  The authors compute the long-time asymptotics for solutions of the NLS
equation just under the assumption that the initial data lies in a weighted
Sobolev space. In earlier work (see e.g. [DZ1],[DIZ]) high orders of decay and
smoothness are required for the initial data. The method here is a further
development of the steepest descent method of [DZ1], and replaces certain
absolute type estimates in [DZ1] with cancellation from oscillations.
