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

Abstract

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 $W_{(0,1)}^s(\Omega )\cap ker\bar{\partial }\to W^s(\Omega )$ for all $s\ge 0$ if there is a corresponding solution operator to the ${\bar{\partial }}_b$-problem with analogous regularity properties.

this version fixes an error stemming from the added correction term to the complex Laplacian in previous version