Characterizations of $A_{\infty }$ Weights in Martingale Spaces

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\'ın-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-Nikodým derivatives with respect to sub$-\sigma -$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.

This paper has now been accepted for publication in The Journal of Geometric Analysis. 22 pages; 22pages