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We show the existence of weak solutions in the extended sense of the Cauchyproblem for the cubic nonlinear Schr\"odinger equation in the modulation space$M_{2,q}^{s}(\mathbb R)$, $1\leq q\leq2$ and $s\geq0.$ In addition, for either$s\geq 0$ and $1\leq q\leq\frac32$ or $\frac32<q\leq 2$ and$s>\frac23-\frac1{q}$ we show that the Cauchy problem is unconditionallywellposed in $M_{2,q}^{s}(\mathbb R).$ It is done with the use of thedifferentiation by parts technique which had been previously used in theperiodic setting.

NLS in the modulation space $M_{2,q}(\mathbb R)$