Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let P be a function smooth up to the boundary on a smooth bounded pseudoconvex domain Omega subset of C-n. We show that, if Omega is convex or the Levi form of the boundary of Q is of rank at least n - 2, then compactness of the Hankel operator H-beta implies that beta is holomorphic "along" analytic discs in the boundary. Furthermore, when Omega is convex in C-2 we show that the condition on is necessary and sufficient for compactness of H-beta. (C) 2009 Elsevier Inc. All rights reserved.