Sharp Caffarelli-Kohn-Nirenberg inequalities on Riemannian manifolds: the influence of curvature

Abstract

We first establish a family of sharp Caffarelli-Kohn-Nirenberg type inequalities on the Euclidean spaces and then extend them to the setting of Cartan-Hadamard manifolds with the same best constant. The quantitative version of these inequalities also are proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli-Kohn-Nirenberg type inequalities with the same sharp constant as in ${\mathbb{R}}^n$ (shortly, sharp CKN inequalities). Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály to a larger class of the sharp CKN inequalities.

arXiv comment21 pages, comment are welcome. This is the second part of my previous preprint arXiv:1708.09306v1