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
or
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.
Keywords
fractional Laplacian, fractional GJMS operator,
Poincaré–Einstein manifold, Robin operator, smooth metric
measure space
