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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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References:
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