Sobolev spaces for the weighted d-bar-Neumann operator

TL;DR

This paper investigates the compactness properties of the weighted d-bar-Neumann operator on complex spaces, introducing weighted Sobolev spaces and a Rellich-Kondrachov lemma to establish compactness estimates.

Contribution

It develops a framework of weighted Sobolev spaces and proves a Rellich-Kondrachov lemma to analyze the compactness of the weighted d-bar-Neumann operator.

Findings

Established conditions for compactness of the weighted d-bar-Neumann operator.

Introduced weighted Sobolev spaces suitable for this analysis.

Proved a Rellich-Kondrachov lemma in the weighted setting.

Abstract

We discuss compactness of the d-bar-Neumann operator in the setting of weighted L^2-spaces on b C^n. In addition we describe an approach to obtain the compactness estimates for the d-bar-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich - Kondrachov lemma.