Subelliptic estimates for some systems of complex vector fields: Quasihomogeneous case
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- by M. Derridj and B. Helffer PDF
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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 \[ L_j = \frac {\partial }{\partial t_j} + i \frac {\partial \varphi }{\partial t_j}(\mathbf {t}) \frac {\partial }{\partial x}\;,\; j=1,\dots , n\;,\; \mathbf {t}\in \Omega , x\in \mathbb R,\] 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.References
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Additional Information
- M. Derridj
- Affiliation: 5 rue de la Juvinière, 78 350 Les loges en Josas, France
- MR Author ID: 56970
- B. Helffer
- Affiliation: Laboratoire de Mathématiques, Univ Paris-Sud and CNRS, F 91 405 Orsay Cedex, France
- MR Author ID: 83860
- Received by editor(s): January 8, 2007
- Received by editor(s) in revised form: July 19, 2007
- Published electronically: November 24, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 2607-2630
- MSC (2000): Primary 35B65; Secondary 32N15
- DOI: https://doi.org/10.1090/S0002-9947-08-04601-1
- MathSciNet review: 2471931