We study the global well-posedness of the Cauchy problem for the KP II equation.
We prove the global well-posedness in the
inhomogeneous-homogeneous anisotropic
Sobolev spaces
$H_{x,y}^{-1/78+\epsilon,0}\cap H_{x,y}^{-17/144,0}$.
Though we require the use of the homogeneous
Sobolev space of negative index,
we obtain the global well-posedness below $L^2$.