On the Cauchy problem for the derivative nonlinear Schroedinger equation   with periodic boundary condition
  It is shown that the Cauchy problem for the DNLS equation in the spatially periodic setting is locally well-posed in Sobolev spaces H^s(T) for s \geq 1/2. Moreover, global well-posedness is shown for s \geq 1 and data with small L^2 norm. 
S. Herr