Compactness of the Complex Green Operator

Let $\mathrm{\Omega }\subset {\text{C}}^n$ be a bounded smooth pseudoconvex domain. We show that compactness of the complex Green operator $G_q$ on $(0,q)$-forms on $b\mathrm{\Omega }$ implies compactness of the $\bar{\mathrm{\partial }}$-Neumann operator $N_q$ on $\mathrm{\Omega }$. We prove that if $1\le q\le n-2$ and $b\mathrm{\Omega }$ satisfies $(P_q)$ and $(P_{n-q-1})$, then $G_q$ is a compact operator (and so is $G_{n-1-q}$). Our method relies on a jump type formula to represent forms on the boundary, and we prove an auxiliary compactness result for an `annulus’ between two pseudoconvex domains. Our results, combined with the known characterization of compactness in the $\bar{\mathrm{\partial }}$-Neumann problem on locally convexifiable domains, yield the corresponding characterization of compactness of the complex Green operator(s) on these domains.