  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.
