Local well-posedness for the modified KdV equation in almost critical ^H^r_s-spaces

We study the Cauchy problem for the modified KdV equation for data u_0 in the space ^H^r_s defined by the norm ||u_0||_{^H^r_s}:=||<\xi>^s u^_0||_{L^r'_\xi}. Local well-posedness of this problem is established in the parameter range 2>=r>1, s>=1/2-1/2r, 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.