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 . When Ψ is of class C3 on , 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 , we show that if ϵ>0 and , then CΨ is bounded on . 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.