Intermediate long wave equation in negative Sobolev spaces

We study the intermediate long wave equation (ILW) in negative Sobolev spaces. In particular, despite the lack of scaling invariance, we identify the regularity $s=-\frac{1}{2}$ as the critical regularity for ILW with any depth parameter, by establishing the following two results. (i) By viewing ILW as a perturbation of the Benjamin-Ono equation (BO) and exploiting the complete integrability of BO, we establish a global-in-time a priori bound on the $H^s$-norm of a solution to ILW for $-\frac{1}{2}<s<0$. (ii) By making use of explicit solutions, we prove that ILW is ill-posed in $H^s$ for $s<-\frac{1}{2}$. Our results apply to both the real line case and the periodic case.