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 β be a function smooth up to the boundary on a smooth bounded pseudoconvex domain Ω⊂Cn. We show that, if Ω is convex or the Levi form of the boundary of Ω is of rank at least n−2, then compactness of the Hankel operator Hβ implies that β is holomorphic “along” analytic discs in the boundary. Furthermore, when Ω is convex in C2 we show that the condition on β is necessary and sufficient for compactness of Hβ.