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arXiv · 1709.06120

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

Van Hoang Nguyen
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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 Krist\'aly to a larger class of the sharp CKN inequalities.

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Van Hoang Nguyen. 2017-09-18. Sharp Caffarelli-Kohn-Nirenberg inequalities on Riemannian manifolds: the influence of curvature. https://arxiv.org/abs/1709.06120

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