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

Abstract

Let D be a bounded pseudoconvex domain in $C^n,n\ge 2,0\le p\le n,$ and $1\le q\le n-1.$ 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 $1\le j\le n$ 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.

9 pages, exposition improved, accepted for publication in JMAA

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