Quadratic Nonlinear Derivative Schr\"odiger Equations - Part 1

In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a local well-posedness result up to the scaling for small initial data with some spherical symmetry structure.