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 Schr\"odinger equation (NLS) with the quadratic nonlinearity \(\overline{u}^2\), posed on one-dimensional and two-dimensional tori. While the relevant bilinear estimate with 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 low regularity by introducing modifications on the \(X^{s, b}\)-space.