Stability results for Sobolev, logarithmic Sobolev, and related inequalities

TL;DR

This paper reviews recent advances in deriving explicit stability estimates for classical functional inequalities such as Sobolev and logarithmic Sobolev inequalities, highlighting new methods that address long-standing open problems.

Contribution

It introduces several modern methods for obtaining explicit stability estimates in fundamental functional inequalities, advancing understanding of their quantitative stability.

Findings

New methods provide explicit stability estimates

Addresses a 30-year open problem in the field

Introduces techniques applicable to multiple inequalities

Abstract

Obtaining explicit stability estimates in classical functional inequalities like the Sobolev inequality has been an essentially open question for 30 years, after the celebrated but non-constructive result of G. Bianchi and H. Egnell in 1991. Recently, new methods have emerged which provide some clues on these fascinating questions. The goal of the course is to give an introduction to the topic for some fundamental functional inequalities and present several methods that can be used to obtain explicit estimates.