New approach to the affine P\'olya-Szeg\"o principle and the stability version of the affine Sobolev inequality

TL;DR

This paper introduces a new, simplified method for the affine Pólya-Szegö principle using the $L_p$ Busemann-Petty centroid inequality, leading to sharper affine Sobolev inequalities and their stability versions.

Contribution

It provides a novel approach that avoids complex inequalities and solves the stability problem for affine Sobolev inequalities, extending the understanding of affine functional inequalities.

Findings

Reproved sharp affine Sobolev-type inequalities with equality conditions

Established a stability estimate for the affine Sobolev inequality

Derived a stability estimate for the affine logarithmic-Sobolev inequality

Abstract

Inspired by a recent work of Haddad, Jim\'enez and Montenegro, we give a new and simple approach to the recently established general affine P\'olya-Szeg\"o principle. Our approach is based on the general $L_p$ Busemann-Petty centroid inequality and does not rely on the general $L_p$ Petty projection inequality or the solution of the $L_p$ Minkowski problem. A Brothers-Ziemer-type result for the general affine P\'olya-Szeg\"o principle is also established. As applications, we reprove some sharp affine Sobolev-type inequalities and settle their equality conditions. We also prove a stability estimate for the affine Sobolev inequality on functions of bounded variation by using our new approach. As a corollary of this stability result, we deduce a stability estimate for the affine logarithmic--Sobolev inequality.