Abstract
Let Omega be a bounded q-pseudoconvex domain in C-n, n >= 2 and let 1 <= q <= n - 1. If Omega is smooth, we find sufficient conditions for the (partial derivative)over-bar-Neumann operator to be compact. If Omega is non-smooth and if q <= p <= n - 1, we show that compactness of the (partial derivative)over-bar-Neumann operator, Np+1, on square integrable (0, p + 1)-forms is equivalent to compactness of the commutators [B-p, (z)over-bar(j)], 1 <= j <= n, on square integrable (partial derivative)over-bar-closed (0, p)-forms, where B-p is the Bergman projection on (0, p)-forms. Moreover, we prove that compactness of the commutator of B-p with bounded functions percolates up in the (partial derivative)over-bar-complex on (partial derivative)over-bar-closed forms and square integrable holomorphic forms. Furthermore, we find a characterization of compactness of the canonical solution operator, Sp+1, of the (partial derivative)over-bar-equation restricted on (0, p+1)-forms with holomorphic coefficients in terms of compactness of commutators [T-p(zj)*, T-p(zj)], 1 <= j <= n, on (0, p)-forms with holomorphic coefficients, where T-p(zj) is the Bergman-Toeplitz operator acting on (0, p)-forms with symbol z(j). This extends to domains which are not necessarily pseudoconvex.