On Higher order Poincar\'e Inequalities with radial derivatives and Hardy improvements on the hyperbolic space

Abstract

In this paper we prove higher order Poincar\'e inequalities involving radial derivatives namely, \begin{equation*} \int_{\mathbb{H}^{N}} |\nabla_{r,\mathbb{H}^{N}}^{k} u|^2 \, {\rm d}v_{\mathbb{H}^{N}} \geq \bigg(\frac{N-1}{2}\bigg)^{2(k-l)} \int_{\mathbb{H}^{N}} |\nabla_{r,\mathbb{H}^{N}}^{l} u|^2 \, {\rm d}v_{\mathbb{H}^{N}} \ \ \text{ for all } u\in H^k(\mathbb{H}^{N}), \end{equation*} where underlying space is $N$-dimensional hyperbolic space $\mathbb{H}^{N}$, $0\leq l<k$ 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.