Carleson embedding on the tri-tree and on the tri-disc (Q2692504)

Summary: 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 \textit{N. Arcozzi} et al. [Math. Ann. 377, No. 1--2, 643--674 (2020; Zbl 1439.42014)] 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 \textit{S.-Y. A. Chang} and \textit{R. Fefferman} [Ann. Math. (2) 112, 179--201 (1980; Zbl 0451.42014); Bull. Am. Math. Soc., New Ser. 12, 1--43 (1985; Zbl 0557.42007)] and the well-known Carleson quilt counterexample [\textit{L. Carleson}, ``A counter example for measures bounded on \(H^p\) for the bi-disc'', Mittag-Leffler Report No. 7 (1974)]. Finally, we explain the obstacle that prevents us from proving our results on poly-discs of dimension four and higher.