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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 [AMPVZ] from bi- tree to tri-tree and from Carleson–Chang condition to Carleson box condition. One of our embedding description is similar to Carleson–Chang–Fefferman condition and involves dyadic open sets. On the other hand, the unusual feature is that embedding on bi-tree and tri-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. Finally, we explain the obstacle that prevents us from proving our results on poly-discs of dimension four and higher.
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Bi-parameter embedding and measures with restricted energy conditions
Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on the bi-tree. In this note we give another proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree in a previous paper of the authors (Arcozzi et al. in Bellman function sitting on a tree, arXiv:1809.03397, 2018), which used the Bellman function technique, the proof here is based on some rather subtle comparisons of energies of measures on the bi-tree.
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- Award ID(s):
- 1900268
- PAR ID:
- 10216035
- Date Published:
- Journal Name:
- Mathematische Annalen
- Volume:
- 377
- ISSN:
- 0025-5831
- Page Range / eLocation ID:
- 643-674
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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