Sharp local well-posedness for KP-I equations in the semilinear regime

We show sharp well-posedness with analytic data-to-solution mapping in the semilinear regime for dispersion-generalized KP-I equations on R² and R × T . On R² we cover the full subcritical range, whereas on R × T the sharp well-posedness is strictly subcritical. We rely on linear and bilinear Strichartz estimates which are proved using decoupling techniques and square function estimates. Nonlinear Loomis-Whitney inequalities are a further 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.