  We show the local in time well-posedness of the Cauchy problem for the
Kadomtsev-Petviashvili II equation for initial data in the non-isotropic
Sobolev space H^{s_1,s_2}(R^2) with s_1 > -1/2 and s_2 \geq 0. On the
H^{s_1,0}(R^2) scale this result includes the full subcritical range without
any additional low frequency assumption on the initial data. More generally, we
prove the local in time well-posedness of the Cauchy problem for a dispersion
generalised KP II type equation. We also deduce a global well-posedness result
for the generalised equation.
