Polynomial Bound and Nonlinear Smoothing for the Benjamin-Ono Equation on the Circle

For initial data in Sobolev spaces $H^s(\mathbb{T})$, $\frac 12 < s \leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t{)}^{3(s-\frac{1}{2})+ϵ}$, $0<ϵ\ll 1$. Key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.