Existence and uniqueness of solution for a generalized nonlinear derivative Schrödinger equation

In this work we study the well-posedness for the initial value problem associated to a generalized derivative Schrödinger equation for small size initial data in weighted Sobolev space. The techniques used include parabolic regularization method combined with sharp linear estimates. An important point in our work is that the contraction principle is likely to fail but gives us inspiration to obtain certain uniform estimates that are crucial to obtain the main result. To prove such uniform estimates we assume smallness on the initial data in weighted Sobolev spaces.