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The -Neumann operator on (0, q)-forms (1 q n) on a bounded convex domain 0 in C n is compact if and only if the boundary of 0 contains no complex analytic (equivalently: affine) variety of dimension greater than or equal to q. 1998Academic Press

Abstract.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.

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