PI: Zhong DMS-9500917 This project treats the almost everywhere behavior and the local smoothing property of the solutions of the Cauchy problem of the time-dependent Schrodinger equation with the initial value in a Sobolev space.To study the almost everywhere behavior of the solutions, by the Stein-Nikishin theory, it suffices to study the mapping properties of the associated maximal operators. By the Kolmogrov-Seliverstov-Plessner method, it suffices to study the mapping properties of the corresponding linearized operators. The local smoothing properties of the solutions are also the mapping properties of some linear operators. To study the mapping properties of the associated linear operators, the main idea of our approach is to estimate the kernel functions first and then apply classical Fourier analysis theories such as the Hardy-Littlewood-Sobolev theorem for fractional integrals, the Sobolev embedding theorem, the Calderon-Zygmund singular integral theory, and Stein's complex interpolation theorem for an analytic family of linear operators. The Tomas-Stein restriction theorem for the Fourier transform is related to these mapping properties. For certain special case, the orthogonality method plays an important role. The main difficulty is in estimating the kernel functions. Some oscillatory integral estimates, such as van der Corput type lemma, can be applied. The ideas of Carleson, Kenig and Ruiz, Sjolin, Vega, Ruiz and Vega seem to be important here. The Schrodinger equation is the basic equation of motion in Quantum Mechanics, which is an important branch of modern Physics. The results about the existence and the regularity of the solutions of the Schrodinger equation can be used to explain some phenomena in Physics. In January, 1926, Schrodinger invented the Schrodinger equation, which describes the discrete states of an atom. Since then, the existence and the regularity of the solutions of the Schrodinger equation have been i mportant research topics in Mathematics and Physics. In this project, we employ some important techniques in modern Mathematics to study the regularity of the solutions of the Schrodinger equation. The complete understanding of these problems will have some important applications in the areas such as Fourier Analysis, Partial Differential Equations and Mathematical Physics.