Compactness of products of Hankel operators on the polydisk and some product domains in C2

Let Dn be the polydisk in Cn and the symbols such that ϕ and ψ are pluriharmonic on any (n−1)-dimensional polydisk in the boundary of Dn. Then is compact on A2(Dn) if and only if for every 1⩽j,k⩽n such that j≠k and any (n−1)-dimensional polydisk D, orthogonal to the zj-axis in the boundary of Dn, either ϕ or ψ is holomorphic in zk on D. Furthermore, we prove a different sufficient condition for compactness of the products of Hankel operators. In C2, our techniques can be used to get a necessary condition on some product domains involving annuli.