On a class of solutions to the generalized derivative Schr\"odinger equations II

In this note we shall continue our study on the initial value problem associated for the generalized derivative Schr\"odinger (gDNLS) equation $$\partial_tu=i\partial_x^2u + \mu\,|u|^{\alpha}\partial_x u, \hskip10pt x,t\in\mathbb{R}, \hskip5pt 0<\alpha \le 1\;\; {\rm and}\;\; |\mu|=1.$$ Inspiring by Cazenave-Naumkin's works we shall establish the local well-posedness for a class of data of arbitrary size in an appropriate weighted Sobolev space, thus removing the size restriction on the data required in our previous work. The main new tool in the proof is the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schr\"odinger equation with lower order variable coefficients established by Kenig-Ponce-Vega.