Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence
Analysis of PDEs
2025-04-02 v5 Classical Analysis and ODEs
Functional Analysis
Abstract
We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement, and refined estimates on the Sobolev functional in the neighborhood of the optimal Aubin--Talenti functions.
Keywords
Cite
@article{arxiv.2209.08651,
title = {Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence},
author = {Jean Dolbeault and Maria J. Esteban and Alessio Figalli and Rupert L. Frank and Michael Loss},
journal= {arXiv preprint arXiv:2209.08651},
year = {2025}
}