Weighted low-regularity solutions of the KP-I initial-value problem

In this paper we establish local well-posedness of the KP-I problem, withinitial data small in the intersection of the natural energy space with thespace of functions which are square integrable when multiplied by the weight y.The result is proved by the contraction mapping principle. A similar (butslightly weaker) result was the main Theorem in the paper " Low regularitysolutions for the Kadomstev-Petviashvili I equation " by Colliander, Kenig andStaffilani (GAFA 13 (2003),737-794 and math.AP/0204244). Ionescu found acounterexample (included in the present paper) to the main estimate used in theGAFA paper, which renders incorrect the proof there. The present paper thusprovides a correct proof of a strengthened version of the main result in theGAFA paper.