The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications

Let U-n be the unit polydisc of C-n and phi = (phi(1),..., phi(n)) a holomorphic self-map of U-n. B-p(U-n), B-o(p) (U-n), and B-o*(p) (U-n) denote the p-Bloch space, little p-Bloch space, and little star p-Bloch space in the unit polydisc U-n, respectively, where p, q > 0. This paper gives the estimates of the essential norms of bounded composition operators C-phi induced by phi between B-p(U-n) (B-o(p) (U-n) or B-o*(p) (U-n)) and B-q(U-n) (B-o(q)(U-n) or B-o*(q) (U-n)). As their applications, some necessary and sufficient conditions for the ( bounded) composition operators C-phi to be compact from B-p(U-n) (B-o(p) (U-n) or B-o*(p) (U-n)) into B-q(U-n) (B-o(q)(U-n) or B-o*(q) (U-n)) are obtained.