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

TL;DR

This paper establishes a connection between the regularity of solutions to the $ar{ ext{d}}$-problem in weakly pseudoconvex domains in $ ext{C}^2$ and the boundary $ar{ ext{d}}_b$-problem, reducing the former to the latter.

Contribution

It demonstrates that the regularity of the $ar{ ext{d}}$-problem solutions depends on the boundary $ar{ ext{d}}_b$-problem, providing a method to construct solution operators with optimal regularity.

Findings

Solution operator for $ar{ ext{d}}$ is bounded on Sobolev spaces if boundary $ar{ ext{d}}_b$-problem has similar regularity.

Reduces the $ar{ ext{d}}$-problem to the boundary $ar{ ext{d}}_b$-problem in weakly pseudoconvex domains.

Establishes regularity transfer from boundary to interior solutions.

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