Multi-parameter Carleson embeddings for $p\neq 2$ on $T^2$ or for $p=2$ on $T^4$, and why the proofs fail

TL;DR

This paper presents counterexamples demonstrating the failure of generalizing bi-parameter embedding results from the well-understood p=2 case to other p values or higher-dimensional trees, highlighting the need for new methods.

Contribution

It provides explicit counterexamples showing the limitations of existing techniques for multi-parameter Carleson embeddings beyond the p=2 case and lower-dimensional settings.

Findings

Counterexamples for p>2 and p<2 cases.

Counterexample to small energy majorization on bi-tree.

Existing methods do not extend to 4-tree without new approaches.

Abstract

This note contains a plethora of counterexamples to attempts to generalize the results of bi-parameter embedding from $p=2$ case to either $p>2$ or $p<2$. This is in striking difference to $p=2$ case that was fully understood in the series of papers \cite{AMPS}, \cite{AMPVZ-K}, \cite{MPVZ1}, \cite{MPVZ2}, \cite{AHMV}, \cite{MPV}. We also build a counterexample to small energy majorization on bi-tree. This counterexample shows that straightforward generalizations of methods of \cite{AMPVZ-K}, \cite{MPVZ1}, \cite{MPVZ2}, \cite{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.