Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

We show the local wellposedness of the Cauchy problem for the cubic nonlinear Schrodinger equation in the modulation space M$^{s}$$_{p,q}$(R) where 1 $a \leq \ $b q < 3, 2 $a \leq \ $b p < 10q0 q0+6 and s $a \geq \ $b 0. This improves [7], where the case p = 2 was considered and the dierentiation by parts technique was introduced to a problem with continuous Fourier variable. Here the same technique is used, but more delicate estimates are necessary for p $\neq$ 2.