Compactness of composition operator on weighted Bergman spaces of the polydisc

We study composition operators induced by a smooth symbol between weighted Bergman spaces of the polydisc.

We first prove a compactness criterion that only requires knowing what happens on the distinguished boundary.

Then we prove simple geometric characterizations of boundedness and compactness on some $A^2_\beta(\mathbb{D}^d)$, particularly for $\beta > d-3$.