Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities

TL;DR

This paper improves stability bounds for key functional inequalities like the logarithmic Sobolev and Gagliardo-Nirenberg inequalities, leading to explicit convergence rates in related diffusion equations.

Contribution

It introduces explicit stability estimates based on scale-aware deficit functionals for several inequalities, enhancing understanding of their equality cases and convergence behavior.

Findings

Explicit stability bounds with constants for Gaussian inequalities

Faster convergence rates in diffusion equations derived from stability results

Enhanced understanding of the structure of optimal functions in inequalities

Abstract

This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit stability measurements, i.e., bound with explicit constants distances to the manifold of optimal functions. Various results are obtained for the Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the Gaussian generalized Poincar{\'e} inequalities and for the Gagliardo-Nirenberg inequalities. As a consequence, faster convergence rates in diffusion equations (fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained.