On unconditional well-posedness of modified KdV

Bourgain(1993) proved that the periodic modified KdV equation (mKdV) is locally well-posed in Sobolev spave H^s(T), s >= 1/2, by introducing new weighted Sobolev spaces X^s,b, where the uniqueness holds conditionally, namely in the intersection of C([0, T]; H^s) and X^s,b. In this paper, we establish unconditional well-posedness of mKdV in H^s(T), s >= 1/2, i.e. we in addition establish unconditional uniqueness in C([0, T]; H^s), s >= 1/2, of solutions to mKdV. We prove this result via differentiation by parts. For the endpoint case s = 1/2, we perform careful quinti- and septi-linear estimates after the second differentiation by parts.