Improved L$^p$-Poincar\'e inequalities on the hyperbolic space

TL;DR

This paper explores improved versions of the $p$-Poincaré inequality on hyperbolic space, including Hardy and Hardy-Maz'ya-type inequalities, providing sharper bounds and new insights into geometric analysis.

Contribution

The authors develop several independent improved inequalities for the $p$-Poincaré inequality on hyperbolic space, including a Poincaré-Hardy inequality and Hardy-Maz'ya-type inequalities.

Findings

Established improved Poincaré-Hardy inequalities on hyperbolic space.

Derived Hardy-Maz'ya inequalities in the Euclidean half-space.

Identified sharper bounds for the $p$-Poincaré inequality with Hardy weights.

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.