We establish improved CR Sobolev inequalities on \(S^{2n+1}\) 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 \(S^{2n+1}\) and handles both the fractional and integral cases. In the same spirit, we derive the classical sharp Sobolev inequalities using commutator identities on \(S^n\) .
