  We prove an endpoint multilinear estimate for the $X^{s,b}$ spaces associated
to the periodic Airy equation. As a consequence we obtain sharp local
well-posedness results for periodic generalized KdV equations, as well as some
global well-posedness results below the energy norm. In particular we prove a
multilinear estimate which completes the proof of global well-posedness for
periodic KdV in a preceding paper (math.AP/0110045) down to the optimal
regularity H^{-1/2}.
