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 φ belongs to Hs for s ≥ 1/2 with ‖φ‖L2 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 [25] where the smallness condition is not needed. We also prove that the real-valued H∞ solutions to mBO equation satisfy a priori local in time Hs bounds in terms of the Hs size of the initial data for s > 1/4.