NLS in the modulation space M$_{2,q}$

We show the local wellposedness of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space M$^{s}$ $_{2,q}$(R), 1 $a \leq \ $b q < 3 and s $a \geq \ $b 0. This improves [3] (for 2 $a \leq \ $b q < 3) where the cases 2 $a \leq \ $b q < unendlich were considered but the solution given there was not persistent. It is done with the use of the differentiation by parts technique and it is the first time that this purely periodic tool is used to attack a problem with a continuous Fourier variable.