Compactness of the complex Green operator
Andrew Raich, Emil J. Sträube · Mathematical Research Letters · 2008
Let Ω ⊂ C n be a bounded smooth pseudoconvex domain.We show that compactness of the complex Green operator Gq on (0, q)-forms on bΩ implies compactness of the ∂-Neumann operator Nq on Ω.We prove that if 1 ≤ q ≤ n-2 and bΩ satisfies (Pq) and (P n-q-1 ), then Gq 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 ∂-Neumann problem on locally convexifiable domains, yield the corresponding characterization of compactness of the complex Green operator(s) on these domains.