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

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