The Frank-Lieb approach to sharp Sobolev inequalities

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.

14 pages

References in corpus (1)

• On a fully nonlinear sharp Sobolev trace inequality