A Class of Domains with noncompact $\bar{\partial}$-Neumann operator

TL;DR

This paper identifies specific complex domains where the $ar{ ext{d}}$-Neumann operator fails to be compact, extending understanding of the operator's behavior in complex analysis.

Contribution

It introduces a class of domains, including higher-dimensional Hartogs triangles, where the $ar{ ext{d}}$-Neumann operator is noncompact, broadening previous results.

Findings

Noncompactness of the $ar{ ext{d}}$-Neumann operator on certain domains

Higher-dimensional analogs of Hartogs triangles exhibit this noncompactness

Embedding generalized Hartogs triangles into domains causes noncompactness

Abstract

The $\bar{\partial}$-Neumann operator (the inverse of the complex Laplacian) is shown to be noncompact on certain domains in complex Euclidean space. These domains are either higher-dimensional analogs of the Hartogs triangle, or have such a generalized Hartogs triangle imbedded appropriately in them.