Geometric Hardy inequalities on the Heisenberg groups via convexity

Abstract: We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group $\mathbb{H}^n$, $n\geq 1$. Our results are based on a geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is also implemented on half-spaces and convex polytopes for the distance defined by the gauge quasi-norm on $\mathbb{H}^n$ related to the fundamental solution of the horizontal Laplacian. 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 $L^2$-Hardy inequality on convex domains. In all cases the constants are shown to be sharp.