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We consider the Cauchy problem for a quadratic derivative nonlinearSchr\"odinger equation whose nonlinearity is a linear combination of$\partial_x (u^2)$ and $\partial_x (|u|^2)$. We prove the local well-posednessin the $L^2$-based Sobolev space $H^s(\mathbb{R})$ for $s\ge 0$ with boundedprimitives. 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.

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