Space-time estimates for null gauge forms and nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation in one space dimension with interaction satisfying null gauge condition. We prove the local well-posedness of the problem in the Sobolev space $H^{1/2}$. The method depends on the nonlinear gauge transformation and on sharp smoothing estimates for the null gauge form.