  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 $s>-\tfrac12$. 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 G\'erard's explicit formula to the full hierarchy; and (iii)
new virial-type identities covering all equations in the hierarchy.
