Characterizations of $A_\infty$ Weights in Martingale Spaces

TL;DR

This paper extends the characterization of $A_ abla$ weights to martingale spaces using conditional expectations, providing new insights into their properties without geometric structures.

Contribution

It introduces new characterizations of $A_ abla$ weights in martingale spaces under regularity assumptions, expanding prior Euclidean and basis-related results.

Findings

Equivalent characterizations of $A_ abla$ weights obtained

One-way implications of different characterizations established

New methods developed for weights without geometric structures

Abstract

Grafakos systematically proved that $A_\infty$ weights have different characterizations for cubes in Euclidean spaces in his classical text book. Very recently, Duoandikoetxea, Mart\'{\i}n-Reyes, Ombrosi and Kosz discussed several characterizations of the $A_{\infty}$ weights in the setting of general bases. By conditional expectations, we study $A_\infty$ weights in martingale spaces. Because conditional expectations are Radon-Nikod\'{y}m derivatives with respect to sub$\hbox{-}\sigma\hbox{-}$fields which have no geometric structures, we need new ingredients. Under a regularity assumption on weights, we obtain equivalent characterizations of the $A_{\infty}$ weights. Moreover, using weights modulo conditional expectations, we have one-way implications of different characterizations.