Products of composition and differentiation operators on the polydisk

We prove the boundedness of the products of composition and differentiation operators acting from the weighted Bergman space on the bidisk, \(A^2_\alpha (\mathbb {D}^2)\) $A_{\alpha }^2({\mathbb{D}}^2)$ , to the weighted Bergman space on the unit disk, \(A^2_\alpha (\mathbb {D})\) $A_{\alpha }^2(\mathbb{D})$ . We shall also obtain conditions on the symbol functions to ensure the boundedness of the composition–differentiation operator acting from the Hardy space on the bidisk, \(H^2(\mathbb {D}^2)\) $H^2({\mathbb{D}}^2)$ , to the Hardy space on the unit disk \(H^2(\mathbb {D})\) $H^2(\mathbb{D})$ .