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