Well-posedness for a quadratic derivative nonlinear Schrödinger system at the critical regularity

We consider the Cauchy problem for a quadratic derivative nonlinear Schrödinger system in Sobolev space HsHs. In this paper, under the null condition and the mass resonance relation, we prove large data local well-posedness if the space dimension d≥3d≥3 and s≥scs≥sc, d=2d=2 and s>scs>sc or d=1d=1 and s≥0s≥0, where scsc is the scaling critical regularity. Moreover, under the same assumptions, we also prove small data global well-posedness and scattering in HsHs, if d≥3d≥3 and s≥scs≥sc. Our proof is based on a contraction argument using the UpUp and VpVp type function spaces.