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

In this paper we introduce a unified and compact method for proving $L^p$Hardy and Rellich-type inequalities. The most important feature of our methodis that it allows us to extend these inequalities beyond the traditionalEuclidean setting. More precisely, the applications of this method span diversecontexts, including the Heisenberg group, general Carnot groups, and operatorsrepresented as the sum of squares of vector fields. Notable examples includethe sub-Laplacian on Carnot groups, the Heisenberg-Greiner operator, and theBaouendi-Grushin operator.