Preprint

Inserted: 6 mar 2026
Last Updated: 6 mar 2026

Year: 2026

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\`anyi gauge and the Carnot--Carath\'eodory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor\`anyi-type homogeneous norm.