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

TL;DR

This paper extends classical inequalities like Hardy, Sobolev, and Caffarelli-Kohn-Nirenberg to submanifolds within a broad class of Riemannian manifolds, including Cartan-Hadamard spaces, broadening their applicability in geometric analysis.

Contribution

It proves new Hardy, weighted Sobolev, and Caffarelli-Kohn-Nirenberg inequalities for submanifolds in various Riemannian manifolds, including Cartan-Hadamard spaces.

Findings

Established Hardy inequalities for submanifolds in Riemannian manifolds.

Derived weighted Sobolev and Caffarelli-Kohn-Nirenberg inequalities.

Extended classical inequalities to a wider geometric setting.

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.