Boundary Operators Associated With the Sixth-Order GJMS Operator

We describe a set of conformally covariant boundary operators associated with the 6th-order Graham-Jenne-Mason-Sparling (GJMS) operator on a conformally invariant class of manifolds that includes compactifications of Poincare-Einstein manifolds. This yields a conformally covariant energy functional for the 6th-order GJMS operator on such manifolds. Our boundary operators also provide a new realization of the fractional GJMS operators of order one, three, and five as generalized Dirichlet-to-Neumann operators. This allows us to prove some sharp Sobolev trace inequalities involving the interior W-3,W-2-seminorm, including an analogue of the Lebedev-Milin inequality on sixdimensional manifolds.

Automated summary

This paper introduces a set of conformally covariant boundary operators associated with the 6th-order Graham-Jenne-Mason-Sparling (GJMS) operator, which can be applied to a conformally invariant class of manifolds such as compactifications of Poincare-Einstein manifolds, providing a conformally covariant energy functional for the 6th-order GJMS operator on these manifolds. Additionally, the boundary operators allow the realization of fractional GJMS operators of order one, three, and five as generalized Dirichlet-to-Neumann operators, leading to the proof of some sharp Sobolev trace inequalities.