spectral gap and weighted Poincar\'e inequalities for some one-dimensional diffusions

TL;DR

This paper investigates classical and weighted Poincaré inequalities for one-dimensional probability measures, linking the optimal constants to spectral gaps of diffusion operators, and provides exact spectral gap values for specific cases.

Contribution

It extends previous multidimensional results to the one-dimensional setting, offering precise spectral gap estimates for weighted Poincaré inequalities.

Findings

Exact spectral gap values obtained for certain measures.

Reformulation of Poincaré constants as spectral gaps.

Extension of multidimensional results to one dimension.

Abstract

We present some classical and weighted Poincar\'e inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors for a class of spherically symmetric probability measures in dimension larger than 2. Our strategy is based on two main ingredients: on the one hand, the optimal constant in the desired weighted Poincar\'e inequality has to be rewritten as the spectral gap of a convenient Markovian diffusion operator, and on the other hand we use a recent result given by the two first authors, which allows to estimate precisely this spectral gap. In particular we are able to capture its exact value for some examples.