Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups

In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: Hu|pdζ ≥ p - 1 pp (ζ)p dist(ζ+)p|u|pdζ,p > 1, which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335-352]. Here, (ζ) =i=1n(X i(ζ),ν)2 + (Y i(ζ),ν)21 2 is the angle function. Also, we obtain a version of the Hardy-Sobolev inequality in a half-space of the Heisenberg group: -p - 1 pp (ζ)p dist(ζ)p|u|pdζ1 p ≥ C +|u|pdζ 1 p, where dist(ζ, +) is the Euclidean distance to the boundary, p:= Qp/(Q - p), and 2 ≤ p < Q. For p = 2, this gives the Hardy-Sobolev-Maz'ya inequality on the Heisenberg group.