Weak Functional Inequalities for Perturbed Measures

TL;DR

This paper extends the study of functional inequalities to weaker forms like weak Poincaré and weighted log-Sobolev inequalities for perturbed measures, with applications to convolution products.

Contribution

It introduces new results on weaker functional inequalities for perturbed measures and explores their applications to convolution products.

Findings

Established weak Poincaré and weighted log-Sobolev inequalities for perturbed measures.

Extended previous results to weaker inequalities, broadening applicability.

Provided applications to convolution product measures.

Abstract

This paper is a follow up to an article by two of the authors dedicated to the study of Poincar\'e and logarithmic Sobolev inequalities for measures of the form $d\mu = e^{-U} d\nu$ where $e^{-U}$ is seen as a perturbation of $d\nu$. Application to the same functional inequalities for convolution products are then discussed. In the present paper we investigate similar problems for weaker functional inequalities, namely weak Poincar\'e, weighted Poincar\'e, weak log-Sobolev and weighted log-Sobolev inequalities.