Weighted Poincaré inequality and Hardy improvements related to some degenerate elliptic differential operators

Abstract: In this paper, we characterize the sharp constant and maximizing functions for weighted Poincar\'e inequalities. These results lead to refinements of Hardy's inequality obtained by adding remainder terms involving (L^p) norms. We use techniques that avoid symmetric rearrangement argument, simplifying the analysis of these inequalities in both Euclidean and non-Euclidean contexts. Specifically, this method applies to a variety of settings, such as the Heisenberg group, various Carnot groups and operators expressed as sums of squares of vector fields. Significant examples include the Heisenberg-Greiner operator and the Baouendi-Grushin operator.