Improved Sobolev inequalities on CR shpere

TL;DR

This paper improves CR Sobolev inequalities on the CR sphere by leveraging higher order moment conditions, providing simpler proofs for minimizers' existence and classification, and deriving classical inequalities through commutator identities.

Contribution

It introduces improved inequalities under moment vanishing conditions and simplifies proofs of minimizers' properties using commutator identities on the CR sphere.

Findings

Established improved CR Sobolev inequalities with moment conditions

Provided simpler proofs for minimizers' existence and classification

Derived classical sharp Sobolev inequalities using commutator identities

Abstract

We establish improved CR Sobolev inequalities on CR sphere under the vanishing of higher order moments of the volume element. As a direct application, we give a simpler proof of the existence and the classification of minimizers of the CR invariant Sobolev inequalities which avoids complicated computation in Frank and Lieb's proof. Our argument relies on nice commutator identities involving the CR intertwining operators on CR sphere and handles both the fractional and integral cases. In the same spirit, we derive the classical sharp Sobolev inequalities using commutator identities on the sphere.