$C^\infty$ in a neighborhood of $\overline{\Omega}$, and we wish to express $\dom\dbar^*$ in terms of $r$ and ${\partial}/{\partial}r$. Then we obtain the following description. There exists a differential operator $L_s$ of order $s$, with $C^\infty (\overline{U})$ coefficients, whose leading term is $({\partial}/{\partial}r)^s$, and such that $$\dom\dbar^* \cap C_{(0,q+1)}^\infty(\bar\Omega)
= \biggl \{ \psi: L_s \bigl(\psi \llcorner\dbar r\bigr)_I 
=0 \text{ on }b\Omega, \text{ for all }I, |I|=q \biggr \} .$$ Indeed, on $U$, $$\pd{}{r} = |\text{grad\,} r|N+ gX,$$ where $g\in C^\infty(\overline{U})$, $g=0$ on $b\Omega$, and $X$ is a vector field on $U$.* ]{}

We set $\dbar^* =\vartheta+\K$, and we want to determine $\K$. Our result is the following.

\[K\] *Let $\Omega$ be a smoothly bounded domain in ${{\Bbb C}^n}$, and let $s$ be a positive integer. Let $\dbar^*$ be the ${W^s(\Omega)}$-Hilbert adjoint of $\dbar$, and $\vartheta$ be the formal adjoint of $\dbar$, respectively. Set $\dbar^*=\vartheta + {\cal K}$. Then, for a $(0,q+1)$ form $\psi$, $\K\psi:=\sum_{|I|=q}\omega_I d\bar z^I$ is the $(0,q)$ form whose components are solutions of the following : $$\label{BV}
\begin{cases}
\sum_{j=0}^{s}(-\Delta)^j \omega_I =0 & \text{ on }\Omega\\
\sum_{j=0}^{s+k-1}T_j N^{s+k-1-j}\omega_I =P_{s+k}^{(I)}\psi  
    & \text{ on }b\Omega ,\, k=1,\dots,s \, .
\end{cases}$$ Here $T_k$ denotes a tangential differential operator of order $\le k$, with 