Hardy-Rellich and second order Poincaré identities on the hyperbolic space via Bessel pairs

Abstract

We prove a family of Hardy-Rellich and Poincaré identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincaré 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.

Improved Hardy-Rellich inequalities

“…Improvements of these inequalities on bounded domains can be found in [27,28,42,47]. Hardy-Rellich inequalities valid on Riemaniann manifolds are investigated in [6,37,46,57]. Further generalizations can be found in [9,12,26].…”