Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group

We prove geometric L-p versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains Omega in the Heisenberg group H-n, where convex is meant in the Euclidean sense. When p = 2 and Omega is the half-space given <xi, nu > > d by this generalizes an inequality previously obtained by Luan and Yang. For such p and the inequality is sharp and takes the form integral(Omega)vertical bar del(Hn) u vertical bar(2) d xi >= 1/4 integral Omega Sigma(n)(i=1) < X-i(xi), nu >(2) + < Y-i(xi), nu >(2)/dist(xi, partial derivative Omega)(2) vertical bar u vertical bar(2) d xi, where dist (., partial derivative Omega) denotes the Euclidean distance from partial derivative Omega.