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>-(1)|(2). The proof rests on a new gauge transfor-mation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional divi-dends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of G & eacute;rard's explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy