Exact regularity of $\bar{\partial}$ on pseudoconvex domains in $\mathbb{C}^2$

TL;DR

This paper proves the existence of a bounded solution operator for the $ar{ ext{d}}$ problem on pseudoconvex domains in $ ext{C}^2$, ensuring regularity in Sobolev spaces for all non-negative orders.

Contribution

It establishes the exact regularity of the $ar{ ext{d}}$ operator on pseudoconvex domains in $ ext{C}^2$, providing a bounded solution operator across all Sobolev spaces.

Findings

Solution operator is bounded for all Sobolev spaces $W^s$ with $s \\ge 0$

Ensures regularity of solutions to $ar{ ext{d}}$ in $ ext{C}^2$ pseudoconvex domains

Advances understanding of $ar{ ext{d}}$ regularity in complex analysis

Abstract

We show there is a solution operator to $\bar{\partial}$ which is bounded as a map $W^{s}_{(0,1)}(\Omega)\cap\mbox{ker }\bar{\partial}\rightarrow W^{s}(\Omega)$ for all $s\ge 0$.