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arXiv · 1509.03857

The Caffarelli-Kohn-Nirenberg Inequality for Submanifolds in Riemannian Manifolds

Márcio Batista·Heudson Mirandola·Feliciano Vitório
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Abstract

After works by Michael and Simon [10], Hoffman and Spruck [9], and White [14], the celebrated Sobolev inequality could be extended to submanifolds in a huge class of Riemannian manifolds. The universal constant obtained depends only on the dimension of the submanifold. A sort of applications to the submanifold theory and geometric analysis have been obtained from that inequality. It is worthwhile to point out that, by a Nash Theorem, every Riemannian manifold can be seen as a submanifold in some Euclidean space. In the same spirit, Carron obtained a Hardy inequality for submanifolds in Euclidean spaces. In this paper, we will prove the Hardy, weighted Sobolev and Caffarelli-Kohn-Nirenberg inequalities, as well as some of their derivatives, as Galiardo-Nirenberg and Heisenberg-Pauli-Weyl inequalities, for submanifolds in a class of manifolds, that include, the Cartan-Hadamard ones.

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Márcio Batista, Heudson Mirandola, Feliciano Vitório. 2015-09-13. The Caffarelli-Kohn-Nirenberg Inequality for Submanifolds in Riemannian Manifolds. https://arxiv.org/abs/1509.03857

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