Gain/Loss of derivatives for complex vector fields

In ${\text{C}}_z\times {\mathbb{R}}_t$ we consider the function $g=g(z)$, set $g_1={\text{di}}_{z}g$, $g_{1\bar{1}}={\text{di}}_z{\text{dib}}_{z}g$ and define the operator $L_g={\text{di}}_z+i{g}_1{\text{di}}_t$. We discuss estimates with loss of derivatives, in the sense of Kohn, for the system $({\bar{L}}_g,f^k{L}_g)$ where $({\bar{L}}_g,L_g)$ is $\frac{1}{2m}$ subelliptic at 0 and $f(0)=0,df(0)\mathrm{\ne }0$. 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 $(g,f^k)=(\mathrm{\mid }z{\mathrm{\mid }}^{2m},{\bar{z}}^k)$ this result was obtained by Kohn and Bove-Derridj-Kohn-Tartakoff for $m=1$ and $m\ge 1$ 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 $(g,f^k)=(\mathrm{\mid }z{\mathrm{\mid }}^{2m},x^k)$, in which the multiplier condition is violated, the loss is not lowered by the type and must be $\ge \frac{k-1}{2}$.