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We obtain the local well-posedness of the one dimensional cubic nonlinearSchr\"odinger Equation for initial data in the modulation space $M_{2, p}$ forall $2\le p<\infty$, which covers all the subcritical cases from the viewpointof scaling. Moreover, in order to approach the endpoint space $M_{2,\infty}$,we will prove the almost global well-posedness in some Orlicz-type space, whichis a natural generalisation of $M_{2,p}$ for large $p$. The new ingredient isan endpoint version of the two dimensional restriction estimate.

On the 1D Cubic Nonlinear Schrodinger Equation in an Almost Critical Space