Higher order Sobolev trace inequalities on balls revisited

Inspired by a recent sharp Sobolev trace inequality of order four on the balls Bn+1 found by Ache and Chang (2017) [2], we propose a different approach to reprove Ache–Chang's trace inequality. To further illustrate this approach, we reprove the classical Sobolev trace inequality of order two on Bn+1 and provide sharp Sobolev trace inequalities of orders six and eight on Bn+1. To obtain all these inequalities up to order eight, and possibly more, we first establish higher order sharp Sobolev trace inequalities on R+n+1, then directly transferring them to the ball via a conformal change. As the limiting case of the Sobolev trace inequalities, Lebedev–Milin type inequalities of order up to eight are also considered.