The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local well-posedness for data u 0 in the space b H s r (T), defined by the normsis shown in the parameter range s ≥ 1 2 , 2 > r > 4 3 . The proof is based on an adaptation of the gauge transform to the periodic setting and an appropriate variant of the Fourier restriction norm method.2000 Mathematics Subject Classification. 35Q55.

The Cauchy problem for the Zakharov-Kuznetsov equation is shown to be locally well-posed in H^s(R^2) for all s>1/2 by using the Fourier restriction norm method and bilinear refinements of Strichartz type inequalities

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces b H r s (R) defined by the norm 2000 Mathematics Subject Classification. 35Q53.

Abstract. We study the Cauchy problem for the modified KdV equationfor data u 0 in the space H r s defined by the normLocal well-posedness of this problem is established in the parameter range, 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.