On the modified scattering for the critical derivative fractional nonlinear Schrödinger equation

We study the global in time existence of small solutions to the Cauchy problem for the critical derivative fractional nonlinear Schrödinger equation { iut−1α|∂x|αu=i∂x(|u|2u), t>0, x∈R,u(0,x)=u0(x), x∈R, where α∈(1,2). We prove a local existence of solutions in a suitable function space by using the parabolic regularization method which is not given in our previous paper and presnt a more simpler proof than that of [15].