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Prescribing analytic singularities for solutions of a class of vector fields on the torus

Authors: Adalberto P. Bergamasco and Sérgio Luís Zani
Journal: Trans. Amer. Math. Soc. 357 (2005), 4159-4174
MSC (2000): Primary 35A20, 35H10
Published electronically: May 20, 2005
MathSciNet review: 2159704
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Abstract: We consider the operator $L=\partial_t+(a(t)+ib(t))\partial_x$ acting on distributions on the two-torus $\mathbb T^2,$ where $a$ and $b$ are real-valued, real analytic functions defined on the unit circle $\mathbb T^1.$We prove, among other things, that when $b$ changes sign, given any subset $\Sigma$ of the set of the local extrema of the local primitives of $b,$ there exists a singular solution of $L$ such that the $t-$projection of its analytic singular support is $\Sigma;$ furthermore, for any $\tau\in\Sigma$ and any closed subset $F$ of $\mathbb T^1_x$ there exists $u\in\mathcal D'(\mathbb T^2)$ such that $Lu\in C^\omega(\mathbb T^2)$ and $\operatorname{sing\, supp_A}(u)=\{\tau\}\times F.$ We also provide a microlocal result concerning the trace of $u$ at $t=\tau.$

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  • [BT] Baouendi, M. S.; Trèves, F., A microlocal version of Bochner's tube theorem, Indiana Univ. Math. J. 31(6) (1982), 885-895. MR 0674873 (84b:35025)
  • [B] Bergamasco, A., Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc. 351 (1999), 4113-4126. MR 1603878 (99m:35032)
  • [BCM] Bergamasco, A.; Cordaro, P.; Malagutti P., Globally hypoelliptic systems of vector fields, J. Funct. Anal. 114 (1993), 267-285. MR 1223704 (94e:35048)
  • [BNZ1] Bergamasco, A.; Nunes, W.; Zani, S., Global analytic hypoellipticity and pseudoperiodic functions, Mat. Contemporanea 18 (2000), 43-57. MR 1812862 (2001m:35062)
  • [BNZ2] Bergamasco, A.; Nunes, W.; Zani S., Global properties of a class of overdetermined systems, Journal of Functional Analysis 200 (2003), 31-64. MR 1974087 (2004c:35295)
  • [DGY] Dickinson, D.; Gramchev, T.; Yoshino, M., Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena, Proc. Edinb. Math. Soc. 45 (2002), 731-759. MR 1933753 (2004h:37020)
  • [G] Greenfield, S., Hypoelliptic vector fields and continued fractions, Proc. Amer. Math. Soc. 31 (1972), 115-118. MR 0301459 (46:617)
  • [GW] Greenfield, S.; Wallach, N., Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc. 31 (1972), 112-114. MR 0296508 (45:5568)
  • [Hi] Himonas, A., Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions, Proc. Amer. Math. Soc. 129 (2001), 2061-2067. MR 1825918 (2002c:35074)
  • [H] Hörmander, L., The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften, 256, Springer-Verlag, Berlin, 1983. MR 0717035 (85g:35002a)
  • [Ho] Hounie, J., Globally hypoelliptic vector fields on compact surfaces, Comm. Partial Differential Equations 7 (1982), no. 4, 343-370. MR 0652813 (83k:35025)
  • [M] Meziani, A., Hypoellipticity of nonsingular closed 1-forms on compact manifolds, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1255-1269. MR 1924466 (2003f:58002)
  • [Sj] Sjöstrand, J., Singularités analytiques microlocales, Astérisque, 95, 1-166, Soc. Math. France, Paris, 1982. MR 0699623 (84m:58151)
  • [T1] Treves, F., Analytic hypoelliptic partial differential equations of principal type, Comm. Pure Appl. Math. 24 (1971), 537-570. MR 0296509 (45:5569)
  • [T2] Treves, F., Hypoanalytic Structures, Princeton University Press, Princeton, NJ, 1992. MR 1200459 (94e:35014)

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Additional Information

Adalberto P. Bergamasco
Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação - USP, Caixa Postal 668, São Carlos, SP, 13560-970 Brasil

Sérgio Luís Zani
Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação - USP, Caixa Postal 668, São Carlos, SP, 13560-970 Brasil

Keywords: Analytic singularities, global analytic hypoellipticity, stationary phase
Received by editor(s): December 9, 2003
Published electronically: May 20, 2005
Additional Notes: The first author was partially supported by CNPq. Both authors were partially supported by FAPESP
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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