Some energy inequalities involving fractional GJMS operators

Abstract

Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2\gamma \in (0,2)$ or $2\gamma \in (2,4)$ and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption is necessary and sufficient for such an inequality to hold. We prove the energy inequalities by introducing conformally covariant boundary operators associated to the weighted conformal Laplacian and weighted Paneitz operator which generalize the Robin operator. As an application, we establish a new sharp weighted Sobolev trace inequality on the upper hemisphere.

26 pages; clarified some definitions (e.g. Definition 2.2 and Definition 3.1)

References in corpus (3)

• On higher order extensions for the fractional Laplacian
• Sobolev-Trace inequalities of order four
• Sharp constants in weighted trace inequalities on Riemannian manifolds

Cited by in corpus (8)

• Existence theorems of the fractional Yamabe problem
• Asymptotic behavior of Palais-Smale sequences associated with fractional Yamabe type equations
• Higher order Sobolev trace inequalities on balls revisited
• Sharp Sobolev trace inequalities for higher order derivatives
• Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian
• Uniformization Theorems: Between Yamabe and Paneitz
• Conformal metrics with prescribed fractional scalar curvature on conformal infinities with positive fractional Yamabe constants
• The Dirichlet principle for the complex $k$-Hessian functional