Local well-posedness for the modified KdV equation in almost critical $\widehat {H^r_s}$-spaces

We study the Cauchy problem for the modified KdV equation \[ u_t + u_{xxx} + (u^3)_x = 0, \hspace {2cm} u(0)=u_0\] for data $u_0$ in the space $\widehat {H_s^r}$ defined by the norm \[ \|u_0\|_{\widehat {H_s^r}} := \|\langle \xi \rangle ^s\widehat {u_0}\| _{L^{r’}_{\xi }}.\] Local well-posedness of this problem is established in the parameter range $2 \ge r >1$, $s \ge \frac {1}{2} - \frac {1}{2r}$, so the case $(s,r)=(0,1)$, which is critical in view of scaling considerations, is almost reached. To show this result, we use an appropriate variant of the Fourier restriction norm method as well as bi- and trilinear estimates for solutions of the Airy equation.