Let \(\Omega\subset \mathbb{C}^2\) be a smoothly bounded
weakly pseudoconvex domain. We discuss the regularity of the
solution to the \(\overline{\partial}\)-equation on \(\Omega\),
i.e. we look to solve \(\overline{\partial}u=f\) for \(f\in
W^s_{(0,1)}(\Omega)\). A result of Barrett shows that the canonical
solution cannot always be used when one is looking for a solution in
\(W^s(\Omega)\).
We construct a solution operator which does exhibit regularity.
Define the space \(A^s_{(0,1)}(\Omega) = W^s_{(0,1)}(\Omega)
\cap\operatorname{ker }\overline{\partial}\) and assign to it the
norm from \(W^s_{(0,1)}(\Omega)\). Our main result is the existence
of a solution operator \(K\) such that \(\overline{\partial} K f =
f\) for all \(f\in A^s_{(0,1)}(\Omega)\) and
\begin{equation*}
K\colon A^s_{(0,1)} (\Omega)\rightarrow W^{s+1/2}(\Omega).
\end{equation*}
for all \(s\ge0\).
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