Geodesic inversion and Sobolev spaces on Heisenberg type groups

Let s be the geodesic inversion on a Heisenberg type group N with homogeneous dimension Q, and denote by S the jacobian of s. We prove that, for -Q/2<a<Q/2, the operators T_a defined by T_a(f)= S^{1/2-a/Q} (f\circ s) are bounded on certain homogeneous Sobolev spaces H^a(N) if and only if N is an Iwasawa N-group.