Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups

Abstract

We establish unweighted Hardy-type inequalities on step-two Carnot groups with one-dimensional vertical layer, with explicit lower bounds for the optimal Hardy constant. The approach is based on a quantitative 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 fully explicit bounds in the Heisenberg group for both the Kor{\`a}nyi gauge and the Carnot--Carath{\'e}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{\`a}nyi-type homogeneous norm.