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We establish improved CR Sobolev inequalities on CR sphere under thevanishing of higher order moments of the volume element. As a directapplication, we give a simpler proof of the existence and the classification ofminimizers of the CR invariant Sobolev inequalities which avoids complicatedcomputation in Frank and Lieb's proof. Our argument relies on nice commutatoridentities involving the CR intertwining operators on CR sphere and handlesboth the fractional and integral cases. In the same spirit, we derive theclassical sharp Sobolev inequalities using commutator identities on the sphere.

Improved Sobolev inequalities on CR shpere