  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.
