2020/01/08 by Pavel Mozolyako, Georgios Psaromiligkos, Mozolyako, Pavel +5
Mathematics · #42B99 #47A99 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #F.2.2 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2001.02373
openalex publication_date 2020/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeAMPVZ 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 \citeAMPVZ 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.