Composition operators induced by smooth self-maps of the unit ball in CN

We study the composition operator CΨ induced by an analytic self-map Ψ of the unit ball BN in CN that extends to be smooth on BN. When Ψ is of class C3 on BN, we extend to weighted Bergman spaces W.R. Wogen’s characterization of when CΨ is bounded on Hp(BN). Next, when Ψ is of class C4 on BN, we show that if >0 and CΨ :A pα (BN)→A p α+1/4− (BN), then CΨ is bounded on A pα (BN). The discrete jump of size 1/4 in the exponent of the weight is sharp. Examples are given that show the assumption Ψ is smooth is essential in these theorems. © 2006 Elsevier Inc. All rights reserved