Weighted composition operators from H∞ to the bloch space on the polydisc

Abstract

Let Dn be the unit polydisc of n, φ(z)=( φ1 (z),...,φn (z)) be a holomorphic self-map of Dn, and ψ(z) a holomorphic function on Dn . Let H( Dn) denote the space of all holomorphic functions with domain Dn, H∞ ( Dn) the space of all bounded holomorphic functions on Dn, and B( Dn) the Bloch space, that is, B( Dn)={ f∈H( Dn) |||f|| B =| f(0) |+ sup z∈Dn ∑ k=1 n | (∂f/ ∂zk)(z) | ( 1- | zk | 2)<+∞ } . We give necessary and sufficient conditions for the weighted composition operator ψCφ induced by φ(z) and ψ(z) to be bounded and compact from H∞ ( Dn) to the Bloch space B( Dn).