A unified approach to $L^{p}$ Hardy and Rellich-type inequalities in Euclidean and non-Euclidean settings

We present a unified and concise method for establishing $L^p$ Hardy and Rellich inequalities for a broad class of subelliptic operators of divergence type. The approach, based on a fundamental algebraic identity, provides explicit control on maximizing sequences and yields sharp constants in several significant cases. It applies beyond the Euclidean framework, covering the Heisenberg and Carnot group settings, and extends to a variety of subelliptic operators such as the Heisenberg–Greiner and Baouendi–Grushin operators.