Compactness of products of Hankel operators on the polydisk and some product domains in ${\mathbb{C}}^2$

Let ${\mathbb{D}}^n$ be the polydisk in ${\mathbb{C}}^n$ and the symbols $\varphi ,\psi \in C(\overline{{\mathbb{D}}^n})$ such that $\varphi$ and $\psi$ are pluriharmonic on any $(n-1)$-dimensional polydisk in the boundary of ${\mathbb{D}}^n\mathrm{.}$ Then $H_{\psi }^{\ast }{H}_{\varphi }$ is compact on $A^2({\mathbb{D}}^n)$ if and only if for every $1\le j,k\le n$ such that $j\mathrm{\ne }k$ and any $(n-1)$-dimensional polydisk $D$, orthogonal to the $z_j$-axis in the boundary of ${\mathbb{D}}^n,$ either $\varphi$ or $\psi$ is holomorphic in $z_k$ on $D\mathrm{.}$ Furthermore, we prove a different sufficient condition for compactnes of the products of Hankel operators. In ${\mathbb{C}}^2,$ our techniques can be used to get a necessary condition on some product domains involving annuli.