Composition operators induced by smooth self-maps of the real or complex unit balls

In this paper, we study the composition operator C-Phi with a smooth but not necessarily holomorphic symbol Phi. A necessary and sufficient condition on Phi for C Phi to be bounded on holomorphic: (respectively harmonic) weighted Bergman spaces of the unit ball in C-n (respectively R-n) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces. (C) 2008 Elsevier Inc. All rights reserved.