Carleson embedding on the tri-tree and on the tri-disc

We prove a multi-parameter dyadic embedding theorem for the Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces in the bi-disc and the tri-disc, this proves the embedding theorem of those Dirichlet spaces of holomorphic functions on the bi-and tri-disc. We completely describe the Carleson measures for such embeddings. The result below generalizes the embedding result of Arcozzi et al. from the bi-tree to the tri-tree and from the Carleson-Chang condition to the Carleson box condition. One of our embedding descriptions is sim-ilar to the Carleson-Chang-Fefferman condition, and involves dyadic open sets. On the other hand, the unusual feature is that the embedding on the bi-tree and the tri-tree turns out to be equivalent to the one box Carleson condition. This is in striking difference to works of Chang-Fefferman and the well-known Carleson quilt counter-example. Finally, we explain the obstacle that prevents us from proving our results on poly-discs of dimension four and higher.