Compactness for the d-bar - Neumann problem - a functional analysis approach

TL;DR

This paper investigates the conditions under which the d-bar-Neumann operator is compact in weighted L^2-spaces on complex n-space, using functional analysis techniques to connect geometric properties with operator compactness.

Contribution

It introduces a functional analysis approach to characterize compactness of the d-bar-Neumann operator and links property (P) to compactness on pseudoconvex domains.

Findings

Relatively compact subsets of weighted L^2-spaces are characterized.

Property (P) implies the compactness of the d-bar-Neumann operator.

An abstract functional analysis criterion for compactness is provided.

Abstract

We discuss compactness of the d-bar-Neumann operator in the setting of weighted L^2-spaces on C^n.$ For this purpose we use a description of relatively compact subsets of L^2- spaces. We also point out how to use this method to show that property (P) implies compactness for the d-bar-Neumann operator on a smoothly bounded pseudoconvex domain and mention an abstract functional analysis characterization of compactness of the d-bar-Neumann operator.