Composition operators between H∞ and a-Bloch spaces on the polydisc

Abstract

Let Un be the unit polydisc of ℂn and φ(z) = (φ1(z),...,φn(z)) a holomorphic self-map of Un. Let H(Un) denote the space of all holomorphic functions on Un, H∞(Un) the space of all bounded holomorphic functions on Un, and Ba(U n), a > 0, the a-Bloch space, i.e., Ba(Un) = {f ∈ H(Un) | ∥f∥Ba = |f(0)| + sup z∈Un ∑k=1n|∂f/∂zk(z)|(1 - |zk|2)a < +∞}. We give a necessary and sufficient condition for the composition operator Cφ induced by φ to be bounded and compact between H∞(Un) and a-Bloch space Ba(Un), when a ≥ 1.