Sharp Sobolev trace inequalities for higher order derivatives

TL;DR

This paper extends sharp Sobolev trace inequalities to higher order derivatives on Euclidean balls and half spaces, providing explicit extremal functions and metrics using scattering theory and Poisson kernels.

Contribution

It introduces higher order sharp Sobolev trace inequalities on Euclidean balls and half spaces, with explicit extremal functions and metrics, advancing the understanding of these inequalities.

Findings

Derived explicit extremal functions for higher order Sobolev trace inequalities.

Established sharp inequalities on Euclidean balls and half spaces.

Computed explicit formulas for the adapted metric on the Euclidean ball.

Abstract

Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic spaces and the generalized Poisson kernel, we obtain the explicit formulas of extremal functions of such inequations. Moreover, we also derive the sharp trace Sobolev inequalities on half spaces for higher order derivatives. Finally, we compute the explicit formulas of adapted metric, introduced by Case and Chang, on the Euclidean unit ball, which is of independent interest.