Abstract Inspired by a recent sharp Sobolev trace inequality of order four on the balls B n + 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 B n + 1 and provide sharp Sobolev trace inequalities of orders six and eight on B n + 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.

中文翻译：

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重新审视球上的高阶 Sobolev 跟踪不等式

摘要 受 Ache 和 Chang (2017) [2] 发现的球 B n + 1 上最近的 4 阶尖锐 Sobolev 迹不等式的启发，我们提出了一种不同的方法来证明 Ache-Chang 迹不等式。为了进一步说明这种方法，我们驳斥了 B n + 1 上二阶的经典 Sobolev 迹不等式，并提供了 B n + 1 上六阶和八阶的尖锐 Sobolev 迹不等式。为了获得所有这些高达八阶甚至更多阶的不等式，我们首先在 R + n + 1 上建立更高阶的尖锐 Sobolev 迹不等式，然后通过保形变化将它们直接转移到球上。作为 Sobolev 迹不等式的极限情况，还考虑了高达 8 阶的 Lebedev-Milin 型不等式。