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Transactions of the American Mathematical Society

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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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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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