A-priori bounds for the 1-d cubic NLS in negative Sobolev spaces
  We consider the cubic Nonlinear Schrodinger Equation in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a-priori local in time H^s bounds in terms of the H^s size of the initial data for s greater than or equal to -1/6. 
Herbert Koch, Daniel Tataru