Improved higher order Poincar\'e inequalities on the hyperbolic space via Hardy-type remainder terms

The paper deals about Hardy-type inequalities associated with the following higher order Poincar'e inequality: $${\left(\frac{N-1}{2}\right)}^{2(k-l)}:=\underset{u\in C_c^{\mathrm{\infty }}\setminus \{0\}}{\inf }\frac{{\int }_{{\mathbb{H}}^N}\mathrm{\mid }{\mathrm{\nabla }}_{{\mathbb{H}}^N}^{k}u{\mathrm{\mid }}^{2}d{v}_{{\mathbb{H}}^N}}{{\int }_{{\mathbb{H}}^N}\mathrm{\mid }{\mathrm{\nabla }}_{{\mathbb{H}}^N}^{l}u{\mathrm{\mid }}^{2}d{v}_{{\mathbb{H}}^N}},$$ where $0\le l<k$ are integers and ${\mathbb{H}}^N$ denotes the hyperbolic space. More precisely, we improve the Poincar'e inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. Furthermore, when $k=2$ and $l=1$ the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms.