Bi-parameter embedding and measures with restriction energy condition

TL;DR

This paper presents a new proof of a bi-parameter Carleson embedding theorem that avoids using bi-tree capacity, employing energy comparison techniques instead of Bellman functions.

Contribution

The authors provide an alternative proof of the bi-parameter Carleson embedding theorem that bypasses bi-tree capacity, using energy comparison methods.

Findings

New proof avoids bi-tree capacity

Utilizes energy comparison techniques

Simplifies understanding of bi-parameter embeddings

Abstract

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 bi-tree. In this note we give one more 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 pervious paper of the authors) that used the Bellman function technique, the proof here is based on some rather subtle comparison of energies of measures on bi-tree.