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Global properties of a class of vector fields in the plane. (English) Zbl 0662.58021
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.
Reviewer: N.Jacob

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
Full Text: DOI
[1] Baouendi, S; Treves, F, A local constancy principle for the solutions of certain overdetermined systems of first order linear partial differential equations, Math. anal. appl. stud., 7A, 245-262, (1981)
[2] Hörmander, L, Pseudodifferential operators of principal type, (), 69-96
[3] Kamke, E; Kamke, E, Über die partielle differentialgleichung f(x, y)zx + g(x, y)zy = h(x, y), II, Math. Z., Math. Z., 42, 287-300, (1936) · Zbl 0015.34804
[4] Nehari, Z, Conformal mapping, (1952), McGraw-Hill New York · Zbl 0048.31503
[5] Nirenberg, L, Lectures on linear partial differential equations, () · Zbl 0267.35001
[6] Springer, G, Introduction to Riemann surfaces, (1957), Addison-Wesley Reading, MA · Zbl 0078.06602
[7] Sjöstrand, Note on a paper of F. treves concerning mizohata type operators, Duke math. J., 41, 3, 601-608, (1980) · Zbl 0471.35076
[8] Treves, F, Remarks about certain first-order linear PDE in two variables, Comm. PDE, 5, 381-425, (1980) · Zbl 0519.35008
[9] Treves, F, Hypoelliptic PDE’s of principal type, sufficient conditions and necessary conditions, Comm. pure appl. math., 24, 631-670, (1971) · Zbl 0234.35019
[10] Treves, F, Approximation and representations of functions and distributions annihilated by a system of complex vector fields, (1981), École Polytech, Centre de Math Palaiseau, France · Zbl 0515.58030
[11] Ważewski, T, Sur un problème de caractère intégral relatif à l’équation Zx + Q(x, y)zy = 0, Mathematica cluj, 8, 103-116, (1934) · Zbl 0008.39403
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