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We study the Cauchy problem for the generalized elliptic and non-ellipticderivative nonlinear Schrodinger equations, the existence of the scatteringoperators and the global well posedness of solutions with small data in Besovspaces and in modulation spaces are obtained. In one spatial dimension, we getthe sharp well posedness result with small data in critical homogeneous Besovspaces. As a by-product, the existence of the scattering operators with smalldata is also shown. In order to show these results, the global versions of theestimates for the maximal functions on the elliptic and non-ellipticSchrodinger groups are established.

Global well posedness and scattering for the elliptic and non-elliptic derivative nonlinear Schrodinger equations with small data