Differences of weighted composition operations on the unit polydisk

Let φ ₁ and φ ₂ be holomorphic self-maps of the unit polydisk \(\mathbb{D}^N\), and let u ₁ and u ₂ be holomorphic functions on \(\mathbb{D}^N\). We characterize the boundedness and compactness of the difference of weighted composition operators W _{φ1, u1} and W _{φ2, u2} from the weighted Bergman space \(A_{\vec \alpha }^p\), 0 < p < ∞, \(\vec \alpha = \left( {\alpha _1 , \ldots ,\alpha _{\rm N} } \right)\), α _j > ?1, j = 1,..., N, to the weighted-type space H _υ ^∞ of holomorphic functions on the unit polydisk \(\mathbb{D}^N\) in terms of inducing symbols φ ₁, φ ₂, u ₁, and u ₂.