$A_\infty$ condition for general bases revisited: complete classification of definitions

TL;DR

This paper thoroughly analyzes twelve different definitions of the $A_$ weight class across various bases, providing a complete classification of their equivalences and implications.

Contribution

It offers a comprehensive classification of the relationships among twelve $A_$ conditions for general bases, completing previous partial results.

Findings

All previously unresolved cases are addressed.

A full diagram of relations between $A_$ conditions is provided.

The study clarifies the structure of $A_$ conditions beyond cubes.

Abstract

We refer to the discussion on different characterizations of the $A_\infty$ class of weights, initiated by Duoandikoetxea, Mart\'in-Reyes, and Ombrosi. Twelve definitions of the $A_\infty$ condition are considered. For cubes in $\mathbb{R}^d$ every two conditions are known to be equivalent, while for general bases we have a trichotomy: equivalence, one-way implication, or no dependency may occur. In most cases the relations between different conditions have already been established. Here all the unsolved cases are treated and, as a result, a full diagram of the said relations is presented.