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On the local solvability of vector fields with critical points

Part of: General first-order equations and systems

Published online by Cambridge University Press:  12 May 2011

François Treves
François Treves
Affiliation:
Department of Mathematics, Rutgers University—Hill Center for the Mathematical Sciences, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA (treves.jeanfrancois@gmail.com)
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Abstract

The article discusses the local solvability (or lack thereof) of various classes of smooth, complex vector fields that vanish on some non-empty subset of the base manifold.

Keywords

vector fieldslocal solvabilityhypoellipticity

MSC classification

Secondary: 35F20: Nonlinear first-order equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Special Issue 3: A Special Issue in Honour of Louis Boutet de Monvel and Pierre Schapira , July 2011 , pp. 785 - 808
Copyright
Copyright © Cambridge University Press 2011

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References

1.Beals, R. and Fefferman, C., On local solvability of linear partial differential equations, Annals Math. 97 (1973), 482–498.Google Scholar
2.Cordaro, P. D. and Gong, X., Normalization of complex-valued planar vector fields which degenerate along a real curve, Adv. Math. 184 (2004), 89–118.Google Scholar
3.Hörmander, L., The analysis of linear partial differential equations, Volume IV (Springer, 1985).Google Scholar
4.Lojasiewicz, S., Sur le problème de la division, Studia Math. 18 (1959), 87–136.Google Scholar
5.Meziani, A., On planar elliptic structures with infinite type degeneracy, J. Funct. Analysis 179 (2001), 333–373.Google Scholar
6.Meziani, A., On the hypoellipticity of differential forms with isolated singularities, preprint.Google Scholar
7.Miwa, T., On the existence of hyperfunction solutions of linear differential equations of the first order with degenerate real principal symbols, Proc. Jpn Acad. A49 (1973), 88–93.Google Scholar
8.Müller, D. H., Local solvability of first order differential operators near a critical point, operators with quadratic symbols and the Heisenberg group, Commun. PDEs 17 (1992), 305–337.Google Scholar
9.Nagano, T., Linear differential systems with singularities and applications to transitive Lie algebras, J. Math. Soc. Jpn 18 (1966), 398–404.Google Scholar
10.Nirenberg, L. and Treves, F., Solvability of a first-order linear partial differential equation, Commun. Pure Appl. Math. 16 (1963), 331–351.CrossRefGoogle Scholar
11.Sussman, H., Orbits of families of vector fields and integrability of distributions, Trans. Am. Math. Soc. 180 (1973), 171–188.Google Scholar
12.Treves, F., Hypoelliptic partial differential equations of principal type: sufficient conditions and necessary conditions, Commun. Pure Appl. Math. 24 (1971), 631–670.Google Scholar
13.Treves, F., Hypo-analytic structures, in Local theory (Princeton University Press, 1992).Google Scholar
14.Treves, F., Topological vector spaces, distributions and kernels (Academic Press, 1967; paperback published by Dover Mineola, New York, 2006).Google Scholar
15.Treves, F., On the solvability of vector fields with real linear coefficients, Proc. Am. Math. Soc. 137 (2009), 4209–4218.Google Scholar
16.Treves, F., On planar vector fields with complex linear coefficients, to appear.Google Scholar