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 $$L_j = \frac{\partial}{\partial t_j} + i \frac {\partial \phi}{\partial t_j}(\t) \frac{\partial}{\partial x}, j=1,..., n, \t\in \Omega, x\in \mathbb R ,$$ with $\phi$ 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 Tr\`eves) of the hypoellipticity of such systems, under the condition that $\phi$ 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. Journ\'e and J.M.Tr\'epreau show by examples that there are very simple systems (with polynomial $\phi$'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 $\phi$ 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.