We investigate the possibility of improving the p-Poincare ́ inequality ∥∇HN u∥p ≥ Λp ∥u∥p on the hyperbolic space, where p > 1 and Λp^p := [(N − 1)/p]p is the
best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincare ́–Hardy inequality,
namely an improvement of the best p-Poincare ́ inequality in terms of the Hardy weight r−p , r being geodesic distance from a given pole. Certain Hardy–Maz’ya-
type inequalities in the Euclidean half-space are also obtained.
