  We prove that the complex-valued modified Benjamin-Ono (mBO) equation is
locally wellposed if the initial data $\phi$ belongs to $H^s$ for $s\geq 1/2$
with $\norm{\phi}_{L^2}$ sufficiently small without performing a gauge
transformation. Hence the real-valued mBO equation is globally wellposed for
those initial datas, which is contained in the results of C. Kenig and H.
Takaoka \cite{KenigT} where the smallness condition is not needed. We also
prove that the real-valued $H^\infty$ solutions to mBO equation satisfy a
priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial
data for $s>1/4$.
