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 $β<7/2$, the solution $u(x,t)$ of the BO remains in the space L²(|x|2β dx) if and only if its data $u(x,0)$ belongs to this space and it is regular enough, i.e. u₀∈ H^β( R).