On Smoothing Properties and Tao's Gauge Transform of the Benjamin-Ono Equation on the Torus

We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space Hs(T,R) for any s ≥ 0. To this end we show that Tao’s gauge transform is a high frequency approximation of the nonlinear Fourier transform Φ for the Benjamin-Ono equation, constructed in our previous work. The results of this paper are manifestations of the quasi-linear character of the Benjamin-Ono equation.