Oscillatory integrals and Lp estimates for Schrödinger equations

This paper is concerned with Schrödinger equations whose principal operators are elliptic. Under certain degenerate condition we show the estimate of an oscillatory integral related to the solution operator in free case, and then employ fractionally integrated groups to obtain the Lp estimate of solutions for the initial data belonging to a dense subspace of Lp in the case of integrable potentials, which improves the corresponding result in [M. Balabane, H.A. Emami-Rad, Lp estimates for Schrödinger evolution equations, Trans. Amer. Math. Soc. 292 (1985) 357–373].