Local well-posedness and a priori bounds for the modified Benjamin-Ono equation

We prove that the complex-valued modified Benjamin-Ono (mBO) equation is analytically locally well posed if the initial data ? be-longs to Hs for s ? 1/2 with ? L2 sufficiently small, without performing a gauge transformation. The key ingredient is that the logarithmic divergence in the high-low frequency interaction can be overcome by a combination of Xs,b structure and smoothing effect structure. We also prove that the real-valued H? solutions to the mBO equation satisfy a priori local-in-time Hs bounds in terms of the Hs size of the initial data for s > 1/4.