Hardy-Rellich and second order Poincar\'e identities on the hyperbolic space via Bessel pairs

TL;DR

This paper establishes new Hardy-Rellich and Poincaré identities and inequalities on hyperbolic space, improving existing results and deriving second order uncertainty principles with sharp constants.

Contribution

It introduces a family of improved Hardy-Rellich and Poincaré identities on hyperbolic space, including radial cases, with applications to second order uncertainty principles.

Findings

New Hardy-Rellich and Poincaré identities with sharp constants

Improved remainder terms over previous literature

Second order uncertainty principles on hyperbolic space

Abstract

We prove a family of Hardy-Rellich and Poincar\'e identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincar\'e inequalities. All remainder terms provided considerably improve those already known in literature, and all identities hold with same constants for radial operators also. Furthermore, as applications of the main results, second order versions of the uncertainty principle on the hyperbolic space are derived.