The Frank-Lieb approach to sharp Sobolev inequalities

TL;DR

This paper presents a new direct proof of sharp Sobolev inequalities and a nonlinear variant involving $\sigma_2$-curvature, using the Frank-Lieb approach and ambient metric techniques, avoiding traditional rearrangement methods.

Contribution

It provides a direct proof of sharp Sobolev inequalities and introduces a novel proof for a nonlinear inequality involving $\sigma_2$-curvature, expanding the Frank-Lieb method.

Findings

Direct proof of sharp Sobolev inequalities without Hardy-Littlewood-Sobolev step

New proof of a nonlinear Sobolev inequality involving $\sigma_2$-curvature

Utilization of ambient metric and commutator identities

Abstract

Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy-Littlewood-Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings $W^{k,2}(\mathbb{R}^n)\hookrightarrow L^{\frac{2n}{n-2k}}(\mathbb{R}^n)$. We show that their argument gives a direct proof of the latter inequalities without passing through Hardy-Littlewood-Sobolev inequalities, and, moreover, a new proof of a sharp fully nonlinear Sobolev inequality involving the $\sigma_2$-curvature. Our argument relies on nice commutator identities deduced using the Fefferman-Graham ambient metric.