Abstract: The mapping of initial data to solution at time t for the free Schrödinger equation is only L^p-bounded for p = 2 in Euclidean space ℝ^d. For other p, boundedness can be restored at the loss of 2d|1/p - 1/2| derivatives and a norm growth in time. This holds also in Hardy spaces for p ≤ 1. What can be said about this property for the Heisenberg group? We shall see that even though the Fourier transform on the Heisenberg group has striking differences from the one on Euclidean space, many of the techniques from harmonic analysis can be carried over to this setting. We shall also mention how the methods can be extended also to the general setting of step 2 stratified Lie groups.
