Global solvability and global hypoellipticity of complex vector fields on surfaces
The problems of the global solvability and hypoellipticity of the linear partial differential operators were intensively studied in the last years, main contributions coming from the Brasilian school, see for example [\textit{A. P. Bergamasco} et al., Commun. Partial Differ. Equations 29, No. 5--6, 785--819 (2004; Zbl 1065.35088); ibid. 37, No. 9, 1517--1527 (2012; Zbl 1387.58037)]. In the present paper, the authors give general results, extending and unifying several previous contributions. Namely, attention is fixed on a complex vector field $L=X+iY$ on a surface. Global solvability implies local solvability, hence the celebrated condition (P) of Nirenberg-Trèves is ussumed as necessary at any point. The main issue is then the analysis of the closed orbits. In particular, global solvability is obtained under the assumption that the commutator of $X$ and $Y$ vanishes of order greater than $1$ on all closed one-dimensional orbits, whereas order $1$ of vanishing contradicts global solvability. The paper also contains a careful analysis of the topological properties of the vector fields $L$, and a related study of global hypoellipticity.
scientific article; zbMATH DE number 7604824
scientific article