Resonant phase-shift and global smoothing of the periodic Korteweg-de Vries in low regularity settings

We show a smoothing effect of near full derivative for low-regularity global-in-time solutions of the periodic Korteweg-de Vries (KdV) equation. The smoothing is given by slightly shifting the space-time Fourier support of the nonlinear solution, which we call \emph{resonant phase-shift}. More precisely, we show that $[\mathcal{S}u](t) - e^{-t\partial_x^3} u(0) \in H^{-s+1-}$ where $u(0) \in H^{-s}$ for $0\leq s <1/2$ where $\mathcal{S}$ is the resonant phase-shift operator described below. We use the normal form method to obtain the result.