On sharpness of the local Kato-smoothing property for dispersive wave equations

Constantin and Saut showed in 1988 that solutions of the Cauchy problem for general dispersive equations $$ w_t +iP(D)w=0,\quad w(x,0)=q (x), \quad x\in \mathbb{R}^n, \ t\in \mathbb{R} , $$ enjoy the local smoothing property $$ q\in H^s (\R ^n) \implies w\in L^2 \Big (-T,T; H^{s+\frac{m-1}{2}}_{loc} \left (\R^n\right )\Big ) , $$ where $m$ is the order of the pseudo-differential operator $P(D)$. This property, now called local Kato smoothing, was first discovered by Kato for the KdV equation and implicitly shown later by Sj\"olin for the linear Schr\"odinger equation. In this paper, we show that the local Kato smoothing property possessed by solutions general dispersive equations in the 1D case is sharp, meaning that there exist initial data $q\in H^s \left (\R \right )$ such that the corresponding solution $w$ does not belong to the space $ L^2 \Big (-T,T; H^{s+\frac{m-1}{2} +\epsilon}_{loc} \left (\R\right )\Big )$ for any $\epsilon >0$.