$$\text{Re} \bigl( N^s (\psi\llcorner\dbar\varrho)_I\bigr)  \ge 1  
\quad\text{on } B(p,\delta)\cap\overline{\Omega},$$ where $B(p,\delta)$ is a small ball center at $p\in\Omega$. For $\varepsilon>0$, consider the collection of $(0,q)$ forms $\phi^{(\varepsilon)}$, $$\phi^{(\varepsilon)}:=(-\varrho)^{s-1}
    (-\varrho+\varepsilon)^{3/4} \chi d\bar z^I ,$$ where $\chi$ is a non-negative $C^\infty$ cut-off function, $\text{supp}\chi\ss B(p,\delta)$, and $\chi=1$ on $B(p,\delta/2)$. Now, an easy calculation shows that $$\| \phi^{(\varepsilon)} \|_s \le C_1$$ independently of $\varepsilon$, while $$\left |\int_{b\Omega} N^s \phi_I^{(\varepsilon)} 
    \cdot\overline{N^s (\psi\llcorner\dbar\varrho)_I} \right | 
\ge C_2 \varepsilon^{-1/4} ,$$ which is unbounded, as $\varepsilon\rightarrow0$. This finishes the proof of the proposition.

*We observe that $\dom\dbar^* \cap C_{(0,q+1)}^\infty(\bar\Omega)$ is dense in $W^s_{(0,q+1)}(\Omega)$. Therefore it suffices to show that for any $\varepsilon>0$ and $\phi\in C^\infty_{(0,q+1)}(\overline{\Omega})$ there exists $\psi\in C^\infty_{(0,q+1)}(\overline{\Omega})$ with $\|\psi\|_s <\varepsilon$ and $\phi-\psi\in\dom\dbar^*$.*

Having fixed $\phi$ and $\varepsilon$, let $\chi\in C^\infty_0 (-1,1)$ and $\chi=1$ in a neighborhood of the origin. Then the form $\psi$ $$\psi:= (
1/s!)(-\varrho)^s \chi(-\varrho/\varepsilon) \bigl( N^s
(\phi\llcorner \dbar\varrho)\bigr) \wedge\dbar\varrho$$ satisfies the required conditions.

[Proof of Proposition 3.3]{} We have set $\dbar^* =\vartheta+\K$, so that for $\psi\in\dom\dbar^*$ we have $$\label{dag}
\la \phi,\dbar^* \psi\ra_s = \la\phi,\vartheta\psi\ra_s
+\la\phi,\K\psi\ra_s .$$ On the other hand by (\[&gt;&gt;&gt;\]) we see that, for $\psi\in\dom\dbar^*$ and $\phi\in C^\infty_{(0,q)
}
(\overline{\Omega})$ we have the equality $$\la\dbar\phi,\psi\ra_s = \la\phi,\vartheta\psi\ra_s 
+\sum_{0\le|\alpha|\le s} \CP
