Compactness of the dbar-Neumann operator and commutators of the Bergman projection with continuous functions

TL;DR

This paper establishes an equivalence between the compactness of the dbar-Neumann operator and certain commutators of the Bergman projection on pseudoconvex domains, revealing new insights into their interplay in complex analysis.

Contribution

It proves that the compactness of the dbar-Neumann operator is equivalent to the compactness of specific commutators of the Bergman projection, and shows how this property extends to continuous functions.

Findings

Compactness of N_{p,q+1} is equivalent to compactness of commutators [P_{p,q}, ar{z}_j).

Compactness of the commutator with continuous functions extends within the dbar-complex.

Percolation of compactness properties in the dbar-closed and holomorphic forms.

Abstract

Let D be a bounded pseudoconvex domain in $C^n, n\geq 2, 0\leq p\leq n,$ and $1\leq q\leq n-1.$ We show that compactness of the dbar-Neumann operator, $N_{p,q+1},$ on square integrable (p,q+1)-forms is equivalent to compactness of the commutators $[P_{p,q}, \bar{z}_j]$ on square integrable dbar-closed (p,q)-forms for $1\leq j\leq n$ where $P_{p,q}$ is the Bergman projection on (p,q)-forms. We also show that compactness of the commutator of the Bergman projection with functions continuous on the closure percolates up in the dbar-complex on dbar-closed forms and square integrable holomorphic forms.