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We study low regularity local well-posedness of the nonlinear Schr\"odingerequation (NLS) with the quadratic nonlinearity $\overline{u}^2$, posed onone-dimensional and two-dimensional tori. While the relevant bilinear estimatewith respect to the $X^{s, b}$-space is known to fail when the regularity $s$is below some threshold value, we establish local well-posedness for such lowregularity by introducing modifications on the $X^{s, b}$-space.

Local well-posedness of the periodic nonlinear Schr\"odinger equation with a quadratic nonlinearity $\overline{u}^2$ in negative Sobolev spaces