Almost sharp Sobolev trace inequalities in the unit ball under constraints

TL;DR

This paper proves almost sharp Sobolev trace inequalities of orders two and four in the unit ball with higher order moment constraints, revealing differences in sharpness between second and fourth order cases and applying the method to generalized Lebedev-Milin inequalities.

Contribution

It introduces new families of almost sharp Sobolev trace inequalities under higher order constraints and constructs smooth test functions to demonstrate their near optimality.

Findings

Established three families of almost sharp Sobolev trace inequalities.

Discovered distinct features in almost sharpness between second and fourth order inequalities.

Applied the construction method to show sharpness of generalized Lebedev-Milin inequalities.

Abstract

We establish three families of Sobolev trace inequalities of orders two and four in the unit ball under higher order moments constraint, and are able to construct \emph{smooth} test functions to show all such inequalities are \emph{almost optimal}. Some distinct feature in \emph{almost sharpness} examples between the fourth order and second order Sobolev trace inequalities is discovered. This has been neglected in higher order Sobolev inequality case in \cite{Hang}. As a byproduct, the method of our construction can be used to show the sharpness of the generalized Lebedev-Milin inequality under constraints.