an:05125010
Zbl 1157.35304
Bergamasco, A. P.; Dattori da Silva, P. L.
Solvability in the large for a class of vector fields on the torus
EN
J. Math. Pures Appl. (9) 86, No. 5, 427-447 (2006).
00188259
2006
j
35A21 35F05 47F05 35A05
closed range; finite-codimensional range; Sussmann orbit
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.