Bilinear space-time estimates for linearised KP-type equations on the three-dimensional torus with applications

A bilinear estimate in terms of Bourgain spaces associated with a linearised Kadomtsev-Petviashvili-type equation on the three-dimensional torus is shown. As a consequence, time localized linear and bilinear space time estimates for this equation are obtained. Applications to the local and global well-posedness of dispersion generalised KP-II equations are discussed. Especially it is proved that the periodic boundary value problem for the original KP-II equation is locally well-posed for data in the anisotropic Sobolev spaces $H_x^s{H}_y^{\text{e}}({\text{T}}^3)$, if $s\ge \frac{1}{2}$ and $\text{e}>0$.