  In $\C_z\times\R_t$ we consider the function $g=g(z)$, set $g_1=\di_z g$,
$g_{1\bar 1}=\di_z\dib_zg$ and define the operator $L_g=\di_z+ig_1\di_t$. We
discuss estimates with loss of derivatives, in the sense of Kohn, for the
system $(\bar L_g,f^kL_g)$ where $(\bar L_g,L_g)$ is $\frac1{2m} $ subelliptic
at 0 and $f(0)=0,\,\,df(0)\neq0$. We prove estimates with a loss
$l=\frac{k-1}{2m} $ if the "multiplier" condition $|f|\simgeq |g_{1\bar
1}|^{\frac1{2(m-1)}}$ is fulfilled. (For estimates without cut-off,
subellipticity can be weakened to compactness and this results in a loss of
$l=\frac [{2(m-1)}$.) For the choice $(g,f^k)=(|z|^{2m},\bar z^k)$ this result
was obtained by Kohn and Bove-Derridj-Kohn-Tartakoff for $m=1$ and $m\geq1$
respectively. Also, the loss $l=\frac{k-1}{2m}$ was proven to be optimal. We
show that it remains optimal for the model $(g,f^k)=(x^{2m},x^k)$. Instead, for
the model $(g,f^k)=(|z|^{2m},x^k)$, in which the multiplier condition is
violated, the loss is not lowered by the type and must be $\geq \frac{k-1}2$.
