Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian

Abstract

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

Corrected Equation (6.3)

References in corpus (4)

• Sobolev-Trace inequalities of order four
• Some energy inequalities involving fractional GJMS operators
• Higher order Sobolev trace inequalities on balls revisited
• Sharp Sobolev trace inequalities for higher order derivatives

Cited by in corpus (1)

• Higher order Sobolev trace inequalities on balls revisited