Composition operators on the polydisc

In this paper we study the boundedness of composition oper-ators on the weighted Bergman spaces and the Hardy space over the polydisc Dn. Studying the volume of sublevel sets we show for which n the necessary conditions obtained by Bayart are sufficient. For arbitrary polydisc we prove the rank suffi-ciency theorem which, in particular, provides us with a simple criterion describing boundedness of composition operators on the spaces over the bidisc. Such a consistent characterization is obtained for the classical Bergman space over the tridisc.(c) 2022 Elsevier Inc. All rights reserved.