On the Cauchy problem for the derivative nonlinear Schrodinger equation with periodic boundary condition

It is shown that the Cauchy problem associated to the derivative nonlinear Schrodinger equation partial derivative(t)u - i partial derivative(2)(x)u = lambda partial derivative(x)(vertical bar u vertical bar(2)u) is locally well-posed for initial data u(0) is an element of H-s(T), if s >= 1/2 and. is real. The proof is based on a variant of the gauge transformation, introduced by Hayashi and Ozawa, adjusted to the periodic setting and sharp multilinear estimates for the gauge equivalent equation in Fourier restriction norm spaces. By the use of a conservation law, the problem is shown to be globally well-posed for s >= 1 and data which is small in L-2.