An improved local wellposedness result for the modified KdV equation

The Cauchy problem for the modified KdV equation is shown to be locally well posed for data u_0 in the space \hat(H^r_s) defined by the norm ||u_0||:=||<\xi>^s \hat(u_0)||_L^r', provided 4/3 < r \le 2, s \ge 1/2 - 1/(2r). For r=2 this coincides with the best possible result on the H^s - scale due to Kenig, Ponce and Vega. The proof uses an appropriate variant of the Fourier restriction norm method and linear as well as bilinear estimates for the solutions of the Airy equation.