For about twenty five years it was a kind of folk theorem that complex vector-fields defined on
Ω
×
R
t
\Omega \times \mathbb R_t
(with
Ω
\Omega
open set in
R
n
\mathbb R^n
) by
\[
L
j
=
∂
∂
t
j
+
i
∂
φ
∂
t
j
(
t
)
∂
∂
x
,
j
=
1
,
…
,
n
,
t
∈
Ω
,
x
∈
R
,
L_j = \frac {\partial }{\partial t_j} + i \frac {\partial \varphi }{\partial t_j}(\mathbf {t})\, \frac {\partial }{\partial x}\;,\; j=1,\dots , n\;,\; \mathbf {t}\in \Omega , x\in \mathbb R,
\]
with
φ
\varphi
analytic, were subelliptic as soon as they were hypoelliptic. This was the case when
n
=
1
n=1
, but in the case
n
>
1
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
\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
φ
\varphi
’s) which are hypoelliptic but not subelliptic in the standard
L
2
L^2
-sense. So it is natural to analyze this problem of subellipticity which is in some sense intermediate (at least when
φ
\varphi
is
C
∞
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.