# zbMATH — the first resource for mathematics

On the global solvability of involutive systems. (English) Zbl 1359.37107
Summary: We consider a class of involutive systems in $$\mathbb{T}^{n + 1}$$ associated with a closed 1-form defined on the torus $$\mathbb{T}^n$$. We prove that, under a geometric condition, the global solvability of this class is equivalent to a diophantine condition involving Liouville forms and the connectedness of all sublevel and superlevel sets of a global primitive associated with the system.

##### MSC:
 37F75 Dynamical aspects of holomorphic foliations and vector fields 32M25 Complex vector fields, holomorphic foliations, $$\mathbb{C}$$-actions 35S05 Pseudodifferential operators as generalizations of partial differential operators 35F05 Linear first-order PDEs
Full Text:
##### References:
 [1] Bergamasco, A. P., Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc., 351, 10, 4113-4126, (1999) · Zbl 0932.35046 [2] Bergamasco, A. P.; Cordaro, P.; Malagutti, P., Globally hypoelliptic systems of vector fields, J. Funct. Anal., 114, 267-285, (1993) · Zbl 0777.58041 [3] Bergamasco, A. P.; Cordaro, P. D.; Petronilho, G., Global solvability for certain classes of underdetermined systems of vector fields, Math. Z., 223, 261-274, (1996) · Zbl 0863.58062 [4] Bergamasco, A. P.; de Medeira, C.; Zani, S. L., Globally solvable systems of complex vector fields, J. Differential Equations, 252, 4598-4623, (2012) · Zbl 1242.35092 [5] Bergamasco, A. P.; Kirilov, A., Global solvability for a class of overdetermined systems, J. Funct. Anal., 252, 603-629, (2007) · Zbl 1158.58011 [6] Bergamasco, A. P.; Kirilov, A.; Nunes, W.; Zani, S. L., On the global solvability for overdetermined systems, Trans. Amer. Math. Soc., 364, 4533-4549, (2012) · Zbl 1275.35004 [7] Bergamasco, A. P.; Kirilov, A.; Nunes, W.; Zani, S. L., Global solutions to involutive systems, Proc. Amer. Math. Soc., 143, 4851-4862, (2015) · Zbl 1330.35078 [8] Bergamasco, A. P.; Nunes, W.; Zani, S. L., Global properties of a class of overdetermined systems, J. Funct. Anal., 200, 1, 31-64, (2003) · Zbl 1034.32024 [9] Bergamasco, A. P.; Petronilho, G., Global solvability of a class of involutive systems, J. Math. Anal. Appl., 233, 314-327, (1999) · Zbl 0942.35011 [10] Berhanu, S.; Cordaro, P.; Hounie, J., An introduction to involutive structures, (2008), Cambridge University Press · Zbl 1151.35011 [11] Cardoso, F.; Hounie, J., Global solvability of an abstract complex, Proc. Amer. Math. Soc., 65, 117-124, (1977) · Zbl 0335.58015 [12] Greenfield, S. J.; Wallach, N. R., Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc., 31, 112-114, (1972) · Zbl 0229.35023 [13] Hounie, J., Globally hypoelliptic and globally solvable first-order evolution equations, Trans. Amer. Math. Soc., 252, 233-248, (1979) · Zbl 0424.35030 [14] Treves, F., Study of a model in the theory of complexes of pseudodifferential operators, Ann. of Math. (2), 104, 269-324, (1976) · Zbl 0354.35067 [15] Treves, F., Hypoanalytic structures (local theory), (1992), Princeton University Press NJ · Zbl 0787.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.