On the Cauchy problem for the NLS equations in R1+1

In this paper, we consider the local and global solutions for the nonlinear Schrödinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to be used are adapted from the Littlewood-Paley trichotomy, the Strichartz type estimate, Kato's smoothing effect and the maximal function (in time) estimate for the Schrödinger equation.