Abstract
Let M be a compact, connected, orientable and real-analytic manifold; consider closed, real-valued, real-analytic 1-forms \(\omega _1, \ldots , \omega _m\) on M and the differential complex over \(M \times \mathbb {T}^m\) naturally associated to the involutive system determined by them. In the real-analytic context, we completely characterize global solvability of the operators in its first (functional setting) and last (distributional setting) levels. Analogous results are obtained simultaneously in the Gevrey framework.
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Notes
Although in the end equivalent to it by virtue of the very same Theorem 1.3.
“Primed” sums are taken only over ordered multi-indices J, being therefore unique.
This is the only place where we employed analyticity of \(\omega _1, \ldots , \omega _m\), for, in that case, \(\mathbb {L}^0_\xi \) is a real-analytic operator.
That is: non \((s, \mathbb {Z}\setminus \{0\})\)-exponential Liouville, in the terminology of Definition 4.4.
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Funding
The three authors were supported by the São Paulo Research Foundation (FAPESP, Grants 2018/12273-5, 2018/14316-3 and 2021/03199-9, respectively). The second author was also partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Grant 313581/2021-5).
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Araújo, G., da Silva, P.L.D. & de Lessa Victor, B. Global Analytic Solvability of Involutive Systems on Compact Manifolds. J Geom Anal 33, 151 (2023). https://doi.org/10.1007/s12220-023-01206-z
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DOI: https://doi.org/10.1007/s12220-023-01206-z