Optimal H\"older Regularity of Solution Operator to the $\bar\partial$-equation on Product Domains

TL;DR

This paper establishes the existence of an optimal solution operator for the ar-equation on product domains that preserves Hf6lder regularity, addressing a key regularity preservation challenge.

Contribution

It introduces a canonical solution operator for the ar-equation on product domains that maintains Hf6lder regularity, which was previously unresolved.

Findings

The solution operator preserves Hf6lder regularity without gain.

It is proven to be optimal in regularity preservation.

The result applies specifically to product domains.

Abstract

This note seeks to prove the existence of a canonical solution operator to the $\bar\partial$-equation that preserves H\"older regularity on product domains. It is a well known fact that such solution operators do not in general gain H\"older regularity, and as such, our solution operator is optimal in this regard.