Sharp affine weighted $L^p$ Sobolev type inequalities

TL;DR

This paper proves new sharp affine weighted $L^p$ Sobolev inequalities using the $L_p$ Busemann-Petty centroid inequality, without relying on Euclidean geometry, and characterizes extremizers in some cases.

Contribution

It introduces a novel approach combining the $L_p$ Busemann-Petty centroid inequality with existing weighted Sobolev inequalities, leading to sharp, affine-invariant results.

Findings

Established sharp affine weighted $L^p$ Sobolev inequalities.

Characterized extremizers in certain cases.

Inequalities do not depend on Euclidean geometric structure.

Abstract

We establish sharp affine weighted $L^p$ Sobolev type inequalities by using the $L_p$ Busemann-Petty centroid inequality proved by Lutwak, Yang and Zhang. Our approach consists in combining in a convenient way the latter one with a suitable family of sharp weighted $L^p$ Sobolev type inequalities obtained by Nguyen and allows to characterize all extremizers in some cases. The new inequalities don't rely on any euclidean geometric structure.