Improved well-posedness for quasilinear and sharp local well-posedness for semilinear KP-I equations

We show new well-posedness results in anisotropic Sobolev spaces for dispersion-generalized KP-I equations with increased dispersion compared to the KP-I equation. We obtain the sharp dispersion rate, below which generalized KP-I equations on R² and on R × T exhibit quasilinear behavior. In the quasilinear regime, we show improved well-posedness results relying on short-time Fourier restriction. In the semilinear regime, we show sharp well-posedness with analytic data-to-solution mapping. On R² we cover the full subcritical range, whereas on R × T the sharp well-posedness is strictly subcritical. Nonlinear Loomis-Whitney inequalities are one ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane R², the cylinder R × T, and the torus T². Finally, we point out that on tori T²_γ, KP-I equations are never semilinear.