A scaling approach to Caffarelli-Kohn-Nirenberg inequality

Abstract

We consider the general Caffarelli-Kohn-Nirenberg inequality in the Euclidean and Remannian setting. From a new parameter introduced, the proof of the former case, follows by simple interpolation arguments and Hölder's inequality. Moreover, the ranges of this convenient parameter completely characterize the inequality. Secondly, the same technics are used to study the Caffarelli-Kohn-Nirenberg inequality in the Riemannian case.