Well-posedness and scattering for the KP-II equation in a critical space

The Cauchy problem for the Kadomtsev-Petviashvili-II equation (u_t+u_{xxx}+uu_x)_x+u_{yy}=0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space \dot H^{-1/2,0}(R^2) is derived. Additionally, it is proved that for arbitrarily large initial data the Cauchy problem is locally well-posed in the homogeneous space \dot H^{-1/2,0}(R^2) and in the inhomogeneous space H^{-1/2,0}(R^2), respectively.