Boundary operators associated to the Paneitz operator

TL;DR

This paper introduces conformally covariant boundary operators linked to the Paneitz operator, establishing their properties, connections to fractional GJMS operators, and applications to sharp Sobolev trace inequalities.

Contribution

It defines new boundary operators associated with the Paneitz operator that are conformally covariant and connects them to fractional GJMS operators and Sobolev inequalities.

Findings

Operators agree with fractional GJMS operators on Poincaré--Einstein manifolds

Established new sharp Sobolev trace inequalities

Defined conformally covariant energy functionals for the Paneitz operator

Abstract

We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conformally covariant pseudodifferential operator. In the setting of Poincar\'e--Einstein manifolds, we show that these operators agree with the fractional GJMS operators of Graham and Zworski. We also use our operators to establish some new sharp Sobolev trace inequalities.