The Frank-Lieb approach to sharp Sobolev inequalities

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 Wk,2(n)-L 2n n-2k(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 σ2-curvature. Our argument relies on nice commutator identities deduced using the Fefferman-Graham ambient metric.