Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schrödinger equation with angular regularity

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in R×R2 with the nonlinearity η|u|2 where η∈C∖{0} and low regularity initial data. If s<−1/4, the ill-posedness result in the Sobolev space Hs(R2) is known. We will prove the well-posedness in Hs(R2) for −1/2