Compact composition operators on the Bloch space in polydiscs

Let U-n be the unit polydisc of O-n and phi = (phi (1), ... , phi (n)) a ho lomorphic self-map of U-n. As the main result of the paper, it shows that t he composition operator C-phi is compact on the Bloch space beta ( U-n ) it and only a for every epsilon > 0, there exists a delta > 0, such that Sigma (n)(k,j=1) /(a)phi1/azk (z)/1-l z(k) l(2)/1-l phi (1)(z) l(2) < e, whenever dist (<phi>(z), partial derivativeU(n)) < <delta>.