Product of quasihomogeneous Toeplitz operators on the pluriharmonic Bergman space of the polydisk

In this article, we first give an essential characterization of Toeplitz operators with quasihomogeneous symbols on the weighted pluriharmonic Bergman space of the unit polydisk. Then we completely characterize when the product of two Toeplitz operators with monomial-type symbols is a Toeplitz operator. As a result, some interesting higher-dimensional phenomena appear on the unit polydisk.