Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian

We establish a family of sharp Sobolev trace inequalities involving the $W^{k,2}({\mathbb{R}}_{+}^{n+1},y^a)$-norm. These inequalities are closely related to the realization of fractional powers of the Laplacian on ${\mathbb{R}}^n=\mathrm{\partial }{\mathbb{R}}_{+}^{n+1}$ 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.