Composition operators induced by symbols defined on a polydisk

Suppose φ is a holomorphic mapping from the polydisk D^m into the polydisk Dⁿ, or from the polydisk D^m into the unit ball B_n, we consider the action of the associated composition operator C_φ on Hardy and weighted Bergman spaces of Dⁿ or B_n. We first find the optimal range spaces and then characterize compactness. As a special case, we show that if{A formula is presented} is a holomorphic self-map of the polydisk Dⁿ, then C_φ maps A_α^p ( Dⁿ ) boundedly into A_β^p ( Dⁿ ), the weight β = n ( 2 + α ) - 2 is best possible, and the operator{A formula is presented} is compact if and only if the function{A formula is presented} tends to 0 as z approaches the full boundary of Dⁿ. This settles an outstanding problem concerning composition operators on the polydisk.