Compactness of Hankel operators with continuous symbols on convex domains

Abstract

Let Ω be a bounded convex domain in C n , n ≥ 2, 1 ≤ q ≤ (n − 1), and φ ∈ C(Ω). If the Hankel operator H q−1 φ on (0, q − 1)-forms with symbol φ is compact, then φ is holomorphic along q-dimensional analytic (actually, affine) varieties in the boundary. We also prove a partial converse: if the boundary contains only 'finitely many' varieties, 1 ≤ q ≤ n, and φ ∈ C(Ω) is analytic along the ones of dimension q (or higher), then H