In this paper, we obtain Hardy, Hardy–Rellich and refined Hardy inequalities on general stratified groups and weighted Hardy inequalities on general homogeneous groups using the factorization method of differential operators, inspired by the recent work of Gesztesy and Littlejohn [9]. We note that some of the obtained inequalities are new also in the usual Euclidean setting. We also obtain analogues of Gesztesy and Littlejohn’s 2-parameter version of the Rellich inequality on stratified groups and on the Heisenberg group, and a new two-parameter estimate on $\mathbb R^n$ which can be regarded as a counterpart to the Gesztesy and Littlejohn estimate. Moreover, we give some applications of Hardy and Rellich type inequalities to the lower semiboundedness and to form boundedness for interactions with countably many strong singularities on stratified groups.

中文翻译：

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分层组的因式分解和Hardy-Rellich不等式

在本文中，我们根据微格算子的因式分解方法，获得了一般分层组的Hardy，Hardy-Rellich和精炼的Hardy不等式以及加权加权Hardy不等式。我们注意到，在通常的欧几里得设置中，某些获得的不等式也是新的。我们还获得了分层组和Heisenberg组上Gesztesy和Littlejohn的Rellich不等式的2参数版本的类似物，以及$ \ mathbb R ^ n $的新的两参数估计，可以将其视为Gesztesy的对应物。和小约翰估计。此外，