Unconditional local well-posedness for periodic NLS

The nonlinear Schr"odinger equations with nonlinearities $\mathrm{\mid }u{\mathrm{\mid }}^{2k}u$ on the $d$-dimensional torus are considered for arbitrary positive integers $k$ and $d$. The solution of the Cauchy problem is shown to be unique in the class $C_t{H}_x^s$ for a certain range of scale-subcritical regularities $s$, which is almost optimal in the case $d\ge 4$ or $k\ge 2$. The proof is based on various multilinear estimates and the infinite normal form reduction argument.