Convex hypersurfaces and Lp estimates for Schrödinger equations

This paper is concerned with Schrödinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the Lp–Lq estimate of the solution operator in the free case. This estimate, combined with the results of fractionally integrated groups, allows us to further obtain the Lp estimate of solutions for the initial data belonging to a dense subset of Lp in the case of integrable potentials.