Global Smooth Effects and Well-Posedness for the Derivative Nonlinear Schr\"odinger Equation with Small Rough Data

\rm We obtain the global smooth effects for the solutions of the linear Schr"odinger equation in anisotropic Lebesgue spaces. Applying these estimates, we study the Cauchy problem for the generalized elliptical and non-elliptical derivative nonlinear Schr"odinger equations (DNLS) and get the global well posedness of solutions with small data in modulation spaces $M_{2,1}^{3\mathrm{/}2}({\mathbb{R}}^n)$. Noticing that $H^{\tilde{s}}\subset M_{2,1}^s$ $(\tilde{s}-s>n\mathrm{/}2)$ is an optimal inclusion, we have shown the global well posedness of DNLS with a class of very rough data.