Haar Type and Carleson Constants

TL;DR

This paper explores how specific collections of dyadic intervals can identify the Haar type of Banach spaces, using Carleson packing conditions to characterize the key dichotomy.

Contribution

It introduces a new characterization of Haar type detection via Carleson packing conditions for collections of dyadic intervals.

Findings

Characterizes collections of dyadic intervals that detect Haar type.

Uses Carleson packing condition to establish the dichotomy.

Provides a new framework for understanding Haar type in Banach spaces.

Abstract

We determine the sub-collections of the dyadic intervals that are able to detect the Haar type of a Banach space. The underlying dichotomy is expressed in terms of the Carleson packing condition.