Application of Hardy Identities and Inequalities on Cartan-Hadamard Manifolds

Abstract

We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vázquez in the Euclidean spaces. As an application on the way of [43], we establish several Hardy type inequalities that show improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces in the literature, with a little bite of change.

Keywords and phrases. Hardy inequality; Hardy-Poincaré-Sobolev; Cartan-Hadamard manifold; Hyperbolic space.