Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces

We consider the initial value problem associated to the regularized Benjamin-Ono equation, rBO. Our aim is to establish local and global well-posedness results in weighted Sobolev spaces via contraction principle. We also prove a unique continuation property that implies that arbitrary polinomial type decay is not preserved yielding sharp results regarding well-posedness of the initial value problem in most weighted Sobolev spaces.