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

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