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: integral(H+) vertical bar del(H)u vertical bar(p)d xi >= (p - 1/p)(p) integral(H+) W(xi)(p)/dist(xi, partial derivative H+)(p) vertical bar u vertical bar(p)d xi, 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, W(xi) = (Sigma(n)(i=1) (2) + (2))(1/2) is the angle function. Also, we obtain a version of the Hardy-Sobolev inequality in a half-space of the Heisenberg group: (integral(H+) vertical bar del(H)u vertical bar(p)d xi - (p - 1/p)(p) integral(H+) W(xi)(p)/dist(xi, partial derivative H+)(p) vertical bar u vertical bar(p)d xi)(1/p) >= C (integral(H+) vertical bar u vertical bar(p)* d xi)(1/p)*, where dist(xi, partial derivative H+) is the Euclidean distance to the boundary, p* := Q(p)/(Q - p), and 2 <= p < Q. For p = 2, this gives the Hardy-Sobolev-Maz'ya inequality on the Heisenberg group