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

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

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

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.