The purpose of this paper is to establish some Adams–Moser–Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space $\mathbb H^n$. First, we prove a sharp Adams’ inequality of order two with the exact growth condition in $\mathbb H^n$. Then we use it to derive a sharp Adams-type inequality and an Adachi–Tanakat-ype inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of $\mathbb H^n$, which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in $\mathbb H^n$. Our proofs rely on the symmetrization method extended to hyperbolic spaces.

中文翻译：

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双曲空间中的Sharp Adams–Moser–Trudinger型不等式

本文的目的是在双曲空间$ \ mathbb H ^ n $中建立一些Adams-Moser-Trudinger不等式，它们是Sobolev嵌入的临界案例。首先，我们证明了精确的增长条件为$ \ mathbb H ^ n $时，亚当斯的二阶尖锐不等式。然后，我们用它得出尖锐的Adams型不等式和Adachi-Tanakat型不等式。我们还证明了在$ \ mathbb H ^ n $的任何有界域上具有Navier边界条件的尖锐Adams型不等式，这将Tarsi的结果推广到双曲空间的设置。最后，我们按照$ \ mathbb H ^ n $的狮子精神建立了Lions型引理和改进的Adams型不等式。我们的证明依赖于扩展到双曲空间的对称化方法。