Composition operators induced by symbols defined on a polydisk

Suppose φ   is a holomorphic mapping from the polydisk DmDm into the polydisk DnDn, or from the polydisk DmDm into the unit ball BnBn, we consider the action of the associated composition operator CφCφ on Hardy and weighted Bergman spaces of DnDn or BnBn. We first find the optimal range spaces and then characterize compactness. As a special case, we show that ifφ(z)=(φ1(z),…,φn(z)),z=(z1,…,zn), is a holomorphic self-map of the polydisk DnDn, then CφCφ maps Aαp(Dn) boundedly into Aβp(Dn), the weight β=n(2+α)−2β=n(2+α)−2 is best possible, and the operatorCφ:Aαp(Dn)→Aβp(Dn) is compact if and only if the function[∏k=1n(1−|zk|2)n]/[∏k=1n(1−|φk(z)|2)] tends to 0 as z   approaches the full boundary of DnDn. This settles an outstanding problem concerning composition operators on the polydisk.