Schrödinger Equations on Damek–Ricci Spaces

In this paper we consider the Laplace–Beltrami operator on Damek–Ricci spaces and derive pointwise estimates for the kernel of e^{\tau\Delta}, when \tau has nonnegative real part. We obtain in particular pointwise estimates of the Schrödinger kernel associated with \Delta. We then prove Strichartz estimates for the Schrödinger 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 Schrödinger equation associated with a distinguished Laplacian on Damek–Ricci spaces, showing that in this case the standard L^1-L^\infty estimate fails while suitable weighted Strichartz estimates hold.