On the solvability of planar complex linear vector fields

Author:
François Treves

Journal:
Trans. Amer. Math. Soc. **364** (2012), 4629-4662

MSC (2010):
Primary 35A01, 35F05; Secondary 35D30, 35H10

DOI:
https://doi.org/10.1090/S0002-9947-2012-05429-8

Published electronically:
April 12, 2012

MathSciNet review:
2922604

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Abstract | References | Similar Articles | Additional Information

Abstract: The article discusses the local solvability, or lack thereof, of vector fields whose coefficients are complex-valued 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.

**[Bergamasco-Meziani, 2005]**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), 77-112. MR**2141289 (2005m:35028)****[Cordaro-Gong, 2004]**Cordaro, P. D. and Gong, X.*Normalization of complex-valued planar vector fields which degenerate along a real curve*, Advances in Math.**184**(2004), 89-118. MR**2047850 (2005a:35004)****[Hörmander, 1959]**Hörmander, L.*On the range of convolution operators,*Ann. of Math. (2)**76**(1962), 148-170. MR**0141984 (25:5379)****[Hörmander, 1969]**Hörmander, L. Linear Partial Differential Operators, Springer-Verlag, Berlin, 1969. MR**0248435 (40:1687)****[Hörmander, 1985]**Hörmander, L. The Analysis of Linear Partial Differential Equations IV, Springer-Verlag, Berlin, 1985. MR**0781537 (87d:35002b)****[Lojasiewiccz, 1965]**Lojasiewicz, S., Notes, Institut Hautes Études, Bures-sur-Yvette, 1965.**[Meziani, 2001]**Meziani, A.*On planar elliptic structures with infinite type degeneracy,*J. Funct. Anal.**179**(2001), 333-373. MR**1809114 (2001k:35122)****[Meziani, 2004]**Meziani, A.*Elliptic vector fields with degeneracies,*Trans. Amer. Math. Soc.**357**(2004) 4225-4248. 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), 88-93. MR**0348236 (50:734)****[Müller, 1992]**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), 305-337. MR**1151265 (93g:35005)****[Nagano, 1966]**Nagano, T.,*Linear differential systems with singularities and applications to transitive Lie algebras,*J. Math. Soc. Japan**18**(1966), 398-404. MR**0199865 (33:8005)****[Nirenberg-Treves, 1963]**Nirenberg, L. and Treves, F.*Solvability of a first-order linear partial differential equation,*Comm. Pure Appl. Math.**16**(1963), 331-351. MR**0163045 (29:348)****[Treves, 1971]**Treves, F.,*Analytic-Hypoelliptic Partial Differential Equations of Principal Type*, Comm. Pure Applied Math.**XXIV**(1971), 537-570. MR**0296509 (45:5569)****[Treves, 1992]**Treves, F., Hypo-Analytic Structures,*Local Theory*, Princeton University Press, Princeton, N. J., 1992. MR**1200459 (94e:35014)****[Treves, 2009]**Treves, F.,*On the solvability of vector fields with real linear coefficients*, Proceedings Amer. Math. Soc.**137**(2009), 4209-4218. 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:
https://doi.org/10.1090/S0002-9947-2012-05429-8

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.