Abstract

Inspired by a recent sharp Sobolev trace inequality of order four on the balls 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 and provide sharp Sobolev trace inequalities of orders six and eight on . To obtain all these inequalities up to order eight, and possibly more, we first establish higher order sharp Sobolev trace inequalities on , 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.