On decay and regularly of solutions of the Benjamin-Ono equation

We study persistence properties of solutions of the Benjamin-Ono equation in weighted Sobolev spaces. Roughly, we show that for $\beta<7/2$, the solution $u(x,t)$ of the BO remains in the space $L^2(|x|^{2\beta} dx)$ if and only if its data $u(x,0)$ belongs to this space and it is regular enough, i.e. $u_0\in H^{\beta}(\mathbb R)$.