Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian

Abstract We establish a family of sharp Sobolev trace inequalities involving the W k , 2 ( R + n + 1 , y a ) -norm. These inequalities are closely related to the realization of fractional powers of the Laplacian on R n = ∂ R + n + 1 as generalized Dirichlet-to-Neumann operators associated to powers of the weighted Laplacian in upper half space, generalizing observations of Caffarelli–Silvestre and of Yang.