Improved L$^p$-Poincaré inequalities on the hyperbolic space

Abstract: 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.