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Global solvability of real analytic involutive systems on compact manifolds. (English) Zbl 1380.35129
Let $$b$$ be a real analytic closed non-exact 1-form defined on a compact and without boundary connected $$n$$-dimensional manifold $$M$$, $$(n > 1)$$. The focus of this work is the smooth global solvability of the differential operator $${\mathbb L}:C^\infty(M\times {\mathbb S}^1)\to \Lambda^1 C^\infty(M\times {\mathbb S}^1)$$ given by $${\mathbb L}u =d_{t_j}u + ib(t) \wedge \partial_x u$$, where $$x\in {\mathbb S}^1$$, and $$d_t$$ is the exterior derivative on $$M$$.
The approach relies on defining an appropriate covering projection $$\tilde M \to M$$ such that the pullback of $$b$$ has a primitive $$\tilde B$$ and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of $$\tilde B$$ are connected. In case of orientable manifolds $$M$$, further charactirizations are made.

##### MSC:
 35N10 Overdetermined systems of PDEs with variable coefficients 58J10 Differential complexes
##### Keywords:
global solvability; linear operators; compact manifolds
Full Text:
##### References:
 [1] Arnol’d, V.I.: Topological and ergodic properties of closed $$1$$-forms with incommensurable periods. Funktsional. Anal. i Prilozhen. 25(2):1-12, 96 (1991) · Zbl 0945.35007 [2] Arnoux, P; Levitt, G, Sur l’unique ergodicité des $$1$$-formes fermées singulières, Invent. Math., 84, 141-156, (1986) · Zbl 0577.58021 [3] Bergamasco, AP; Cordaro, PD; Malagutti, PA, Globally hypoelliptic systems of vector fields, J. Funct. Anal., 114, 267-285, (1993) · Zbl 0777.58041 [4] Bergamasco, AP; Cordaro, PD; Petronilho, G, Global solvability for certain classes of underdetermined systems of vector fields, Math. Z., 223, 261-274, (1996) · Zbl 0863.58062 [5] Bergamasco, AP; Medeira, C; Zani, SL, Globally solvable systems of complex vector fields, J. Differ. Equ., 252, 4598-4623, (2012) · Zbl 1242.35092 [6] Bergamasco, AP, Global solvability for a class of overdetermined systems, J. Funct. Anal., 252, 603-629, (2007) · Zbl 1158.58011 [7] Bergamasco, AP; Kirilov, A; Nunes, WVL; Zani, SL, On the global solvability for overdetermined systems, Trans. Am. Math. Soc., 364, 4533-4549, (2012) · Zbl 1275.35004 [8] Bergamasco, AP; Nunes, WVL; Zani, SL, Global properties of a class of overdetermined systems, J. Funct. Anal., 200, 31-64, (2003) · Zbl 1034.32024 [9] Bergamasco, AP; 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.D., Hounie, J.: An introduction to involutive structures. New Mathematical Monographs, vol. 6. Cambridge University Press, Cambridge (2008) · Zbl 1151.35011 [11] Bierstone, E; Milman, PD, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67, 5-42, (1988) · Zbl 0674.32002 [12] Cardoso, F; Hounie, J, Global solvability of an abstract complex, Proc. Am. Math. Soc., 65, 117-124, (1977) · Zbl 0335.58015 [13] Chanillo, S; Treves, F, Local exactness in a class of differential complexes, J. Am. Math. Soc., 10, 393-426, (1997) · Zbl 0867.35005 [14] Chen, W; Chi, MY, Hypoelliptic vector fields and almost periodic motions on the torus $$T^n$$, Commun. Partial Differ. Equ., 25, 337-354, (2000) · Zbl 0945.35007 [15] Cordaro, P; Hounie, J, On local solvability of underdetermined systems of vector fields, Am. J. Math., 112, 243-270, (1990) · Zbl 0708.58025 [16] Cordaro, P; Trèves, F, Homology and cohomology in hypo-analytic structures of the hypersurface type, J. Geom. Anal., 1, 39-70, (1991) · Zbl 0724.32009 [17] Cordaro, PD; Hounie, J, Local solvability for top degree forms in a class of systems of vector fields, Am. J. Math., 121, 487-495, (1999) · Zbl 0964.58020 [18] Cordaro, PD; Hounie, JG, Local solvability for a class of differential complexes, Acta Math., 187, 191-212, (2001) · Zbl 1004.58012 [19] Cordaro, P.D., Trèves, F.: Hyperfunctions on hypo-analytic manifolds. Annals of Mathematics Studies, vol. 136. Princeton University Press, Princeton (1994) · Zbl 0817.32001 [20] Cordaro, PD; Trèves, F, Necessary and sufficient conditions for the local solvability in hyperfunctions of a class of systems of complex vector fields, Invent. Math., 120, 339-360, (1995) · Zbl 0834.35004 [21] Greenfield, SJ; Wallach, NR, Global hypoellipticity and Liouville numbers, Proc. Am. Math. Soc., 31, 112-114, (1972) · Zbl 0229.35023 [22] Hanges, N; Jacobowitz, H, Involutive structures on compact manifolds, Am. J. Math., 117, 491-522, (1995) · Zbl 0826.32014 [23] Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002) · Zbl 1044.55001 [24] Hironaka, H.: Subanalytic sets. In: Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, pp. 453-493. Kinokuniya, Tokyo (1973) · Zbl 0708.58025 [25] Hounie, J, Globally hypoelliptic vector fields on compact surfaces, Commun. Partial Differ. Equ., 7, 343-370, (1982) · Zbl 0588.35064 [26] Hounie, J; Malagutti, P, Local integrability of mizohata structures, Trans. Am. Math. Soc., 338, 337-362, (1993) · Zbl 0801.35091 [27] Maire, H-M, Hypoelliptic overdetermined systems of partial differential equations, Commun. Partial Differ. Equ., 5, 331-380, (1980) · Zbl 0436.35024 [28] Mendoza, GA; Trèves, F, Local solvability in a class of overdetermined systems of linear PDE, Duke Math. J., 63, 355-377, (1991) · Zbl 0794.35002 [29] Meziani, A, Classification of germs of mizohata structures, Commun. Partial Differ. Equ., 20, 499-539, (1995) · Zbl 0824.58004 [30] Meziani, A, The mizohata complex, Trans. Am. Math. Soc., 349, 1029-1062, (1997) · Zbl 0869.35023 [31] Meziani, A.: Pseudoconvex Mizohata structures on compact manifolds. In: Baklouti, A., El Kacimi, A., Kallel, S., Mir, N. (eds.) Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol. 127, pp. 241-266. Springer, Switzerland (2015) · Zbl 1332.32043 [32] Palais, RS, Natural operations on differential forms, Trans. Am. Math. Soc., 92, 125-141, (1959) · Zbl 0092.30802 [33] Teissier, B, Appendice: sur trois questions de finitude en géométrie analytique réelle, Acta Math., 151, 39-48, (1983) [34] Treves, F.: Study of a model in the theory of complexes of pseudodifferential operators. Ann. Math. (2) 104(2), 269-324 (1976) · Zbl 0354.35067 [35] Trèves, F, On the local solvability and the local integrability of systems of vector fields, Acta Math., 151, 1-38, (1983) · Zbl 0534.35009 [36] Trèves, F.: Hypo-analytic structures. Local theory, Princeton Mathematical Series, vol. 40. Princeton University Press, Princeton (1992) · Zbl 0577.58021 [37] Zugliani, G.A.: Global solvability of systems on compact surfaces. PhD thesis, Universidade de São Paulo (2014). http://www.teses.usp.br/teses/disponiveis/55/55135/tde-29092014-155847/ · Zbl 0794.35002
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