On the Cauchy-problem for generalized Kadomtsev-Petviashvili-II equations

The Cauchy-problem for the generalized Kadomtsev-Petviashvili-II equation $$u_t+u_{xxx}+{\mathrm{\partial }}_x^{-1}{u}_{yy}=(u^l{)}_x,l\ge 3,$$ is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines local smoothing and maximal function estimates as well as bilinear refinements of Strichartz type inequalities via multilinear interpolation in $X_{s,b}$-spaces.