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For about twenty five years it was a kind of folk theorem that complexvector-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 werehypoelliptic. This was the case when $n=1$ but in the case $n>1$, an inaccuratereading of the proof given by Maire (see also Tr\`eves) of the hypoellipticityof such systems, under the condition that $\phi$ does not admit any localmaximum or minimum (through a non standard subelliptic estimate), wassupporting the belief for this folk theorem. Quite recently, J.L. Journ\'e andJ.M.Tr\'epreau show by examples that there are very simple systems (withpolynomial $\phi$'s) which were hypoelliptic but not subelliptic in thestandard $L^2$-sense. So it is natural to analyze this problem ofsubellipticity which is in some sense intermediate (at least when $\phi$ is$C^\infty$) between the maximal hypoellipticity (which was analyzed byHelffer-Nourrigat and Nourrigat) and the simple local hypoellipticity (or localmicrohypoellipticity) and to start first with the easiest non trivial examples.The analysis presented here is a continuation of a previous work by the firstauthor and is devoted to the case of quasihomogeneous functions.

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