Discrete Fourier restriction phenomenon associated with some periodic dispersive equations

The thesis consists of six chapters. In Chapter 1, we will briefly introduce the background of the topic, as well as some results we already know. The next five chapters can be divided into two parts. The first part is about the discrete Fourier restriction phenomenon. In Chapter 2, we consider the discrete Fourier restriction phenomenon associated with Schrodinger equations. We study the size of the Fourier transform of a periodic function on a truncated discrete paraboloid. We develop two ways to tackle the problem, and the second one recovers Bourgain's level set result on Strichartz estimates associated with periodic Schrodinger equations. Some sharp estimates on $L^\frac{2(d+2)}{d}$ norms of certain exponential sums in higher dimensional cases are established. In Chapter 3 we further discuss the discrete Fourier restriction problem associated with higher order dispersive equations, with the method developed in Chapter 2. We obtain some sharp bound on the size of the Fourier transform of a function for large indices. Some new Strichartz estimates of this type are obtained. Also, we can use the method to prove some exponential sum estimates, which are classic in number theory. The second part of the thesis is about the local well-posedness of some dispersive equations. In Chapter 4, we prove the local well-posedness of the periodic gKdV equations. The method we apply here is a generalization of Bourgain's "denominator manipulation". With this idea, we further discuss a more general type of KdV equation in Chapter 5. We establish the local well-posedness of the periodic KdV equations with nonlinear terms $F(u)u_x$, provided $F\in C^5$ and the initial data $u_0\in H^s$ with $s>1/2$ (1/2 is sharp). In Chapter 6 we focus on the local well-posedness of the periodic fifth order KdV type dispersive equations with nonlinear terms $P_1(u)u_x + P_2(u)u_x^2$, provided the initial data $u_0\in H^s$ with $s>1$. Here $P_1(u)$ and $P_2(u)$ are polynomials of $u$. Some Strichartz estimates derived in Chapter 3 are used in the proof. Also, a couple of counterexamples are given to exhibit the sharpness of the indices