The paper deals with the Kadomtsev-Petviashvili I equation. There are obtained local existence and suitable uniqueness and continuous dependence for small data in the intersection of the energy space and a natural weighted \(L^2\) space.
There are several results on local and global existence of solutions, but not a satisfactory well-posedness theory for data with less than two derivatives in \(L^2\). The energy space is the natural space where the Hamiltonian is defined. In the above framework, the authors refine the previous local well-posedness results to reduce the number of the derivatives needed on the initial data to bring it to a space which is close to the energy space.