  In this article we study the generalized dispersion version of the
Kadomtsev-Petviashvili II equation, on $\T \times \R$ and $\T \times \R^2$. We
start by proving bilinear Strichartz type estimates, dependent only on the
dimension of the domain but not on the dispersion. Their analogues in terms of
Bourgain spaces are then used as the main tool for the proof of bilinear
estimates of the nonlinear terms of the equation and consequently of local
well-posedness for the Cauchy problem.
