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

Let D-n be the unit polydisc of C-n,phi(z)=(phi(1)( z), . . . ,phi(n)(z)) be a holomorphic self-map of Dn, and psi( z) a holomorphic function on D-n. Let H-infinity(D-n) be the space of all bounded holomorphic functions on D-n and B-alpha(D-n) the alpha-Bloch space. Necessary and suffcient conditions for the weighted composition operator psi C-phi induced by phi(z) and psi( z) to be bounded and compact from B-alpha(D-n)to H-infinity(D-n)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-infinity and the Bloch space, Taiwanese.