Some energy inequalities involving fractional GJMS operators

TL;DR

This paper proves energy inequalities for fractional GJMS operators on Poincaré--Einstein manifolds under spectral conditions, introducing new boundary operators and deriving a sharp weighted Sobolev trace inequality.

Contribution

It establishes necessary and sufficient spectral conditions for energy inequalities involving fractional GJMS operators and introduces conformally covariant boundary operators.

Findings

Proves energy inequalities under spectral assumptions.

Introduces conformally covariant boundary operators.

Derives a sharp weighted Sobolev trace inequality.

Abstract

Under a spectral assumption on the Laplacian of a Poincar\'e--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.