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.

中文翻译：

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关于带有径向导数的高阶Poincaré不等式和双曲空间上的Hardy改进

在本文中，我们证明了涉及径向导数的高阶Poincaré不等式，即

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

其中基础空间是N维双曲空间\（\ mathbb {H} ^ {N} \），\（0 \ le l <k \）是整数，常数\（\ big（\ frac {N-1} {2} \ big）^ {2（kl）} \）很锋利。此外，我们通过添加Hardy型余项来改善上述不等式，并讨论了一些常数的敏锐度。