On the Well-posedness of the 1-D Quadratic Semilinear Schrödinger Equations

This paper deals with a slight improvement on the results of the 1-D semilinear Schrdinger equations with quadratic nonlinearities.We study the local well-posedness of the initial value problem in particular function spaces containing the Sobolev spaces H s with s > -1/4 for the nonlinearity u, and with s > -3/4 for u 2 or 2 , in which the local well-posedness was proved by Kenig, Ponce and Vega.Our improvement lies in the estimate of the Fourier restriction norm with a homogeneous weight || s .It makes the behavior of the initial data at = 0 in the phase space less restrictive.