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Global hypoellipticity of planar complex vector fields. (English) Zbl 1430.35062
Summary: We study the global hypoellipticity property for non-singular planar complex vector fields. The results obtained are related to condition \((\mathcal{P})\) of Nirenberg-Treves and the boundedness of certain subsets of the plane where the real and the imaginary parts of the vector field under consideration are linearly dependent. We also prove the existence of a globally defined first integral when the vector field is globally hypoelliptic and has only one orbit.
35H10 Hypoelliptic equations
Full Text: DOI
[1] Bergamasco, A.; Cordaro, P. D.; Hounie, J., Global properties of a class of vector fields in the plane, J. Differ. Equ., 74, 2, 179-199 (1988) · Zbl 0662.58021
[2] Bergamasco, A. P.; Cordaro, P. D.; Malagutti, P. A., Globally hypoelliptic systems of vector fields, J. Funct. Anal., 114, 2, 267-285 (1993) · Zbl 0777.58041
[3] Bergamasco, A. P.; Mendoza, G. A.; Zani, S. L., On global hypoellipticity, Commun. Partial Differ. Equ., 37, 9, 1517-1527 (2012) · Zbl 1387.58037
[4] Berhanu, S.; Cordaro, P. D.; Hounie, J., An Introduction to Involutive Structures (2008), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1151.35011
[5] Daverman, R., Decompositions of Manifolds (1986), Academic Press: Academic Press Orlando, Florida 32887 · Zbl 0608.57002
[6] Folland, G. B., Real Analysis: Modern Techniques and Their Applications (1999), Wiley: Wiley New York · Zbl 0924.28001
[7] Godbillon, C., Dynamical Systems on Surfaces (1983), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0502.58002
[8] Guillemin, V., Geometric Asymptotics (1990), American Mathematical Society: American Mathematical Society Providence, R.I.
[9] Guillemin, V.; Pollack, A., Differential Topology (1974), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J. · Zbl 0361.57001
[10] Hörmander, L., Pseudodifferential operators of principal type, (Singularities in Boundary Value Problems. Singularities in Boundary Value Problems, Proc. NATO Adv. Study Inst., Maratea, 1980. Singularities in Boundary Value Problems. Singularities in Boundary Value Problems, Proc. NATO Adv. Study Inst., Maratea, 1980, NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 65 (1981), Reidel: Reidel Dordrecht-Boston, Mass.), 69-96
[11] Hounie, J., Minimal sets of families of vector fields on compact surfaces, J. Differ. Geom., 16, 4, 739-744 (1981) · Zbl 0471.58019
[12] Hounie, J., Globally hypoelliptic vector fields on compact surfaces, Commun. Partial Differ. Equ., 7, 4, 343-370 (1982) · Zbl 0588.35064
[13] Hounie, J., Global Cauchy problems modulo flat functions, Adv. Math., 51, 3, 240-252 (1984) · Zbl 0565.35019
[14] Lee, J. M., Submanifolds, (Introduction to Smooth Manifolds (2013), Springer: Springer New York), 98-124
[15] Moore, R. L., Concerning upper semi-continuous collections of continua, Trans. Am. Math. Soc., 27, 4, 416-428 (1925) · JFM 51.0464.03
[16] Sussmann, H. J., Orbits of families of vector fields and integrability of distributions, Trans. Am. Math. Soc., 180, 171-188 (06 1973)
[17] Treves, F., Basic Linear Partial Differential Equations (1975), Academic Press: Academic Press New York · Zbl 0305.35001
[18] Treves, F., Hypo-Analytic Structures: Local Theory (1992), Princeton University Press: Princeton University Press Princeton, N.J., ISBN 0-691-08744-x · Zbl 0787.35003
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