Necessary and sufficient conditions for the local solvability in hyperfunctions of a class of systems of complex vector fields

Summary A necessary and sufficient condition is proved for the validity of the Poincaré Lemma in degreeq1, in the differential complex attached to a locally integrable structure of codimension one, in spaces of hyperfunctions. The base manifold is only assumed to be smooth. The hyperfunctions are defined in the hypo-analytic structure associated to a smooth first integral Z. The condition is that the singular homology of the fibres of the map Z be trivial in dimensionq-1. By the approximation formula of [BT]|the germ of this fibration at a point is independent of the choice of the first integral Z.Oblatum 25-III-1994 & 19-X-1994The research of Cordaro was supported by CNPq Grant # 304825/89-1. The work of Treves was supported by NSF Grants DMS-9201980 and No INT-9103833 (US-Brazil Cooperative Research)