Global properties of a class of vector fields in the plane (Q1113504)

The paper is concerned with the global solvability of the problem \(Ln=0\), dn\(\neq 0\) on \({\mathbb{R}}^ 2\), where L is a complex vector field without singularities. First it is shown that for a suitable class of vector fields L the Mizohata operator \(\partial_ t-it\partial y\) is a model operator in a neighborhood of the characteristic set of L. Then several integrability conditions are discussed and some global range theorems for the Mizohata operator are given. In an appendix relations to hyperelliptic vector fields are considered.