Sobolev inequalities and the d-bar-Neumann operator

TL;DR

This paper explores a complex-valued Sobolev inequality and its connection to the compactness of the d-bar-Neumann operator, using abstract compactness criteria and discussing extensions under subelliptic estimates.

Contribution

It introduces a complex-valued Sobolev inequality framework and links it to the compactness and extension properties of the d-bar-Neumann operator.

Findings

Established a characterization of compactness in L^2-spaces.

Connected Sobolev inequalities to the compactness of the d-bar-Neumann operator.

Showed the operator's continuous extension under subelliptic estimates.

Abstract

We study a complex valued version of the Sobolev inequalities and its relationship to compactness of the d-bar-Neumann operator. For this purpose we use an abstract characterization of compactness derived from a general description of precompact subsets in L^2-spaces. Finally we remark that the d-bar-Neumann operator can be continuously extended provided a subelliptic estimate holds.