Abstract
We study low regularity local well-posedness of the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity u¯2 , posed on one-dimensional and two-dimensional tori. While the relevant bilinear estimate with respect to the Xs,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 Xs,b -space.