The Caffarelli-Kohn-Nirenberg Inequalities on Complete Manifolds

We find a new sharp Caffarelli-Kohn-Nirenberg inequality and show that the Euclidean spaces are the only complete non-compact Riemannian manifolds of non-negative 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.

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Published 1 January 2007