Improved higher-order Sobolev inequalities on CR sphere

TL;DR

This paper advances higher-order CR Sobolev inequalities on the sphere by incorporating moment conditions, providing new proofs for minimizer classification, and extending results to general CR manifolds with applications to CR Yamabe problems.

Contribution

It introduces improved higher-order CR Sobolev inequalities under moment vanishing conditions and extends sharp inequalities to general CR manifolds, with implications for CR Yamabe-type problems.

Findings

Established improved higher-order CR Sobolev inequalities with moment conditions.

Provided a new proof for classification of minimizers of CR invariant inequalities.

Proved existence of minimizers for CR Yamabe-type problems when certain Yamabe constants are favorable.

Abstract

We improve higher-order CR Sobolev inequalities on $S^{2n+1}$ 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 $C^{2k}(N)$ of higher-order CR Yamabe-type problems when $Y_k(N)<Y_k(\mathbb{H}^n)$.