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We prove that the low energy parts of the wave operators for Schrödinger operators on are bounded in for and are unbounded for if has resonances at the threshold. If has eigenfunctions only at the threshold, it has recently been proved that they are bounded in for in general and for if all threshold eigenfunctions satisfy for . We prove in this case that they are unbounded in for unless the latter condition is satisfied. It is long known that the high energy parts are bounded in for all and that the same holds for if has no eigenfunctions nor resonances at the threshold.