Weighted Composition Operators from H∞ to the Bloch Space on the Polydisc

Abstract

Let Dn be the unit polydisc of ℂn, 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)|+supz∈Dn∑k=1n|(∂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).