Composition operators induced by smooth self-maps of the unit ball in C^N

We study the composition operator C_Ψ induced by an analytic self-map Ψ of the unit ball B_N in C^N that extends to be smooth on over(B, -)_N. When Ψ is of class C³ on over(B, -)_N, we extend to weighted Bergman spaces W.R. Wogen's characterization of when C_Ψ is bounded on H^p (B_N). Next, when Ψ is of class C⁴ on over(B, -)_N, we show that if ε{lunate} > 0 and C_Ψ : A_α^p (B_N) → A_{α + 1 / 4 - ε{lunate}}^p (B_N), then C_Ψ is bounded on A_α^p (B_N). 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.