Improved higher-order Sobolev inequalities on CR sphere

We improve higher-order CR Sobolev inequalities on S2n+1 under the vanishing of higher order moments of the volume element. As an application, we give a new and direct proof of the classification of minimizers of the CR invariant higher-order Sobolev inequalities. In the same spirit, we prove almost sharp Sobolev inequalities for GJMS operators to general CR manifolds, and obtain the existence of minimizers in C2k(N) of higher-order CR Yamabe-type problems when Yk(N) < Yk(Hn).(c) 2023 Elsevier Inc. All rights reserved.