  We continue the study of the theory of scattering for some long range Hartree
equations with potential |x|^-gamma, performed in a previous paper, denoted as
I, in the range 1/2 < gamma < 1. Here we extend the results to the range 1/3 <
gamma < 1/2. More precisely, we study the local Cauchy problem with infinite
initial time, which is the main step in the construction of the modified wave
operators. We solve that problem without loss of regularity between the
asymptotic state and the solution, as in I, but in contrast to I, we are no
longer able to cover the entire subcritical range of regularity of the
solutions. The method is an extension of that of I, using a better asymptotic
form of the solutions, obtained as the next step of a natural procedure of
successive approximations.
