2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/164013We prove that the Benjamin-Ono equation is globally well-posed in $ H^s(\T) $ for $ s\ge 0 $. Moreover we show that the associated flow-map is Lipschitz on every bounded set of $ {\dot H}^s(\T) $, $s\ge 0$, 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 $ H^\infty(\T) $) cannot be of class $ C^{1+α} $, $α>0 $, from $ {\dot H}^s(\T) $ into $ {\dot H}^s(\T) $ as soon as $ s< 0 $.47 pagesAnalysis of PDEs35A05; 35Q53text