  In this paper we show that the floow map of the Benjamin-Ono equation on the
line is weakly continuous in L2(R), using "local smoothing" estimates. L2(R) is
believed to be a borderline space for the local well-posedness theory of this
equation. In the periodic case, Molinet [27] has recently proved that the flow
map of the Benjamin-Ono equation is not weakly continuous in L2(T). Our results
are in line with previous work on the cubic nonlinear Schrodinger equation,
where Goubet and Molinet [11] showed weak continuity in L2(R) and Molinet [28]
showed lack of weak continuity in L2(T).
