of (\[\*\]) can be treated in the same way: $$\begin{gathered}
\sum_{0\le|\alpha|\le s} \CP \int_{b\Omega} \biggl( 
\sum_{\ell=0}^{|\alpha|}
T_{\alpha,|\alpha|-\ell} N^\ell \phi_J\biggr)\overline{\biggl(
\sum_{kK} \e{K}{kJ} D^\alpha \psi_K \pd{\varrho}{z_k} \biggr)}\\  
= \sum_{\ell=0}^{s} \int_{b\Omega} N^\ell \phi_J \cdot 
\overline{ \sum_{\ell
\le|\alpha|\le s} \CP
T_{\alpha,|\alpha|-\ell}^* \biggl(
\sum
_{kK} \e{K}{kJ} D^\alpha \psi_K \pd{\varrho}{z_k} \biggr)} .
\label{DDAG}\end{gathered}$$ Notice that the top order term vanishes since $N^s \phi_J$ is paired with $$\sum_{|\alpha|=s} \CP 
T_{\alpha,0}^* \biggl( \sum_{kK} \e{K}{kJ} D^\alpha
\psi_K \pd{\varrho}{z_k} \biggr) = \sum_{kK}\e{K}{kJ} N^s \psi_K 
\pd{\varrho}{z_k} ,$$ which equals $0$ on $b\Omega$, because $\psi\in\dom\dbar^*$. &gt;From these calculations, and by equating the right hand sides of (\[DAG\]) and (\[DDAG\]), we obtain the $s$ boundary equations. Set $$\sum_{kK}\e{K}{kJ}D^\alpha\psi_K \pd
{\varrho}{z_k} 
= (L_\alpha \psi)_J .$$ Then, on $b\Omega$, we have $$\begin{aligned}
N^s (\K\psi)_J 
& = \sum_{s-1\le|\alpha|\le s} \CP T_{\alpha,|\alpha|-s+1}^*
    (L_\alpha \psi)_J \\
\lefteqn{\sum_{s-2\le|\alpha|\le s-1} \CP 
T_{\alpha,|\alpha|-s+2}^* ND^\alpha
\biggl( 
\sum_{j=0}^{s-1-|\alpha|}(-\Delta)^{j} (\K\psi)_J \biggr)
}  
  \hbox{\qquad \qquad \qquad \qquad \qquad} \\ 
& \qquad = \sum_{s-2\le|\alpha|\le s} \CP
T_{\alpha,|\alpha|-s+2}^* (L_\alpha \psi)_J\end{aligned}$$ $$\qquad \qquad \qquad \qquad \cdots \qquad \qquad  \qquad \qquad
\cdots \\$$ $$\begin{aligned}
\lefteqn{\sum_{0\le|\alpha|\le s-1} \CP T_{\alpha,|\alpha|}^*
ND^\alpha 
\biggl( 
\sum_{j=0}^{s-1-|\alpha|}(-\Delta)^{j} (\K\psi)_J \biggr)} 
\hbox{\qquad \qquad \qquad \qquad \qquad} \\
& = \sum_{0\le|\alpha|\le s} \CP 
T_{\alpha,|\alpha|}^* (L_\alpha
 \psi)_J .\end{aligned}$$ Thus we have $s$ boundary equations in $(\K\psi)_J$. Notice that the 