On the existence of periodic solutions to the modified Korteweg-de Vries equation below $H^{1\mathrm{/}2}(\mathbb{T})$

Existence and a priori estimates for real-valued periodic solutions to the modified Korteweg-de Vries equation with initial data in $H^s$ are established for $s>0$. The short-time Fourier restriction norm method is employed to overcome the derivative loss. Further, non-existence of solutions below $L^2$ is proved conditional upon conjectured linear Strichartz estimates.