Unweighted Hardy inequalities on the Heisenberg group and in step-two Carnot groups

We establish a projected Hardy-type inequality on step-two Carnot groups with one-dimensional vertical layer, which is sharp when the group has no Euclidean factor and under an additional symmetry assumption, and use it to derive quantitative lower bounds for the optimal full-horizontal-gradient Hardy constant. The approach is based on an integration-by-parts mechanism that replaces the non-horizontal Euler vector field by a suitably constructed horizontal vector field with controlled norm. As applications, we obtain explicit lower bounds in the Heisenberg group for both the Korányi gauge and the Carnot–Carathéodory distance, including closed-form estimates in specified parameter regimes, we extend the construction to non-isotropic structures and to groups with a Euclidean factor, and we obtain closed-form constants on arbitrary groups of Heisenberg type, such as the quaternionic Heisenberg group.