Gauge transformation for the kinetic derivative nonlinear Schr\"odinger equation on the torus

We consider the kinetic derivative nonlinear Schr\"odinger equation, which is a one-dimensional nonlinear Schr\"odinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. In our previous work, we proved small-data global well-posedness of the Cauchy problem on the torus in Sobolev space $H^s$ for $s>1/2$ by combining the Fourier restriction norm method with the parabolic smoothing effect, which is available in the periodic setting. In this article, we improve the regularity range to $s>1/4$ for the global well-posedness by constructing an effective gauge transformation. Moreover, we remove the smallness assumption by making use of the dissipative nature of the equation.