Improved Lp -Poincaré inequalities on the hyperbolic space

We investigate the possibility of improving the optimal Lp-Poincaré inequality on the hyperbolic space, where p>1. We prove several different, and independent, improved inequalities, one of which is a Poincaré–Hardy inequality, namely an improvement of the best Lp-Poincaré inequality in terms of a Hardy weight related to geodesic distance from a given pole. Certain Hardy–Maz’ya-type inequalities in the Euclidean half-space are also obtained