A∞ condition for general bases revisited: complete classification of definitions
Abstract
We refer to the discussion on different characterizations of the
A∞ class of weights, initiated by Duoandikoetxea, Martín-Reyes, and Ombrosi
[Math. Z. 282 (2016), pp. 955–972]. Twelve definitions of the A∞ condition are
considered. For cubes in Rd 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.