Schr\"odinger equation on Damek-Ricci spaces

In this paper we consider the Laplace-Beltrami operator \Delta on Damek-Ricci spaces and derive pointwise estimates for the kernel of exp(\tau \Delta), when \tau \in C* with Re(\tau) \geq 0. When \tau \in iR*, we obtain in particular pointwise estimates of the Schr\"odinger kernel associated with \Delta. We then prove Strichartz estimates for the Schr\"odinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schr\"odinger equation associated with a distinguished Laplacian on Damek-Ricci spaces, showing that in this case the standard dispersive estimate fails while suitable weighted Strichartz estimates hold.