Carleson embedding on tri-tree and on tri-disc

TL;DR

This paper establishes multi-parameter dyadic embedding theorems for Hardy operators on multi-trees and applies these results to characterize Carleson measures for Dirichlet spaces on bi- and tri-discs, extending previous bi-tree results.

Contribution

It generalizes the embedding theorem from bi-tree to tri-tree and provides a Carleson measure characterization involving dyadic open sets, revealing unexpected equivalences.

Findings

Embedding on tri-tree is equivalent to a one box Carleson condition.

Carleson measures are fully described for Dirichlet spaces on bi- and tri-discs.

Results highlight differences between bi-tree, tri-tree, and higher-dimensional polydiscs.

Abstract

We prove multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces in bi-disc and tri-disc this proves the embedding theorem of those Dirichlet spaces of holomorphic function on bi- and tri-disc. We completely describe the Carleson measures for such embeddings. The result below generalizes embedding result of \cite{AMPVZ} from bi-tree to tri-tree. One of our embedding description is similar to Carleson--Chang--Fefferman condition and involves dyadic open sets. On the other hand, the unusual feature of \cite{AMPVZ} was that embedding on bi-tree turned out to be equivalent to one box Carleson condition. This is in striking difference to works of Chang--Fefferman and well known Carleson quilt counterexample. We prove here the same unexpected result for the tri-tree. Finally, we explain the obstacle that prevents us from proving our results on polydiscs of dimension four and higher.