Conformally Covariant Boundary Operators and Sharp Higher Order Sobolev Trace Inequalities on Poincar\'e-Einstein Manifolds

In this paper we introduce conformally covariant boundary operators for Poincar\'e-Einstein manifolds satisfying a mild spectral assumption. Using these boundary operators we set up higher order Dirichlet problems whose solutions are such that, when applied to by our boundary operators, they recover the fractional order GJMS operators on the conformal infinity of the manifold. We moreover obtain all related higher order trace Sobolev inequalities on these manifolds. In conjunction with Beckner's fractional Sobolev inequalities on the sphere, we obtain as an application the sharp higher order Sobolev trace inequalities on the ball.