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

Abstract

We study the nonlinear Schrödinger equation associated with the sublaplacian L on the unit sphere $S^{2n+1}$ in $C^{n+1}$ equipped with its natural CR structure. We first prove Strichartz estimates with fractional loss of derivatives for the solutions of the free Schrödinger equation and we then deduce some local in time well-posedness results. Our results are stated in terms of certain Sobolev-type spaces, that measure the regularity of functions on $S^{2}n+1$ 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 $N^2$.

32 pages. Final version, to appear in Transactions of the American Mathematical Society

References in corpus (2)

• L^p-summability of Riesz means for the sublaplacian on complex spheres
• Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary