Remark on Well-posedness of Quadratic Schr\"odinger equation with nonlinearity $u\bar u$ in $H^{-1/4}(\R)$

Abstract

In this remark, we give another approach to the local well-posedness of quadratic Schr\"odinger equation with nonlinearity $u\bar u$ in $H^{-1/4}$, which was already proved by Kishimoto \cite{kis}. Our resolution space is $l^1$-analogue of $X^{s,b}$ space with low frequency part in a weaker space $L^{\infty}_{t}L^2_x$. Such type spaces was developed by Guo. \cite{G} to deal the KdV endpoint $H^{-3/4}$ regularity.