In this paper, we study the critical Hardy inequalities with Bessel pairs with exact remainder terms and their applications. We then establish the relations between the critical Hardy inequalities and the subcritical Hardy inequalities on higher dimensional spaces. We also set up some Hardy identities that can be used to deduce the critical Hardy equality via a limiting process.

中文翻译：

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关于临界哈代不等式的一些说明

在本文中，我们研究了具有精确余项的 Bessel 对的临界 Hardy 不等式及其应用。然后我们在高维空间上建立临界哈代不等式和亚临界哈代不等式之间的关系。我们还建立了一些哈代恒等式，可用于通过限制过程推导出临界哈代等式。