$C^\infty$ coefficients, $T_0 =(-1)^{k-1} \cdot \hbox{id}$ on $b\Omega$, and $P_{s+k}^{(I)}$ is a differential operator of order $s+k$ with $C^\infty$ coefficients and acting on the components of $\psi$.*

As a consequence, from the theory of elliptic boundary value problems [@LiMa], we shall obtain the next result.

\[K-order-1\] Let $s$ a be positive integer. Then, $\K$ is a well defined operator of order $1$. More precisely, for all $t>s+1/2$. there exists a positive constant $C_t >0$ such that we have the estimate $$\|\K \psi\|_{t-1} \le C_t \|\psi\|_t$$ for all $\psi\in C^\infty_{(0,q+1)}(\overline{\Omega})$. Furthermore, when restricted to purely tangential forms, $\K$ is of order $0$, i.e. for all $t>s+1/2$ there exists $C_t >0$ such that if $\psi\llcorner\dbar\varrho=0$ in a neighborhood of $b\Omega$, then $$\|\K\psi\|_{t-1} \le C_t \|\psi\|_{t-1} .$$

As a consequence of these facts, we obtain the following representation for the $\dbars$-Neumann problem. We set $G_s :=\dbar\K+\K\dbar$. With the notation above, the $\dbars$- Neumann problem is equivalent to the boundary value problem $$\begin{cases}
(\Box+G_s)u=f &\quad\text{on }\Omega\\
N^s (u\llcorner\dbar\varrho)=0 &\quad\text{on }b\Omega\\
N^s (\dbar u\llcorner\dbar\varrho)=0 &\quad\text{on }b\Omega \, . 
\end{cases}$$

Here $\Box
:=\dbar\vt+\vt\dbar$ is the complex Laplacian, and it equals $-4\Delta$ on $\Omega\ss{\Bbb C}^n$. Notice that $G_s$ is the singular Green’s operator we mentioned earlier. The 