# Strong Closed Range Estimates: Necessary Conditions and Applications

## TL;DR

This paper develops necessary geometric and potential theoretic conditions for strong closed range estimates of the $ar ext{-}\partial$ operator in complex domains, extending $L^2$ theory beyond pseudoconvex cases and exploring applications to the $ar ext{-}\partial$-Neumann problem.

## Contribution

It introduces a family of strong closed range estimates and generalizes Kohn's weighted theory through elliptic regularization for non-pseudoconvex domains.

## Key findings

- Established necessary conditions for strong closed range estimates.
- Analyzed implications for compactness in the $ar ext{-}\partial$-Neumann problem.
- Extended weighted theory via elliptic regularization for broader classes of domains.

## Abstract

The $L^2$ theory of the $\bar\partial$ operator on domains in $\mathbb{C}^n$ is predicated on establishing a good basic estimate. Typically, one proves not a single basic estimate but a family of basic estimates that we call a family of strong closed range estimates. Using this family of estimates on $(0,q)$-forms as our starting point, we establish necessary geometric and potential theoretic conditions. The paper concludes with several applications. We investigate the consequences for compactness estimates for the $\bar\partial$-Neumann problem, and we also establish a generalization of Kohn's weighted theory via elliptic regularization. Since our domains are not necessarily pseudoconvex, we must take extra care with the regularization.