Local Well-posedness and a priori bounds for the modified Benjamin-Ono equation without using a gauge transformation

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 $s>1/4$.