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Subelliptic estimates for some systems of complex vector fields: Quasihomogeneous case

Authors: M. Derridj and B. Helffer
Journal: Trans. Amer. Math. Soc. 361 (2009), 2607-2630
MSC (2000): Primary 35B65; Secondary 32N15
Published electronically: November 24, 2008
MathSciNet review: 2471931
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Abstract: For about twenty five years it was a kind of folk theorem that complex vector-fields defined on $ \Omega\times \mathbb{R}_t$ (with $ \Omega$ open set in $ \mathbb{R}^n$) by

$\displaystyle L_j = \frac{\partial}{\partial t_j} + i \frac{\partial\varphi}{... ...ial}{\partial x}\;,\; j=1,\dots, n\;,\; \mathbf{t}\in \Omega, x\in \mathbb{R},$

with $ \varphi$ analytic, were subelliptic as soon as they were hypoelliptic. This was the case when $ n=1$, but in the case $ n>1$, an inaccurate reading of the proof given by Maire (see also Trèves) of the hypoellipticity of such systems, under the condition that $ \varphi$ does not admit any local maximum or minimum (through a nonstandard subelliptic estimate), was supporting the belief for this folk theorem. Quite recently, J.L. Journé and J.M. Trépreau show by examples that there are very simple systems (with polynomial $ \varphi$'s) which are hypoelliptic but not subelliptic in the standard $ L^2$-sense. So it is natural to analyze this problem of subellipticity which is in some sense intermediate (at least when $ \varphi$ is $ C^\infty$) between the maximal hypoellipticity (which was analyzed by Helffer-Nourrigat and Nourrigat) and the simple local hypoellipticity (or local microhypoellipticity) and to start first with the easiest nontrivial examples. The analysis presented here is a continuation of a previous work by the first author and is devoted to the case of quasihomogeneous functions.

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Additional Information

M. Derridj
Affiliation: 5 rue de la Juvinière, 78 350 Les loges en Josas, France

B. Helffer
Affiliation: Laboratoire de Mathématiques, Univ Paris-Sud and CNRS, F 91 405 Orsay Cedex, France

Received by editor(s): January 8, 2007
Received by editor(s) in revised form: July 19, 2007
Published electronically: November 24, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.