Remark on the scattering operator for the cubic nonlinear Dirac equation in three space dimensions

This paper is concerned with the scattering operator S for the three-dimensional Dirac equation with a cubic nonlinearity. It follows from known results that S   is well-defined on a neighborhood of 0 in the Sobolev space Hκ(R3;C4)Hκ(R3;C4) for any κ>1κ>1. In the present paper, we prove that for any M∈NM∈N and s≥max⁡{κ,M}s≥max⁡{κ,M}, there exists some neighborhood U   of 0 in the weighted Sobolev space Hs,M(R3;C4)Hs,M(R3;C4) such that S(U)⊂Hs,M(R3;C4)S(U)⊂Hs,M(R3;C4).