Global well-posedness in L 2 for the periodic Benjamin-Ono equation

We prove that the Benjamin-Ono equation is globally well-posed in Hˢ( T) for s≥ 0. Moreover we show that the associated flow-map is Lipschitz on every bounded set of Hˢ₀( T), s≥ 0, and even real-analytic in this space for small times. This result is sharp in the sense that the flow-map (if it can be defined and coincides with the standard flow-map on H₀^∞( T)) cannot be of class C1+α, α>0, from H₀ˢ( T) into H₀ˢ( T) as soon as $ s< 0 $.