Short-range scattering of Hartree type fractional NLS

In this paper we consider scattering problem for Hartree type fractional NLS with |∇|α|∇|α (1<α<21<α<2) and potential V∼|x|−γV∼|x|−γ. We show small data scattering in a weighted space for the short range 6−2α4−α<γ<2. The difficulty arises from the non-locality and non-smoothness of |∇|α|∇|α. To overcome it we utilize the method of commutator estimate based on Balakrishnan's formula.