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The Frank–Lieb approach to sharp Sobolev inequalities
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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 [Formula: see text]. 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 [Formula: see text]-curvature. Our argument relies on nice commutator identities deduced using the Fefferman–Graham ambient metric.
Title: The Frank–Lieb approach to sharp Sobolev inequalities
Description:
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 [Formula: see text].
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 [Formula: see text]-curvature.
Our argument relies on nice commutator identities deduced using the Fefferman–Graham ambient metric.
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