  We prove smoothing properties of the solutions of the Benjamin-Ono equation
in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s\ge 0$. To this
end we show that Tao's gauge transform is a high frequency approximation of the
nonlinear Fourier transform $\Phi$ 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.
