A quantitative log-Sobolev inequality for a two parameter family of functions

TL;DR

This paper establishes a sharp, dimension-free stability result for a specific two-parameter family of functions in the classical logarithmic Sobolev inequality, with applications to entropy bounds.

Contribution

It introduces a new stability result for a two-parameter family of log C^{1,1} functions, extending the classical inequality with dimension-free constants.

Findings

Proves a sharp stability result for the log-Sobolev inequality.

Provides bounds on entropy for the considered function family.

Shows how to enlarge the function space with controlled constants.

Abstract

We prove a sharp, dimension-free stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log $C^{1,1}$ functions. Moreover, we show how to enlarge this space at the expense of the dimensionless constant and the sharp exponent. As an application we obtain new bounds on the entropy.