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

Let Ω be a bounded pseudoconvex domain in Cn,n≥2,0≤p≤n, and 1≤q≤n−1. We show that compactness of the ∂¯-Neumann operator, Np,q+1, on square integrable (p,q+1)-forms is equivalent to compactness of the commutators [Pp,q,z¯j] on square integrable ∂¯-closed (p,q)-forms for 1≤j≤n where Pp,q is the Bergman projection on (p,q)-forms. We also show that compactness of the commutator of the Bergman projection with bounded functions percolates up in the ∂¯-complex on ∂¯-closed forms and square integrable holomorphic forms.