Necessary and sufficient conditions for the local solvability in hyperfunctions of a class of systems of complex vector fields
A necessary and sufficient condition is proved for the local exactness at degree \(q\geq 1\) of the differential complex attached to a locally integrable structure of codimension one, in spaces of hyperfunctions with respect to the hypo-analytic structure associated to a smooth ``first integral'' \(Z\) on a general smooth manifold (cf. [\textit{F. Treves}, Hypo- analytic structures: local theory. Princeton Mathematical Series. Princeton, NJ: Princeton University Press (1992; Zbl 0787.35003)]). The condition is that the singular homology of the fibres of the map \(Z\) be trivial in dimension \(q-1\). This is considered to be an analogue of the Poincaré lemma for the present situation, but the true content is a kind of local solvability for some systems of vector fields with complex coefficients.
scientific article; zbMATH DE number 784005