L 2-estimates with Levi-singular weight, and existence foroverline partial

We give a symplectic proof of the link between pseudoconvexity of domains ofC ⁿ and of their boundaries (cf. [7, Th. 2.6.12]). Our approach also allows us to treat boundaries of codimension >1. We then extend the estimates by Hörmander in [7, Ch. 4, 5] and [6] toL ²-norms which haveC ¹ but notC ² weights and under a less restrictive assumption of weakq-pseudoconvexity. (A special trick is needed as a substitute for the method of thelowest positive eigenvalue of [6].)