Improved Lp-Poincaré inequalities on the hyperbolic space

We investigate the possibility of improving the p-Poincaré inequality ‖∇HNu‖pp≥Λp‖u‖pp on the hyperbolic space, where p>1 and Λp:=[(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 Poincaré-Hardy inequality, namely an improvement of the best p-Poincaré 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.