Characterizations of Weights in Martingale Spaces

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