Exact regularity of the bar{partial}-problem with dependence on the bar{partial}_b-problem on weakly pseudoconvex domains in mathbb{C}²

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