We prove Lᵖ-Hardy inequalities with distance to the boundary for domains in the Heisenberg group Hⁿ, n 1. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is then implemented for the distance defined by the gauge quasi-norm related to the fundamental solution of the horizontal Laplacian when the domain is a half-space or a convex polytope. Finally it is implemented for the Carnot-Carathéodory distance on half-spaces and arbitrary bounded convex domains of Hⁿ. In all cases the constant ( (p-1) /p) ᵖ is obtained. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak H-concavity of the Euclidean distance to the boundary, thus obtaining an alternative proof for the L²-Hardy inequality on convex domains.
Barbatis et al. (Tue,) studied this question.