#### Vol. 10, No. 2, 2017

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Some energy inequalities involving fractional GJMS operators

### Jeffrey S. Case

Vol. 10 (2017), No. 2, 253–280
##### 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 $2\gamma \in \left(0,2\right)$ or $2\gamma \in \left(2,4\right)$ 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
##### Mathematical Subject Classification 2010
Primary: 58J32
Secondary: 53A30, 58J40