Well-posedness for a nonlinear Schr\"{o}dinger equation with quadratic derivative nonlinearities for bounded primitive initial data

We consider the Cauchy problem for a quadratic derivative nonlinear Schr\"odinger equation whose nonlinearity is a linear combination of $\partial_x (u^2)$ and $\partial_x (|u|^2)$. 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.