Nonlinear Schr\"odinger equation, differentiation by parts and modulation spaces

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr"odinger equation in the modulation space $M_{p,q}^s(\mathbb{R})$ where $1\le q\le 2$, $2\le p<\frac{10q^{\mathrm{\prime }}}{q^{\mathrm{\prime }}+6}$ and $s\ge 0$. Moreover, for either $1\le q\le \frac{3}{2},s\ge 0$ and $2\le p\le 3$ or $\frac{3}{2}<q\le \frac{18}{11},s>\frac{2}{3}-\frac{1}{q}$ and $2\le p\le 3$ or $\frac{18}{11}<q\le 2,s>\frac{2}{3}-\frac{1}{q}$ and $2\le p<\frac{10q^{\mathrm{\prime }}}{q^{\mathrm{\prime }}+6}$ we show that the Cauchy problem is unconditionally wellposed in $M_{p,q}^s(\mathbb{R})\mathrm{.}$ This improves \cite{NP}, where the case $p=2$ was considered and the differentiation 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 $p\mathrm{\ne }2$.