Evolution on the backbone: Apocynaceae phylogenomics and new perspectives on growth forms, flowers, and fruits

Fishbein, Mark; Livshultz, Tatyana; Straub, Shannon C.; Simões, André O.; Boutte, Julien; McDonnell, Angela; Foote, Abbey

doi:10.1002/ajb2.1067

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.

More Like this