In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) related to the radial derivation (i.e., the derivation along the geodesic curves) on the Cartan–Hadamard manifolds. By Gauss lemma, our new Hardy inequalities are stronger than the classical ones. We also establish the improvements of these inequalities in terms of sectional curvature of the underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequalities on the hyperbolic space ℍn. Especially, we show that our new Rellich inequalities are indeed stronger than the classical ones on the hyperbolic space ℍn.

中文翻译：

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Cartan-Hadamard 流形上新的尖锐 Hardy 和 Rellich 型不等式及其改进

在本文中，我们证明了与 Cartan 上的径向推导（即沿测地曲线的推导）相关的几个新的 Hardy 型不等式（例如加权 Hardy 不等式、加权 Rellich 不等式、临界 Hardy 不等式和临界 Rellich 不等式）——哈达玛歧管。根据高斯引理，我们的新哈代不等式比经典不等式更强。我们还根据底层流形的截面曲率确定了这些不等式的改进，这说明了曲率对这些不等式的影响。此外，我们在双曲空间上获得了 Hardy 和 Rellich 不等式的一些改进 ℍn. 特别是，我们证明了我们的新 Rellich 不等式确实比双曲空间上的经典不等式更强 ℍn.