Local well posedness for modified Kadomstev-Petviashvili equations

In this paper we consider the Kadomstev-Petivashvili equation and also the modified Kadomstev-Petviashvili equation, with nonlinearity $\partial_x(u^3).$ We improve on previous results of Iório and Nunes [5], and also on previous work of the authors, [13]. For the modified $(KP-II)$ equation we give optimal (up to endpoint) maximal function type estimates for the solution of the associated linear initial-value problem. These estimates enable us to obtain a local well-posedness result via the contraction mapping principle. For modified $(KP-I)$ we use methods developed by Kenig in [9], which use an energy estimate together with Strichartz estimates and "interpolation inequalities." We give some counterexamples to well posedness via the contraction mapping principle, for both the Kadomstev-Petviashvili equation and the modified equation.