\le c\|f\|_s.$$*

The proof is in two steps. In the first, we rely heavily on Kohn’s estimates [@Kohn], to construct and estimate the [*canonical solutions*]{} in $W^s$ to the equations $\dbar u=f$ and $\dbar^* v=g$. In the second step we prove the solvabilty of the $(\dbar,s)$-Neumann problem. In the course of the proof, by [*orthogonal*]{} we shall always mean orthogonality in the $W^s$ inner product.

By (3.21) in [@Kohn], since the $\dbar$-cohomology is trivial on a pseudoconvex domain $\Omega\ss {\Bbb C}^n$, we have that $\rg \dbar_{(0,q-1)} = \ker \dbar_{(0,q)}$. This equality implies that $\rg\dbar_{(0,q-1)}$ is closed in ${W^s_{(0,q)}(\Omega)}$. Now Lemma 4.1.1 in [@HO], applied with $F=\rg\dbar_{(0,q-1)}$, gives that $$\| f\|_s \le c \|\dbar^*_{(0,q)} f\|_s$$ for all $f\in \rg\dbar_{(0,q-1)}\cap\dom\dbar^*_{(0,q)}$. This in turn, by Lemma 4.1.2 in [@HO], implies that for all $v$ in the orthogonal complement of $\ker\dbar_{(0,q-1)}$, i.e. in the closure of $\rg\dbar^*_{(0,q)}$, there exists $f\in\dom\dbar^*_{(0,q)}$ such that $\dbar^*_{(0,q)} f=v$. Hence, $\rg\dbar_{(0,q)}^*$ is closed as well, and therefore we have the estimate $\| f\|_s \le C\|\dbar f\|_s$ for all $f\in\rg\dbar^*_{(0,q)}\cap\dom\dbar_{(0,q-1)}$. Moreover, we have the strong orthogonal decomposition $${W^s_{(0,q)}(\Omega)}=\rg\dbar_{(0,q+1)}^* \oplus\rg\dbar_{(
0,q-1)} .$$ Now, given any $g\in{W^s_{(0,q)}(\Omega)}$, with $\dbar_{(0,q)} g=0$, i.e. $g\in\rg\dbar_{(0,q-1)}$, we can find $v\in\dom\dbar_{(0,q-1)}$, orthogonal to $\ker\dbar_{(0,q-1)}$, such that $\dbar 