Higher order Sobolev trace inequalities on balls revisited

Inspired by a recent sharp Sobolev trace inequality of order four on the balls $\mathbb B^{n+1}$ found by Ache and Chang [AC15], we propose a slightly different approach to reprove Ache-Chang's trace inequality. To illustrate this approach, we reprove the classical Sobolev trace inequality of order two on $\mathbb B^{n+1}$ and provide sharp Sobolev trace inequalities of orders six and eight on $\mathbb B^{n+1}$. As the limiting case of the Sobolev trace inequality, a Lebedev-Milin type inequality of order up to eight is also considered.