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

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 2γ (0,2) or 2γ (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.

fractional Laplacian, fractional GJMS operator, Poincaré–Einstein manifold, Robin operator, smooth metric measure space
Mathematical Subject Classification 2010
Primary: 58J32
Secondary: 53A30, 58J40
Received: 6 October 2015
Revised: 6 October 2016
Accepted: 28 November 2016
Published: 23 February 2017
Jeffrey S. Case
109 McAllister Building
Penn State University
University Park, PA 16802
United States