On the global well-posedness of the Benjamin-Ono equation

The Cauchy problem for the Benjamin-Ono equation is considered. It is shown that this problem is globally well-posed in H'(lli) for any s:::: 3/2. It is also established that for such values of s, local and global smoothing effects are present in the solution. These smoothing effects which are the main tools in the proof of the extremal case (s = 3/2) are reminiscent of the dispersive character of the associated linear equation.