arXiv:2108.04789v1 [math.CA] 10 Aug 2021 MULTI-PARAMETER CARLESON EMBEDDINGS FOR p =2 ON T 2 OR FOR p =2 ON T 4 AND WHY THE PROOFS FAIL PAVEL MOZOLYAKO, GEORGIOS PSAROMILIGKOS, AND ALEXANDER VOLBERG Abstract. This note contains a plethora of counterexamples to at- tempts to generalize the results of bi-parameter embedding from p =2 case to either p> 2 or p< 2. This is in striking juxtaposition to p = 2 case that was fully understood in the series of papers [AMPS], [AMPVZ-K], [MPVZ1], [MPVZ2], [AHMV], [MPV]. We also build a counterexample to small energy majorization on bi-tree. This coun- terexample shows that straightforward generalizations of methods of [AMPVZ-K], [MPVZ1], [MPVZ2], [AHMV] from 2-tree T 2 or 3-tree T 3 to 4-tree T 4 will not work even for p = 2 unless some new approach is invented. 1. Introduction Embedding theorems on graphs are interesting in particular because they are related to the structure of spaces of holomorphic functions. For Dirichlet space on a disc this fact has been explored in [ARSW14] [ARS02] [ARSW11], and for Dirichlet space on bi-disc in [AMPS], [AMPVZ-K], [AHMV]. Bi-disc case is much harder as the corresponding graph has cycles. One particular interesting case see in [Saw1] (a small piece of bi-tree is considered). The difference between one parameter theory (graph is a tree) and two parameter theory (graph is a bi-tree) is huge. One explanation is that in a multi-parameter theory all the notions of singular integrals, para-products, BMO, Hardy classes etc become much more subtle than in one parameter settings. There are many examples of this effect. It was demonstrated in results of S.Y. A. Chang, R. Fefferman and L. Carleson, see [Carl74], [Chang79], [ChF80],[RF1], [Tao]. Another difference between one- and two-parameter embeddings is that in one parameter case the results for L p are the same as for L 2 . This seems not to be the case for the two parameter theory. 2010 Mathematics Subject Classification. 42B20, 42B35, 47A30. PM is supported by the Russian Science Foundation grant 17-11-01064. AV is partially supported by the NSF grant DMS 1900268 and by Alexander von Humboldt foundation. 1