Regularity of $\bar{\partial}$ and $\bar{\partial}_b$ on pseudoconvex domains in $\mathbb{C}^2$

We reduce the problem of constructing a linear solution operator to the \dbar-equation on smoothly bounded weakly pseudoconvex domains, $\Omega$, in $\mathbb{C}^2$ to the problem of the boundary $\mdbar_b$-equation. We show there is a solution operator to $\mdbar$ which is bounded as a map $W^{s}_{(0,1)}(\Omega)\cap{ker}\mdbar \rightarrow W^{s}(\Omega)$ for all $s>1/2$ if there is a corresponding solution operator to the $\mdbar_b$-problem with analogous regularity properties.