  A filtered Lie splitting scheme is proposed for the time integration of the
cubic nonlinear Schr\"odinger equation on the two-dimensional torus
$\mathbb{T}^2$. The scheme is analyzed in a framework of discrete Bourgain
spaces, which allows us to consider initial data with low regularity; more
precisely initial data in $H^s(\mathbb{T}^2)$ with $s>0$. In this way, the
usual stability restriction to smooth Sobolev spaces with index $s>1$ is
overcome. Rates of convergence of order $\tau^{s/2}$ in $L^2(\mathbb{T}^2)$ at
this regularity level are proved. Numerical examples illustrate that these
convergence results are sharp.
