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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.

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)
Full Text: DOI
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