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

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.