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On the local solvability for top-degree forms in hypo-analytic structures. (English) Zbl 0763.35022
The author extends to the general \({\mathcal C}^ \infty\) case the local solvability result of P. Cordaro and J. Hounie [Am. J. Math. 112, No. 2, 243-270 (1990; Zbl 0708.58025)] which states that the nonexistence of hypo-analytic functions \(h\) whose real part has local extrema ensures the local solvability of \(Lu=f\) in an open set \(\Omega\subset\mathbb{R}^{m+n}\) equipped with a hypo-analytic structure, where \(L\) is the differential operator in the differential complex defined by the hypo-analytic structure, at least in the cases where either the tangent or the cotangent structure bundles have rank exactly equal to one and are real-analytic bundles. When the rank of the cotangent structure is equal to 1 this is established by modifying the method of Cordaro-Hounie [loc. cit.], in which case the existence of hypo-analytic functions whose real parts have local extrema is seen to be equivalent to the nonvanishing of the \((n-1)\)-dimensional homology of the fibers of the structure. When it is the rank of the tangent structure which is equal to 1, the result is derived by reinterpreting Moyer’s proof of the necessity of condition \((\Psi)\).

MSC:
35G05 Linear higher-order PDEs
58J10 Differential complexes
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