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We prove that the initial value problem associated to a nonlocal perturbationof the Benjamin-Ono equation is locally and globally well-posed in Sobolevspaces $H^s(\mathbb{R})$ for any $s>-3/2$ and we establish that our result issharp 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 thesolution flow in the weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb{R})\capL^2(|x|^{2r}\,dx)$ for $s\geq r >0$. We also prove some unique continuationproperties of the solution flow in these spaces.

The IVP for a nonlocal perturbation of the Benjamin-Ono equation in classical and weighted Sobolev spaces