Hardy's Identities and Inequalities on Cartan-Hadamard Manifolds

TL;DR

This paper investigates Hardy identities and inequalities on Cartan-Hadamard manifolds, providing new insights and improvements over existing inequalities, with implications for extremal functions and related inequalities in hyperbolic spaces.

Contribution

It introduces Hardy identities on Cartan-Hadamard manifolds using Bessel pairs, offering detailed information on extremal functions and improving known Hardy inequalities.

Findings

Hardy identities yield detailed extremal function information.

New Hardy inequalities improve upon existing results.

Applications to Hardy-Poincaré-Sobolev inequalities on hyperbolic spaces.

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-V\'{a}zquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincar\'{e}-Sobolev type inequalities on hyperbolic spaces in the literature.