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The Benjamin--Ono equation is shown to be well-posed, both on the line and onthe circle, in the Sobolev spaces $H^s$ for $s>-\tfrac12$. The proof rests on anew gauge transformation and benefits from our introduction of a modified Laxpair representation of the full hierarchy. As we will show, these developmentsyield important additional dividends beyond well-posedness, including (i) theunification of the diverse approaches to polynomial conservation laws; (ii) ageneralization of G\'erard's explicit formula to the full hierarchy; and (iii)new virial-type identities covering all equations in the hierarchy.

Sharp well-posedness for the Benjamin--Ono equation