Lp - boundedness of the wave operator for the one dimensional Schroedinger operator

Given a one dimensional perturbed Schrödinger operator H = ?d2/dx2 + V(x), we consider the associated wave operators W±, defined as the strong L2 limits of U(s)U0(-s) as s goes to infinity. We prove that W± are bounded operators on L p for all 1 < p < ?, provided V(x) belongs to a suitable weighted L1 space. For p = infinity we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.