Almost sharp weighted Sobolev trace inequalities in the unit ball under constraints

Abstract

In this paper, we establish some improved weighted Sobolev trace inequalities [Formula: see text] under the zero higher order moments constraint via the concentration compactness principle, where [Formula: see text] is a defining function of [Formula: see text] and [Formula: see text]. This relates to the fractional (conformal) Laplacians and related problems in conformal geometry. We construct some test functions and show that the inequality is almost optimal when [Formula: see text].