Bounded Composition Operators on Holomorphic Lipschitz and Bloch Spaces of the Polydisk

Let 0 ≤ α < 1. Denote the open unit polydisk in C by ∆. We obtain a purely function-theoretic condition characterizing the holomorphic self-maps of ∆ that induce bounded composition operators on (1−α)-Bloch spaces, which we define on ∆. As a consequence, we extend Madigan’s characterization of bounded composition operators on analytic α-Lipschitz spaces over ∆ for 0 < α < 1 to an analogous characterization in ∆. Our proof depends on an auxiliary result that the holomorphic α-Lipschitz and (1−α)Bloch spaces of the polydisk are equal and norm equivalent when 0 < α < 1, thus extending a classical result of Hardy/Littlewood on ∆. Mathematics Subject Classification (2000). Primary 47B33; Secondary 47A13, 32A16, 26A16, 32A26, 32A37, 32A38, 32H02.