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

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dc.creatorDerridj, Makhlouf
dc.creatorHelffer, Bernard
dc.date2006-11-29
dc.date.accessioned2026-07-07T07:33:30Z
dc.date.available2026-07-07T07:33:30Z
dc.descriptionFor about twenty five years it was a kind of folk theorem that complex vector-fields defined on $Ω\times \mathbb R_t$ (with $Ω$ open set in $\mathbb R^n$) by $$ L_j = \frac{\partial}{\partial t_j} + i \frac {\partial ϕ}{\partial t_j}(\t) \frac{\partial}{\partial x}, j=1,..., n, \t\in Ω, x\in \mathbb R ,$$ with $ϕ$ 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 $ϕ$ does not admit any local maximum or minimum (through a non standard 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 $ϕ$'s) which were 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 $ϕ$ 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 non trivial 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.
dc.description35 pages, 1 figure
dc.identifierhttps://arxiv.org/abs/math/0611926
dc.identifierhttp://arxiv.org/abs/math/0611926
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/119479
dc.subjectAnalysis of PDEs
dc.titleSubelliptic estimates for some systems of complex vector fields : quasihomogeneous case
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