Composition operators on weighted Bergman spaces of the polydisc

We study composition operators between weighted Bergman spaces of the polydisc induced by smooth symbols. We prove a general result of continuity which only involves the behaviour of the symbol on the polytorus. We deduce from this several consequences about the automatic continuity of the induced operator. We study in depth the case of the tridisc and exhibit several examples showing that a characterization of continuity using only derivatives seems impossible.