\sum_{|J|=q} \biggl(\int_{b\Omega} \phi_J
\overline{N(\K\psi)_J} + \sum_{1\l
e|\beta|\le s-1} \CPb
\int_{b\Omega}  D^\beta \phi_J \overline{ND^\beta(\K\psi)_J} \\
& \qquad\qquad 
-  \la \phi_J , \Delta(\K\psi)_J \ra_{s-1} + \dots \biggr ) , \end{aligned}$$ where the dots stand for terms that do not contribute to any boundary expression.

We iterate this calculation on the last term on the right in the above chain of equalities to obtain that $$\la\phi,\K\psi\ra_s  = \sum_{|J|=q} 
\sum_{i=0}^{s-1} \sum_{|\alpha|\le i}\CP \int_{b\Omega}
(D^\alpha\phi_J)\overline{ND^\alpha(
-\Delta)^{s-1-i}(\K\psi)_J} +
\dots ,$$ where the dots have the same meaning as before. &gt;From this equation and (\[bnry-eq-K\]) it follows that, for all $J$, $$\begin{gathered}
\label{*}
\sum_{i=0}^{s-1} \sum_{|\alpha|\le i} \CP
\int_{b\Omega}(D^\alpha\phi_J) 
\overline{ND^\alpha(-\Delta)^{s-1-i}(\K\psi)_J} \\
= \sum_{0\le|\alpha|\le s} \CP \sum_{kK} \e{K}{kJ} \int_{b\Omega}
D^\alpha \phi_J \overline{D^\alpha \psi_K} \pd{\varrho}{\bar z_k} . \end{gathered}$$ This equation must hold true for all $\phi\in C^\infty_{(0,q)}
(\overline{\Omega})$. Thus we need to isolate the terms containing $N^\ell \phi_J$ for $\ell=0,1,\dots,s-1$, and for all $J$.

Now observe that, if $f$ and $g$ are smooth functions on the boundary, then $$\begin{aligned}
\sum_{j=1}^{2n} \int_{b\Omega} D_j f \overline{D
_j g}
= & \sum_j \int_{b\Omega}(Y_j +\nu_j N)f
            \overline{(Y_j +\nu_j N)g}\\
= & \sum_j \int_{b\Omega} Y_j f \overline{Y_j g}
        +\int_{b\Omega} Nf \overline{N g}, \end{aligned}$$ where we have used the fact that $\sum_j \nu_j Y_j =0$. Now 