Local Well-Posedness of the Periodic Nonlinear Schrödinger Equation with a Quadratic Nonlinearity $$\overline{u}^2$$ in Negative Sobolev Spaces

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