Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schr\"odinger equations

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schr"odinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first author obtained some well-posedness results in the Sobolev space $H^s$. But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using the nonlinear version of the classical Loomis-Whitney inequality, and prove the well-posedness in $H^s$ for $s\ge 1\mathrm{/}2$ if $d=2$, and $s>1\mathrm{/}2$ if $d=3$.