The initial value problem for the 1-D semilinear Schrodinger equation in Besov spaces

We define a class of Besov type spaces which is a generalization of that defined by Kenig-Ponce-Vega ([4], [5]) in their study on KdV equation and nonlinear Schrodinger equation. Using these spaces, we prove the following results: the 1-dimendional semilinear Schrodinger equation with the nonlinear term c(1)u(2) + c(2)(u) over bar (2) has a unique local-in-time solution for the initial data is an element of B-2,1(-3/4), and that with cu (u) over bar has a unique local-in-time solution for the initial data is an element of B-2,1(-1/4#). Note that B-2,1(-1/4,#) (R) superset of B-2,1(-1/4) (R) superset of H-s(R) for any s > -1/4.