  We prove that the initial value problem associated to a nonlocal perturbation
of the Benjamin-Ono equation is locally and globally well-posed in Sobolev
spaces $H^s(\mathbb{R})$ for any $s>-3/2$ and we establish that our result is
sharp in the sense that the flow map of this equation fails to be $C^2$ in
$H^s(\mathbb{R})$ for $s<-3/2$. Finally, we study persistence properties of the
solution flow in the weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb{R})\cap
L^2(|x|^{2r}\,dx)$ for $s\geq r >0$. We also prove some unique continuation
properties of the solution flow in these spaces.
