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

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