Some energy inequalities involving fractional GJMS operators

Case, Jeffrey S.

doi:10.2140/apde.2017.10.253

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Some energy inequalities involving fractional GJMS operators

Jeffrey S. Case

Vol. 10 (2017), No. 2, 253–280

DOI: 10.2140/apde.2017.10.253

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 $2 γ \in (0, 2)$ or $2 γ \in (2, 4)$ 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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 58J32

Secondary: 53A30, 58J40

Milestones

Received: 6 October 2015

Revised: 6 October 2016

Accepted: 28 November 2016

Published: 23 February 2017

Authors

Jeffrey S. Case
109 McAllister Building Penn State University University Park, PA 16802 United States

Vol. 10, No. 2, 2017