Sharp well-posedness for the Benjamin–Ono equation

The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces \(H^{s}\) for \(s>-\tfrac{1}{2}\). The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.