$\dbar u$ belong to $\dom\dbar^*$. While doing this we use the particular choice of the inner product (\[Sobolev\]) to obtain reasonably clean equations and formulas. We then conclude with some remarks about what lies ahead. In a forthcoming paper we give estimates for the above problem in the special case of a strongly pseudoconvex domain, and with $s=1$. The foundations for the present work, studied in the real variable context of the de Rham complex, were laid in the papers [@FKP1], [@FKP2].

We thank H. Boas for making several useful remarks and comments on an earlier version of this paper. We also thank the referee for making helpful suggestions. Work of the second author at MSRI was supported by NSF Grant DMS-9022140.

Solvability of the $\dbars$-Neumann problem
===========================================

The aim of the present section is to prove the following theorem.

\[solvability\] *Let $\Omega$ be a smoothly bounded pseudoconvex domain in ${{\Bbb C}^n}$. Let $s,q$ be positive integers, $0<q\le n$. Let $f\in{W^s_{(0,q)}(\Omega)}$. Then there exists a unique $u\in{W^s_{(0,q)}(\Omega)}$ that solves the $\dbars$-Neumann problem $$\begin{cases}
(\dbar\dbar^* +\dbar^* \dbar)u=f & \text{ on }\Omega\\
u,\, \dbar u\in\dom \dbar^* \, . &
\end{cases}$$ Moreover, there exists a constant $c>0$, independent of $f$, such that $$\|u\|_s 