Modified Wave Operators for the Hartree Equation with Data,Image and Convergence in the Same Space,II

We study modified wave operators for the Hartree equation with a long-range potential |x|^-n |x|^{-nu} , extending the result in [12] to the whole range of the Dollard type 1/2 < n nu < 1. We construct the modified wave operators in the whole space of (1 + |x|)^-sL² (1 + |x|)^{-s}L^2 . We also have the image, strong continuity and strong asymptotic approximation in the same space. The lower bound $ s > 1 - nu / 2 $ s > 1 - nu / 2 of the weight is sharp from the scaling argument. Those maps are homeomorphic onto open subsets, which implies in particular asymptotic completeness for small data.