Solvability in the large for a class of vector fields on the torus
AP Bergamasco, PLD da Silva - Journal de mathématiques pures et …, 2006 - Elsevier
Journal de mathématiques pures et appliquées, 2006•Elsevier
We study a class of complex vector fields defined on the two-torus of the form L=∂/∂ t+ (a
(x, t)+ ib (x, t))∂/∂ x, a, b∈ C∞(T2; R), b≢ 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 (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 …
(x, t)+ ib (x, t))∂/∂ x, a, b∈ C∞(T2; R), b≢ 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 (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 …
We study a class of complex vector fields defined on the two-torus of the form L=∂/∂t+(a(x,t)+ib(x,t))∂/∂x, a,b∈C∞(T2;R), b≢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 (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.
Elsevier
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