Weighted composition operators from α-Bloch space to H ∞ on the polydisc

Abstract

Let double struk D signn be the unit polydisc of ℂn, φ(z) = (φ1(z), . . . , φn(z)) be a holomorphic self-map of double struk D sign n, and φ(z) a holomorphic function on double struk D sign n. Let H∞(double struk D signn) be the space of all bounded holomorphic functions on double struk D signn and script B sign(double struk D signn) the -Bloch space. Necessary and sufficient conditions for the weighted composition operator φC φ induced by φ(z) and φ(z) to be bounded and compact from script B sign(double struk D signn) to H∞(double struk D signn) are given in this paper. One of our results corrects and extends Theorem 1 in the paper by S. Ohno, Weighted composition operators between H∞ and the Bloch space, Taiwanese J. Math. 2001; 5:555-563.