The Caffarelli-kohn-nirenberg Inequalities on Complete Manifolds

@inproceedings{Xia2007TheCI,
  title={The Caffarelli-kohn-nirenberg Inequalities on Complete Manifolds},
  author={Changyu Xia},
  year={2007}
}
We find a new sharp Caffarelli-Kohn-Nirenberg inequality and show that the Euclidean spaces are the only complete non-compact Riemannian manifolds of nonnegative Ricci curvature satisfying this inequality. We also show that a complete open manifold with non-negative Ricci curvature in which the optimal Nash inequality holds is isometric to a Euclidean space.