obtain an unbounded, densely defined, closed operator $$\dbar:{W^s_{(0,q)}(\Omega)}\rightarrow W^s_{(0,q+1)}(\Omega) .$$ Thus, in particular, $\ker \dbar$ is a closed subspace in ${W^s_{(0,q)}(\Omega)}$. Sometimes we shall use the notation $\dbar_{(0,q)}$ to stress the fact that the operator $\dbar$ is acting on $(0,q)$ forms.

Consider now the ${W^s(\Omega)}$-Hilbert space adjoint $\dbar^*$ of $\dbar$. We want to study the boundary value problem $$\label{dbar-neumann}
\begin{cases}
(\dbar\dbar^* +\dbar^* \dbar)u=f &\text{ on }\Omega\\
u,\, \dbar u\in \dom \dbar^* \, , &   
\end{cases}$$ where $f$ is a given $(0,q)$ form. When appropriate, we shall refer to this problem as [**(3,s)**]{} in order to emphasize that the topology is coming from the $W^s$ inner product. The condition that $u$ and $\overline{\partial} u$ lie in the domain of $\overline{\partial}^*$ leads to the [*$\overline{\partial}$-Neumann $s$-order boundary conditions*]{}. We shall refer below to the $(\dbar,s)$-Neumann conditions, and the $\dbars$-Neumann problem. Notice that if the Hilbert space under consideration is $L^2
(\Omega)$ (that is, $s=0$) with respect to the Lebesgue measure, then the problem [**(3,s)**]{} reduces to the classical $\dbar$-Neumann problem.

J. J. Kohn solved the $\dbar
$-Neumann ($= (\dbar,0)$-Neumann) problem in a series of papers in 1963-4 (see [@Folland-Kohn] and references therein). This work has proved important in the theory of partial 