Strichartz estimates for the Schrödinger equation for the sublaplacian on complex spheres

Abstract

In this paper we consider the sublaplacian L \mathcal L on the unit complex sphere S 2 n + 1 ⊂ C n + 1 S^{2n+1}\subset {\mathbf {C}}^{n+1} , equipped with its natural CR structure, and derive Strichartz estimates with fractional loss of derivatives for the solutions of the free Schrödinger equation associated with L \mathcal L . Our results are stated in terms of certain Sobolev-type spaces that measure the regularity of functions on S 2 n + 1 S^{2n+1} differently according to their spectral localization. Stronger conclusions are obtained for particular classes of solutions, corresponding to initial data whose spectrum is contained in a proper cone of N 2 {\mathbf {N}}^2 .