Global well-posedness of the Benjamin-Ono equation in H^1(R)

We show that the Benjamin-Ono equation is globally well-posed in $H^s(\mathbb{R})$ for $s\ge 1$. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in $H^s$ for any $s$. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.