  We study the initial value problem associated to the Benjamin-Ono equation.
The aim is to establish persistence properties of the solution flow in the
weighted Sobolev spaces $Z_{s,r}=H^s(\R)\cap L^2(|x|^{2r}dx)$, $s\in\R, \,s\geq
1$ and $s\geq r$. We also prove some unique continuation properties of the
solution flow in these spaces. In particular, these continuation principles
demostrate that our persistence properties are sharp.
