Regularity of dbar on worm domains

TL;DR

This paper constructs a solution operator for the $ar{ ext{d}}$-equation on unbounded worm domains, demonstrating it preserves regularity and depends on domain symmetries, advancing understanding of complex analysis on such domains.

Contribution

It introduces a new solution operator for the $ar{ ext{d}}$-equation on unbounded worm domains with proven regularity preservation, considering domain symmetries.

Findings

Operator preserves regularity of data

Operator acts as a continuous mapping on specific Sobolev subspaces

Regularity estimates depend on domain rotational invariance

Abstract

A solution operator to the $\bar{\partial}$-equation is constructed on unbounded worm domains, $D_{\beta}$. Regularity estimates are proven showing the operator preserves regularity of the data. The operator may be viewed as a continuous mapping among appropriate subpaces of $W^s(D_{\beta})$, which depend on a rotational invariance of the domains.