Well-posedness for a nonlinear Schrödinger equation with quadratic derivative nonlinearities for bounded primitive initial data

We consider the Cauchy problem for a quadratic derivative nonlinear Schrödinger equation whose nonlinearity is a linear combination of ${\mathrm{\partial }}_x(u^2)$ and ${\mathrm{\partial }}_x(\mathrm{\mid }u{\mathrm{\mid }}^2)$. We prove the local well-posedness in the $L^2$-based Sobolev space $H^s(\mathbb{R})$ for $s\ge 0$ with bounded primitives. Moreover, we prove the global well-posedness in $H^s(\mathbb{R})$ for $s\ge 1$ and a special case of the coefficients of nonlinearities.