Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian

We establish a family of sharp Sobolev trace inequalities involving the W-k,W-2 (R-+(n+1), y(a))-norm. These inequalities are closely related to the realization of fractional powers of the Laplacian on R-n = partial derivative 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. (C) 2020 Elsevier Inc. All rights reserved.