Higher order Sobolev trace inequalities on balls revisited

TL;DR

This paper revisits higher order Sobolev trace inequalities on balls, offering a new proof approach and establishing sharp inequalities of orders two, six, and eight, including a Lebedev-Milin type inequality.

Contribution

It introduces a novel method to reprove and extend Sobolev trace inequalities of higher orders on balls, including sharp inequalities of orders six and eight.

Findings

Reproved classical Sobolev trace inequality of order two.

Established sharp Sobolev trace inequalities of orders six and eight.

Derived a Lebedev-Milin type inequality as a limit case.

Abstract

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.