On the solvability of planar complex linear vector fields
Author:
François Treves
Journal:
Trans. Amer. Math. Soc. 364 (2012), 46294662
MSC (2010):
Primary 35A01, 35F05; Secondary 35D30, 35H10
Published electronically:
April 12, 2012
MathSciNet review:
2922604
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Abstract: The article discusses the local solvability, or lack thereof, of vector fields whose coefficients are complexvalued linear functions in (here called complex linear). It is proved that such a vector field is locally solvable in if and only if it is locally solvable and does not have compact orbits in the complement of its critical set or, equivalently, its Meziani number is different from zero.
 [BergamascoMeziani, 2005]
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Meziani, Solvability near the characteristic set for a class of
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 Bergamasco, A. P. and Meziani, A. Solvability near the characteristic set for a class of planar vector fields of infinite type, Ann. Inst. Fourier, Grenoble 55 (2005), 77112. MR 2141289 (2005m:35028)
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 Cordaro, P. D. and Gong, X. Normalization of complexvalued planar vector fields which degenerate along a real curve, Advances in Math. 184 (2004), 89118. MR 2047850 (2005a:35004)
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 Hörmander, L. On the range of convolution operators, Ann. of Math. (2) 76 (1962), 148170. MR 0141984 (25:5379)
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 Hörmander, L. Linear Partial Differential Operators, SpringerVerlag, Berlin, 1969. MR 0248435 (40:1687)
 [Hörmander, 1985]
 Hörmander, L. The Analysis of Linear Partial Differential Equations IV, SpringerVerlag, Berlin, 1985. MR 0781537 (87d:35002b)
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 Lojasiewicz, S., Notes, Institut Hautes Études, BuressurYvette, 1965.
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 Meziani, A. On planar elliptic structures with infinite type degeneracy, J. Funct. Anal. 179 (2001), 333373. MR 1809114 (2001k:35122)
 [Meziani, 2004]
 Meziani, A. Elliptic vector fields with degeneracies, Trans. Amer. Math. Soc.357 (2004) 42254248. MR 2159708 (2006f:35107)
 [Miwa, 1973]
 Miwa, T. On the existence of hyperfunction solution of Linear Differential Equations of the first order with degenerate real principal symbol, Proc. Japan Acad. 49 (1973), 8893. MR 0348236 (50:734)
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 Müller, D. H. Local solvability of first order differential operators near a critical point, operators with quadratic symbols and the Heisenberg group, Comm. P. D. E. 17 (1992), 305337. MR 1151265 (93g:35005)
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 Nagano, T., Linear differential systems with singularities and applications to transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398404. MR 0199865 (33:8005)
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 Nirenberg, L. and Treves, F. Solvability of a firstorder linear partial differential equation, Comm. Pure Appl. Math. 16 (1963), 331351. MR 0163045 (29:348)
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 Treves, F., AnalyticHypoelliptic Partial Differential Equations of Principal Type, Comm. Pure Applied Math. XXIV (1971), 537570. MR 0296509 (45:5569)
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 [Treves, 2009]
 Treves, F., On the solvability of vector fields with real linear coefficients, Proceedings Amer. Math. Soc. 137 (2009), 42094218. MR 2538582 (2010k:35060)
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Additional Information
François Treves
Affiliation:
12002 Spruce Canyon Circle, Golden, Colorado 80403
Email:
treves.jeanfrancois@gmail.com
DOI:
http://dx.doi.org/10.1090/S000299472012054298
Received by editor(s):
November 29, 2009
Received by editor(s) in revised form:
July 22, 2010
Published electronically:
April 12, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
