Nonlinear smoothing for the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity

We consider the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity. We establish the nonlinear smoothing properties of these equations, according to which the difference between the solution and the linear evolution is smoother than the initial data. In addition, we establish new local well-posedness results for these equations when the dispersion is sufficiently large. Our method also improves known local well-posedness results for a class of non-integrable fifth-order KdV equations.