We investigate the possibility of improving the $p$-Poincar\'e inequality $\|\nabla_{ \mathbb{ H}^N} u\|_p \ge \Lambda_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $\Lambda_p:=[(N-1)/p]^{ p}$ is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincar\'e-Hardy inequality, namely an improvement of the best $p$-Poincar\'e 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.
Categories: math.FA math.DG