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

Abstract

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 similar 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 counterexample. Finally, we explain the obstacle that prevents us from proving our results on poly-discs of dimension four and higher.