Error estimates at low regularity of splitting schemes for NLS

We study a filtered Lie splitting scheme for the cubic nonlinear Schr"{o}dinger equation. We establish error estimates at low regularity by using discrete Bourgain spaces. This allows us to handle data in $H^s$ with $0<s<1$ overcoming the standard stability restriction to smooth Sobolev spaces with index $s>1\mathrm{/}2$ . More precisely, we prove convergence rates of order ${\tau }^{s\mathrm{/}2}$ in $L^2$ at this level of regularity.