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Solvability in the large for a class of vector fields on the torus. (English) Zbl 1157.35304
Summary: We study a class of complex vector fields defined on the two-torus of the form \(L=\partial/\partial t+(a(x,t)+ib(x,t))\partial/\partial x\), \(a,b\in C^\infty(\mathbb{T}^2;\mathbb{R})\), \(b\not\equiv 0\). We view \(L\) as an operator acting on smooth functions and present conditions for \(L\) to have either a closed range or a finite-codimensional range. Our results involve, besides condition \(({\mathcal P})\) of Nirenberg and Treves, the behavior of \(a+ib\) near each one-dimensional Sussmann orbit homotopic to the unit circle. One of the main goals of our work is to provide some clarification about the role played by the coefficient \(a\) in the validity of the above properties of the range.

MSC:
35A21 Singularity in context of PDEs
35F05 Linear first-order PDEs
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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[1] Bergamasco, A.P.; Cordaro, P.D.; Petronilho, G., Global solvability for a class of complex vector fields on the two-torus, Comm. partial differential equations, 29, 785-819, (2004) · Zbl 1065.35088
[2] Bergamasco, A.P.; Meziani, A., Semiglobal solvability of a class of planar vector fields of infinite type, Mat. contemporânea, 18, 31-42, (2000) · Zbl 0983.35036
[3] Bergamasco, A.P.; Meziani, A., Solvability near the characteristic set for a class of planar vector fields of infinite type, Ann. inst. Fourier (Grenoble), 55, 77-112, (2005) · Zbl 1063.35051
[4] Bergamasco, A.P.; Nunes, W.V.L.; Zani, S.L., Global properties of a class of overdetermined systems, J. funct. anal., 200, 31-64, (2003) · Zbl 1034.32024
[5] Bergamasco, A.P.; Dattori da Silva, P.L., Global solvability for a special class of vector fields on the torus, Contemp. math., 400, 11-20, (2006) · Zbl 1108.35026
[6] Berhanu, S.; Mendoza, G.A., Orbits and global unique continuation for systems of vector fields, J. geom. anal., 7, 173-194, (1997) · Zbl 0917.58004
[7] Cordaro, P.D.; Gong, X., Normalization of complex-valued planar vector fields which degenerate along a real curve, Adv. math., 184, 89-118, (2004) · Zbl 1129.35419
[8] Epstein, D.B.A., Curves on 2-manifolds and isotopies, Acta math., 115, 83-107, (1966) · Zbl 0136.44605
[9] Godin, P., Propagation des singularités pour LES opérateurs différentiels de type principal, localement résolubles, à coefficients analytiques, en dimension 2, Ann. inst. Fourier (Grenoble), 29, 2, 223-245, (1979) · Zbl 0365.58019
[10] Helffer, B., Addition de variables et applications à la régularité, Ann. inst. Fourier (Grenoble), 28, 2, 221-231, (1978) · Zbl 0365.35012
[11] Hirsch, M.W., Differential topology, Graduate texts in mathematics, vol. 33, (1976), Springer New York · Zbl 0121.18004
[12] Hörmander, L., Propagation of singularities and semi-global existence theorems for (pseudo-) differential operators of principal type, Ann. math., 108, 569-609, (1978) · Zbl 0396.35087
[13] Hörmander, L., Pseudo-differential operators of principal type, (), 69-96
[14] Hörmander, L., The analysis of linear partial differential operators IV, (1984), Springer-Verlag New York
[15] Hounie, J., Globally hypoelliptic and globally solvable first order evolutions equations, Trans. amer. math. soc., 252, 233-248, (1979) · Zbl 0424.35030
[16] Hounie, J., Minimal sets of families of vector fields on compact surfaces, J. diferential geom., 16, 739-744, (1981) · Zbl 0471.58019
[17] Hounie, J., Globally hypoelliptic vector fields on compact surfaces, Comm. partial differential equations, 7, 343-370, (1982) · Zbl 0588.35064
[18] Hounie, J., A note on global solvability of vector fields, Proc. amer. math. soc., 94, 61-64, (1985) · Zbl 0559.58032
[19] Köthe, G., Topological vector spaces II, (1979), Springer-Verlag New York · Zbl 0417.46001
[20] Meziani, A., Elliptic planar vector fields with degeneracies, Trans. amer. math. soc., 357, 4225-4248, (2005) · Zbl 1246.35090
[21] Sussmann, H.J., Orbits of families of vector fields and integrability of distributions, Trans. amer. math. soc., 180, 171-188, (1973) · Zbl 0274.58002
[22] Schwartz, A., A generalization of the poincaré-bendixson theorem to closed two-dimensional manifolds, Amer. J. math., 85, 453-458, (1963) · Zbl 0116.06803
[23] Treves, F., Approximation and representation of functions and distributions annihilated by a system of complex vector fields, (1981), École Polytechnique, Centre de Mathématiques Palaiseau · Zbl 0515.58030
[24] Wermer, J., Banach algebras and several complex variables, (1976), Springer-Verlag Berlin · Zbl 0336.46055
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