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 $\beta <7\mathrm{/}2$, the solution $u(x,t)$ of the BO remains in the space $L^2(\mathrm{\mid }x{\mathrm{\mid }}^{2\beta }dx)$ if and only if its data $u(x,0)$ belongs to this space and it is regular enough, i.e. $u_0\in H^{\beta }(\mathbb{R})$.