  The Cauchy problem for the derivative nonlinear Schr\"odinger equation with
periodic boundary condition is considered. Local well-posedness for periodic
initial data u_0 in the space ^H^s_r, defined by the norms
||u_0||_{^H^s_r}=||<xi>^s ^u_0||_{l^r'} is 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.
