  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.
