On a system of quadratic nonlinear Schrödinger equations and scale invariant spaces in 2D

We consider the initial value problem of a system of 2D nonlinear Schrödinger equations with quadratic nonlinearities in homogeneous Sobolev spaces which are close to the scale invariant spaces. We show the global solution, which is not necessarily in $\mathbf{L}^{2}( \mathbb{R}^{2})$, satisfies uniform time decay of order $|t|^{-1}$ in $\mathbf{L}^{\infty }( \mathbb{R}^{2})$.