The IVP for the Benjamin–Ono equation in weighted Sobolev spaces

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 Zs,r = Hs (R) ∩ L2(|x|2r dx), s ∈ R, s 1 and s r. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence properties are sharp. © 2010 Elsevier Inc. All rights reserved.