The sharp affine L 2 Sobolev trace inequality and variants

Abstract

We establish a sharp affineL p Sobolev trace inequality by using the L p Busemann–Petty centroid inequality. For p= 2 , our affine version is stronger than the famous sharp L 2 Sobolev trace inequality proved independently by Escobar and Beckner. Our approach allows also to characterize all extremizers in this case. For this new inequality, no Euclidean geometric structure is needed.