On higher order Poincaré inequalities with radial derivatives and Hardy improvements on the hyperbolic space

Abstract

In this paper we prove higher order Poincaré inequalities involving radial derivatives namely,

$$\begin{aligned} \int _{\mathbb {H}^{N}} |\nabla _{r,\mathbb {H}^{N}}^{k} u|^2 \, \mathrm{d}v_{\mathbb {H}^{N}} \ge \bigg (\frac{N-1}{2}\bigg )^{2(k-l)} \int _{\mathbb {H}^{N}} |\nabla _{r,\mathbb {H}^{N}}^{l} u|^2 \, \mathrm{d}v_{\mathbb {H}^{N}} \ \ \text { for all } u\in H^k(\mathbb {H}^{N}), \end{aligned}$$

where underlying space is N-dimensional hyperbolic space $\mathbb {H}^{N}$, $0\le l<k$ are integers and the constant $\big (\frac{N-1}{2}\big )^{2(k-l)}$ is sharp. Furthermore we improve the above inequalities by adding Hardy-type remainder terms and the sharpness of some constants is also discussed.